statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
set_like.homogeneous_submonoid [add_monoid ι] [monoid R]
(A : ι → S) [set_like.graded_monoid A] : submonoid R | { carrier := { a | set_like.is_homogeneous A a },
one_mem' := set_like.is_homogeneous_one A,
mul_mem' := λ a b, set_like.is_homogeneous.mul } | def | set_like.homogeneous_submonoid | algebra | src/algebra/graded_monoid.lean | [
"algebra.group.inj_surj",
"data.list.big_operators.basic",
"data.list.fin_range",
"group_theory.group_action.defs",
"group_theory.submonoid.basic",
"data.set_like.basic",
"data.sigma.basic"
] | [
"add_monoid",
"monoid",
"set_like.graded_monoid",
"set_like.is_homogeneous",
"set_like.is_homogeneous.mul",
"set_like.is_homogeneous_one",
"submonoid"
] | When `A` is a `set_like.graded_monoid A`, then the homogeneous elements forms a submonoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghas_smul [has_add ι] | (smul {i j} : A i → M j → M (i + j)) | class | graded_monoid.ghas_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [] | A graded version of `has_smul`. Scalar multiplication combines grades additively, i.e.
if `a ∈ A i` and `m ∈ M j`, then `a • b` must be in `M (i + j)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghas_mul.to_ghas_smul [has_add ι] [ghas_mul A] : ghas_smul A A | { smul := λ _ _, ghas_mul.mul } | instance | graded_monoid.ghas_mul.to_ghas_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [] | A graded version of `has_mul.to_has_smul` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ghas_smul.to_has_smul [has_add ι] [ghas_smul A M] :
has_smul (graded_monoid A) (graded_monoid M) | ⟨λ (x : graded_monoid A) (y : graded_monoid M), ⟨_, ghas_smul.smul x.snd y.snd⟩⟩ | instance | graded_monoid.ghas_smul.to_has_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"graded_monoid",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_smul_mk [has_add ι] [ghas_smul A M] {i j} (a : A i) (b : M j) :
mk i a • mk j b = mk (i + j) (ghas_smul.smul a b) | rfl | lemma | graded_monoid.mk_smul_mk | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gmul_action [add_monoid ι] [gmonoid A] extends ghas_smul A M | (one_smul (b : graded_monoid M) : (1 : graded_monoid A) • b = b)
(mul_smul (a a' : graded_monoid A) (b : graded_monoid M) : (a * a') • b = a • a' • b) | class | graded_monoid.gmul_action | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"add_monoid",
"graded_monoid",
"one_smul"
] | A graded version of `mul_action`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gmonoid.to_gmul_action [add_monoid ι] [gmonoid A] :
gmul_action A A | { one_smul := gmonoid.one_mul,
mul_smul := gmonoid.mul_assoc,
..ghas_mul.to_ghas_smul _ } | instance | graded_monoid.gmonoid.to_gmul_action | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"add_monoid",
"one_smul"
] | The graded version of `monoid.to_mul_action`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gmul_action.to_mul_action [add_monoid ι] [gmonoid A] [gmul_action A M] :
mul_action (graded_monoid A) (graded_monoid M) | { one_smul := gmul_action.one_smul,
mul_smul := gmul_action.mul_smul } | instance | graded_monoid.gmul_action.to_mul_action | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"add_monoid",
"graded_monoid",
"mul_action",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.has_graded_smul {S R N M : Type*} [set_like S R] [set_like N M]
[has_smul R M] [has_add ι] (A : ι → S) (B : ι → N) : Prop | (smul_mem : ∀ ⦃i j : ι⦄ {ai bj}, ai ∈ A i → bj ∈ B j → ai • bj ∈ B (i + j)) | class | set_like.has_graded_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"has_smul",
"set_like"
] | A version of `graded_monoid.ghas_smul` for internally graded objects. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.ghas_smul {S R N M : Type*} [set_like S R] [set_like N M]
[has_smul R M] [has_add ι] (A : ι → S) (B : ι → N) [set_like.has_graded_smul A B] :
graded_monoid.ghas_smul (λ i, A i) (λ i, B i) | { smul := λ i j a b, ⟨(a : R) • b, set_like.has_graded_smul.smul_mem a.2 b.2⟩ } | instance | set_like.ghas_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"graded_monoid.ghas_smul",
"has_smul",
"set_like",
"set_like.has_graded_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.coe_ghas_smul {S R N M : Type*} [set_like S R] [set_like N M]
[has_smul R M] [has_add ι] (A : ι → S) (B : ι → N) [set_like.has_graded_smul A B]
{i j : ι} (x : A i) (y : B j) :
(@graded_monoid.ghas_smul.smul ι (λ i, A i) (λ i, B i) _ _ i j x y : M) = ((x : R) • y) | rfl | lemma | set_like.coe_ghas_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"has_smul",
"set_like",
"set_like.has_graded_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_like.has_graded_mul.to_has_graded_smul [add_monoid ι] [monoid R]
{S : Type*} [set_like S R] (A : ι → S) [set_like.graded_monoid A] :
set_like.has_graded_smul A A | { smul_mem := λ i j ai bj hi hj, set_like.graded_monoid.mul_mem hi hj, } | instance | set_like.has_graded_mul.to_has_graded_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"add_monoid",
"monoid",
"set_like",
"set_like.graded_monoid",
"set_like.has_graded_smul"
] | Internally graded version of `has_mul.to_has_smul`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_like.is_homogeneous.graded_smul [has_add ι] [has_smul R M] {A : ι → S} {B : ι → N}
[set_like.has_graded_smul A B] {a : R} {b : M} :
set_like.is_homogeneous A a → set_like.is_homogeneous B b → set_like.is_homogeneous B (a • b) | | ⟨i, hi⟩ ⟨j, hj⟩ := ⟨i + j, set_like.has_graded_smul.smul_mem hi hj⟩ | lemma | set_like.is_homogeneous.graded_smul | algebra | src/algebra/graded_mul_action.lean | [
"algebra.graded_monoid"
] | [
"has_smul",
"set_like.has_graded_smul",
"set_like.is_homogeneous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator {M} [has_zero M] (s : set α) (f : α → M) : α → M | | x := by haveI := classical.dec_pred (∈ s); exact if x ∈ s then f x else 0 | def | set.indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec_pred"
] | `indicator s f a` is `f a` if `a ∈ s`, `0` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator (s : set α) (f : α → M) : α → M | | x := by haveI := classical.dec_pred (∈ s); exact if x ∈ s then f x else 1 | def | set.mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec_pred"
] | `mul_indicator s f a` is `f a` if `a ∈ s`, `1` otherwise. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
piecewise_eq_mul_indicator [decidable_pred (∈ s)] :
s.piecewise f 1 = s.mul_indicator f | funext $ λ x, @if_congr _ _ _ _ (id _) _ _ _ _ iff.rfl rfl rfl | lemma | set.piecewise_eq_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_apply (s : set α) (f : α → M) (a : α) [decidable (a ∈ s)] :
mul_indicator s f a = if a ∈ s then f a else 1 | by convert rfl | lemma | set.mul_indicator_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_of_mem (h : a ∈ s) (f : α → M) :
mul_indicator s f a = f a | by { letI := classical.dec (a ∈ s), exact if_pos h } | lemma | set.mul_indicator_of_mem | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_of_not_mem (h : a ∉ s) (f : α → M) :
mul_indicator s f a = 1 | by { letI := classical.dec (a ∈ s), exact if_neg h } | lemma | set.mul_indicator_of_not_mem | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_eq_one_or_self (s : set α) (f : α → M) (a : α) :
mul_indicator s f a = 1 ∨ mul_indicator s f a = f a | begin
by_cases h : a ∈ s,
{ exact or.inr (mul_indicator_of_mem h f) },
{ exact or.inl (mul_indicator_of_not_mem h f) }
end | lemma | set.mul_indicator_eq_one_or_self | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_apply_eq_self :
s.mul_indicator f a = f a ↔ (a ∉ s → f a = 1) | by letI := classical.dec (a ∈ s); exact ite_eq_left_iff.trans (by rw [@eq_comm _ (f a)]) | lemma | set.mul_indicator_apply_eq_self | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_eq_self : s.mul_indicator f = f ↔ mul_support f ⊆ s | by simp only [funext_iff, subset_def, mem_mul_support, mul_indicator_apply_eq_self, not_imp_comm] | lemma | set.mul_indicator_eq_self | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"not_imp_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_eq_self_of_superset (h1 : s.mul_indicator f = f) (h2 : s ⊆ t) :
t.mul_indicator f = f | by { rw mul_indicator_eq_self at h1 ⊢, exact subset.trans h1 h2 } | lemma | set.mul_indicator_eq_self_of_superset | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_apply_eq_one :
mul_indicator s f a = 1 ↔ (a ∈ s → f a = 1) | by letI := classical.dec (a ∈ s); exact ite_eq_right_iff | lemma | set.mul_indicator_apply_eq_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec",
"ite_eq_right_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_eq_one :
mul_indicator s f = (λ x, 1) ↔ disjoint (mul_support f) s | by simp only [funext_iff, mul_indicator_apply_eq_one, set.disjoint_left, mem_mul_support,
not_imp_not] | lemma | set.mul_indicator_eq_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint",
"not_imp_not",
"set.disjoint_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_eq_one' :
mul_indicator s f = 1 ↔ disjoint (mul_support f) s | mul_indicator_eq_one | lemma | set.mul_indicator_eq_one' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_apply_ne_one {a : α} :
s.mul_indicator f a ≠ 1 ↔ a ∈ s ∩ mul_support f | by simp only [ne.def, mul_indicator_apply_eq_one, not_imp, mem_inter_iff, mem_mul_support] | lemma | set.mul_indicator_apply_ne_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"not_imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_mul_indicator :
function.mul_support (s.mul_indicator f) = s ∩ function.mul_support f | ext $ λ x, by simp [function.mem_mul_support, mul_indicator_apply_eq_one] | lemma | set.mul_support_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"function.mem_mul_support",
"function.mul_support"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_of_mul_indicator_ne_one (h : mul_indicator s f a ≠ 1) : a ∈ s | not_imp_comm.1 (λ hn, mul_indicator_of_not_mem hn f) h | lemma | set.mem_of_mul_indicator_ne_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | If a multiplicative indicator function is not equal to `1` at a point, then that point is in the
set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_mul_indicator : eq_on (mul_indicator s f) f s | λ x hx, mul_indicator_of_mem hx f | lemma | set.eq_on_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_mul_indicator_subset : mul_support (s.mul_indicator f) ⊆ s | λ x hx, hx.imp_symm (λ h, mul_indicator_of_not_mem h f) | lemma | set.mul_support_mul_indicator_subset | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_mul_support : mul_indicator (mul_support f) f = f | mul_indicator_eq_self.2 subset.rfl | lemma | set.mul_indicator_mul_support | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_range_comp {ι : Sort*} (f : ι → α) (g : α → M) :
mul_indicator (range f) g ∘ f = g ∘ f | by letI := classical.dec_pred (∈ range f); exact piecewise_range_comp _ _ _ | lemma | set.mul_indicator_range_comp | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec_pred"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_congr (h : eq_on f g s) :
mul_indicator s f = mul_indicator s g | funext $ λx, by { simp only [mul_indicator], split_ifs, { exact h h_1 }, refl } | lemma | set.mul_indicator_congr | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_univ (f : α → M) : mul_indicator (univ : set α) f = f | mul_indicator_eq_self.2 $ subset_univ _ | lemma | set.mul_indicator_univ | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_empty (f : α → M) : mul_indicator (∅ : set α) f = λa, 1 | mul_indicator_eq_one.2 $ disjoint_empty _ | lemma | set.mul_indicator_empty | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_empty' (f : α → M) : mul_indicator (∅ : set α) f = 1 | mul_indicator_empty f | lemma | set.mul_indicator_empty' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_one (s : set α) :
mul_indicator s (λx, (1:M)) = λx, (1:M) | mul_indicator_eq_one.2 $ by simp only [mul_support_one, empty_disjoint] | lemma | set.mul_indicator_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_one' {s : set α} : s.mul_indicator (1 : α → M) = 1 | mul_indicator_one M s | lemma | set.mul_indicator_one' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_mul_indicator (s t : set α) (f : α → M) :
mul_indicator s (mul_indicator t f) = mul_indicator (s ∩ t) f | funext $ λx, by { simp only [mul_indicator], split_ifs, repeat {simp * at * {contextual := tt}} } | lemma | set.mul_indicator_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_inter_mul_support (s : set α) (f : α → M) :
mul_indicator (s ∩ mul_support f) f = mul_indicator s f | by rw [← mul_indicator_mul_indicator, mul_indicator_mul_support] | lemma | set.mul_indicator_inter_mul_support | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_mul_indicator (h : M → β) (f : α → M) {s : set α} {x : α}
[decidable_pred (∈ s)] :
h (s.mul_indicator f x) = s.piecewise (h ∘ f) (const α (h 1)) x | by letI := classical.dec_pred (∈ s); convert s.apply_piecewise f (const α 1) (λ _, h) | lemma | set.comp_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec_pred"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_comp_right {s : set α} (f : β → α) {g : α → M} {x : β} :
mul_indicator (f ⁻¹' s) (g ∘ f) x = mul_indicator s g (f x) | by { simp only [mul_indicator], split_ifs; refl } | lemma | set.mul_indicator_comp_right | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_image {s : set α} {f : β → M} {g : α → β} (hg : injective g)
{x : α} : mul_indicator (g '' s) f (g x) = mul_indicator s (f ∘ g) x | by rw [← mul_indicator_comp_right, preimage_image_eq _ hg] | lemma | set.mul_indicator_image | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_comp_of_one {g : M → N} (hg : g 1 = 1) :
mul_indicator s (g ∘ f) = g ∘ (mul_indicator s f) | begin
funext,
simp only [mul_indicator],
split_ifs; simp [*]
end | lemma | set.mul_indicator_comp_of_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_mul_indicator_const (c : M) (f : M → N) (hf : f 1 = 1) :
(λ x, f (s.mul_indicator (λ x, c) x)) = s.mul_indicator (λ x, f c) | (mul_indicator_comp_of_one hf).symm | lemma | set.comp_mul_indicator_const | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_preimage (s : set α) (f : α → M) (B : set M) :
(mul_indicator s f)⁻¹' B = s.ite (f ⁻¹' B) (1 ⁻¹' B) | by letI := classical.dec_pred (∈ s); exact piecewise_preimage s f 1 B | lemma | set.mul_indicator_preimage | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec_pred"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_one_preimage (s : set M) :
t.mul_indicator 1 ⁻¹' s ∈ ({set.univ, ∅} : set (set α)) | begin
classical,
rw [mul_indicator_one', preimage_one],
split_ifs; simp
end | lemma | set.mul_indicator_one_preimage | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_const_preimage_eq_union (U : set α) (s : set M) (a : M)
[decidable (a ∈ s)] [decidable ((1 : M) ∈ s)] :
U.mul_indicator (λ x, a) ⁻¹' s = (if a ∈ s then U else ∅) ∪ (if (1 : M) ∈ s then Uᶜ else ∅) | begin
rw [mul_indicator_preimage, preimage_one, preimage_const],
split_ifs; simp [← compl_eq_univ_diff]
end | lemma | set.mul_indicator_const_preimage_eq_union | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_const_preimage (U : set α) (s : set M) (a : M) :
U.mul_indicator (λ x, a) ⁻¹' s ∈ ({set.univ, U, Uᶜ, ∅} : set (set α)) | begin
classical,
rw [mul_indicator_const_preimage_eq_union],
split_ifs; simp
end | lemma | set.mul_indicator_const_preimage | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_one_preimage [has_zero M] (U : set α) (s : set M) :
U.indicator 1 ⁻¹' s ∈ ({set.univ, U, Uᶜ, ∅} : set (set α)) | indicator_const_preimage _ _ 1 | lemma | set.indicator_one_preimage | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_preimage_of_not_mem (s : set α) (f : α → M)
{t : set M} (ht : (1:M) ∉ t) :
(mul_indicator s f)⁻¹' t = f ⁻¹' t ∩ s | by simp [mul_indicator_preimage, pi.one_def, set.preimage_const_of_not_mem ht] | lemma | set.mul_indicator_preimage_of_not_mem | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"pi.one_def",
"set.preimage_const_of_not_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_range_mul_indicator {r : M} {s : set α} {f : α → M} :
r ∈ range (mul_indicator s f) ↔ (r = 1 ∧ s ≠ univ) ∨ (r ∈ f '' s) | by simp [mul_indicator, ite_eq_iff, exists_or_distrib, eq_univ_iff_forall, and_comm, or_comm,
@eq_comm _ r 1] | lemma | set.mem_range_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"exists_or_distrib",
"ite_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_rel_mul_indicator {r : M → M → Prop} (h1 : r 1 1)
(ha : a ∈ s → r (f a) (g a)) :
r (mul_indicator s f a) (mul_indicator s g a) | by { simp only [mul_indicator], split_ifs with has has, exacts [ha has, h1] } | lemma | set.mul_indicator_rel_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_union_mul_inter_apply (f : α → M) (s t : set α) (a : α) :
mul_indicator (s ∪ t) f a * mul_indicator (s ∩ t) f a =
mul_indicator s f a * mul_indicator t f a | by by_cases hs : a ∈ s; by_cases ht : a ∈ t; simp * | lemma | set.mul_indicator_union_mul_inter_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_union_mul_inter (f : α → M) (s t : set α) :
mul_indicator (s ∪ t) f * mul_indicator (s ∩ t) f = mul_indicator s f * mul_indicator t f | funext $ mul_indicator_union_mul_inter_apply f s t | lemma | set.mul_indicator_union_mul_inter | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_union_of_not_mem_inter (h : a ∉ s ∩ t) (f : α → M) :
mul_indicator (s ∪ t) f a = mul_indicator s f a * mul_indicator t f a | by rw [← mul_indicator_union_mul_inter_apply f s t, mul_indicator_of_not_mem h, mul_one] | lemma | set.mul_indicator_union_of_not_mem_inter | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_union_of_disjoint (h : disjoint s t) (f : α → M) :
mul_indicator (s ∪ t) f = λa, mul_indicator s f a * mul_indicator t f a | funext $ λa, mul_indicator_union_of_not_mem_inter (λ ha, h.le_bot ha) _ | lemma | set.mul_indicator_union_of_disjoint | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_mul (s : set α) (f g : α → M) :
mul_indicator s (λa, f a * g a) = λa, mul_indicator s f a * mul_indicator s g a | by { funext, simp only [mul_indicator], split_ifs, { refl }, rw mul_one } | lemma | set.mul_indicator_mul | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_mul' (s : set α) (f g : α → M) :
mul_indicator s (f * g) = mul_indicator s f * mul_indicator s g | mul_indicator_mul s f g | lemma | set.mul_indicator_mul' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_compl_mul_self_apply (s : set α) (f : α → M) (a : α) :
mul_indicator sᶜ f a * mul_indicator s f a = f a | classical.by_cases (λ ha : a ∈ s, by simp [ha]) (λ ha, by simp [ha]) | lemma | set.mul_indicator_compl_mul_self_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_compl_mul_self (s : set α) (f : α → M) :
mul_indicator sᶜ f * mul_indicator s f = f | funext $ mul_indicator_compl_mul_self_apply s f | lemma | set.mul_indicator_compl_mul_self | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_self_mul_compl_apply (s : set α) (f : α → M) (a : α) :
mul_indicator s f a * mul_indicator sᶜ f a = f a | classical.by_cases (λ ha : a ∈ s, by simp [ha]) (λ ha, by simp [ha]) | lemma | set.mul_indicator_self_mul_compl_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_self_mul_compl (s : set α) (f : α → M) :
mul_indicator s f * mul_indicator sᶜ f = f | funext $ mul_indicator_self_mul_compl_apply s f | lemma | set.mul_indicator_self_mul_compl | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_mul_eq_left {f g : α → M}
(h : disjoint (mul_support f) (mul_support g)) :
(mul_support f).mul_indicator (f * g) = f | begin
refine (mul_indicator_congr $ λ x hx, _).trans mul_indicator_mul_support,
have : g x = 1, from nmem_mul_support.1 (disjoint_left.1 h hx),
rw [pi.mul_apply, this, mul_one]
end | lemma | set.mul_indicator_mul_eq_left | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint",
"mul_one",
"pi.mul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_mul_eq_right {f g : α → M}
(h : disjoint (mul_support f) (mul_support g)) :
(mul_support g).mul_indicator (f * g) = g | begin
refine (mul_indicator_congr $ λ x hx, _).trans mul_indicator_mul_support,
have : f x = 1, from nmem_mul_support.1 (disjoint_right.1 h hx),
rw [pi.mul_apply, this, one_mul]
end | lemma | set.mul_indicator_mul_eq_right | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint",
"one_mul",
"pi.mul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_mul_compl_eq_piecewise
[decidable_pred (∈ s)] (f g : α → M) :
s.mul_indicator f * sᶜ.mul_indicator g = s.piecewise f g | begin
ext x,
by_cases h : x ∈ s,
{ rw [piecewise_eq_of_mem _ _ _ h, pi.mul_apply, set.mul_indicator_of_mem h,
set.mul_indicator_of_not_mem (set.not_mem_compl_iff.2 h), mul_one] },
{ rw [piecewise_eq_of_not_mem _ _ _ h, pi.mul_apply, set.mul_indicator_of_not_mem h,
set.mul_indicator_of_mem (set.mem_c... | lemma | set.mul_indicator_mul_compl_eq_piecewise | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"mul_one",
"one_mul",
"pi.mul_apply",
"set.mem_compl",
"set.mul_indicator_of_mem",
"set.mul_indicator_of_not_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_hom {α} (M) [mul_one_class M] (s : set α) : (α → M) →* (α → M) | { to_fun := mul_indicator s,
map_one' := mul_indicator_one M s,
map_mul' := mul_indicator_mul s } | def | set.mul_indicator_hom | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"mul_one_class"
] | `set.mul_indicator` as a `monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_smul_apply (s : set α) (r : α → M) (f : α → A) (x : α) :
indicator s (λ x, r x • f x) x = r x • indicator s f x | by { dunfold indicator, split_ifs, exacts [rfl, (smul_zero (r x)).symm] } | lemma | set.indicator_smul_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_smul (s : set α) (r : α → M) (f : α → A) :
indicator s (λ (x : α), r x • f x) = λ (x : α), r x • indicator s f x | funext $ indicator_smul_apply s r f | lemma | set.indicator_smul | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_const_smul_apply (s : set α) (r : M) (f : α → A) (x : α) :
indicator s (λ x, r • f x) x = r • indicator s f x | indicator_smul_apply s (λ x, r) f x | lemma | set.indicator_const_smul_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_const_smul (s : set α) (r : M) (f : α → A) :
indicator s (λ (x : α), r • f x) = λ (x : α), r • indicator s f x | funext $ indicator_const_smul_apply s r f | lemma | set.indicator_const_smul | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_inv' (s : set α) (f : α → G) :
mul_indicator s (f⁻¹) = (mul_indicator s f)⁻¹ | (mul_indicator_hom G s).map_inv f | lemma | set.mul_indicator_inv' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"map_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_inv (s : set α) (f : α → G) :
mul_indicator s (λa, (f a)⁻¹) = λa, (mul_indicator s f a)⁻¹ | mul_indicator_inv' s f | lemma | set.mul_indicator_inv | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_div (s : set α) (f g : α → G) :
mul_indicator s (λ a, f a / g a) =
λ a, mul_indicator s f a / mul_indicator s g a | (mul_indicator_hom G s).map_div f g | lemma | set.mul_indicator_div | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"map_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_div' (s : set α) (f g : α → G) :
mul_indicator s (f / g) = mul_indicator s f / mul_indicator s g | mul_indicator_div s f g | lemma | set.mul_indicator_div' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_compl (s : set α) (f : α → G) :
mul_indicator sᶜ f = f * (mul_indicator s f)⁻¹ | eq_mul_inv_of_mul_eq $ s.mul_indicator_compl_mul_self f | lemma | set.mul_indicator_compl | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"eq_mul_inv_of_mul_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_compl {G} [add_group G] (s : set α) (f : α → G) :
indicator sᶜ f = f - indicator s f | by rw [sub_eq_add_neg, indicator_compl'] | lemma | set.indicator_compl | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_diff (h : s ⊆ t) (f : α → G) :
mul_indicator (t \ s) f = mul_indicator t f * (mul_indicator s f)⁻¹ | eq_mul_inv_of_mul_eq $ by { rw [pi.mul_def, ←mul_indicator_union_of_disjoint, diff_union_self,
union_eq_self_of_subset_right h], exact disjoint_sdiff_self_left } | lemma | set.mul_indicator_diff | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint_sdiff_self_left",
"eq_mul_inv_of_mul_eq",
"pi.mul_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_diff {G : Type*} [add_group G] {s t : set α} (h : s ⊆ t) (f : α → G) :
indicator (t \ s) f = indicator t f - indicator s f | by rw [indicator_diff' h, sub_eq_add_neg] | lemma | set.indicator_diff | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mul_indicator_subset_of_eq_one [has_one N] (f : α → N)
(g : α → N → M) {s t : finset α} (h : s ⊆ t) (hg : ∀ a, g a 1 = 1) :
∏ i in s, g i (f i) = ∏ i in t, g i (mul_indicator ↑s f i) | begin
rw ← finset.prod_subset h _,
{ apply finset.prod_congr rfl,
intros i hi,
congr,
symmetry,
exact mul_indicator_of_mem hi _ },
{ refine λ i hi hn, _,
convert hg i,
exact mul_indicator_of_not_mem hn _ }
end | lemma | set.prod_mul_indicator_subset_of_eq_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"finset",
"finset.prod_congr",
"finset.prod_subset"
] | Consider a product of `g i (f i)` over a `finset`. Suppose `g` is a
function such as `pow`, which maps a second argument of `1` to
`1`. Then if `f` is replaced by the corresponding multiplicative indicator
function, the `finset` may be replaced by a possibly larger `finset`
without changing the value of the sum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_mul_indicator_subset (f : α → M) {s t : finset α} (h : s ⊆ t) :
∏ i in s, f i = ∏ i in t, mul_indicator ↑s f i | prod_mul_indicator_subset_of_eq_one _ (λ a b, b) h (λ _, rfl) | lemma | set.prod_mul_indicator_subset | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"finset"
] | Taking the product of an indicator function over a possibly larger `finset` is the same as
taking the original function over the original `finset`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.finset.prod_mul_indicator_eq_prod_filter
(s : finset ι) (f : ι → α → M) (t : ι → set α) (g : ι → α) [decidable_pred (λ i, g i ∈ t i)]:
∏ i in s, mul_indicator (t i) (f i) (g i) = ∏ i in s.filter (λ i, g i ∈ t i), f i (g i) | begin
refine (finset.prod_filter_mul_prod_filter_not s (λ i, g i ∈ t i) _).symm.trans _,
refine eq.trans _ (mul_one _),
exact congr_arg2 (*)
(finset.prod_congr rfl $ λ x hx, mul_indicator_of_mem (finset.mem_filter.1 hx).2 _)
(finset.prod_eq_one $ λ x hx, mul_indicator_of_not_mem (finset.mem_filter.1 hx).2... | lemma | finset.prod_mul_indicator_eq_prod_filter | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"congr_arg2",
"finset",
"finset.prod_congr",
"finset.prod_eq_one",
"finset.prod_filter_mul_prod_filter_not",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_finset_prod (I : finset ι) (s : set α) (f : ι → α → M) :
mul_indicator s (∏ i in I, f i) = ∏ i in I, mul_indicator s (f i) | (mul_indicator_hom M s).map_prod _ _ | lemma | set.mul_indicator_finset_prod | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"finset",
"map_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_finset_bUnion {ι} (I : finset ι)
(s : ι → set α) {f : α → M} : (∀ (i ∈ I) (j ∈ I), i ≠ j → disjoint (s i) (s j)) →
mul_indicator (⋃ i ∈ I, s i) f = λ a, ∏ i in I, mul_indicator (s i) f a | begin
classical,
refine finset.induction_on I _ _,
{ intro h, funext, simp },
assume a I haI ih hI,
funext,
rw [finset.prod_insert haI, finset.set_bUnion_insert, mul_indicator_union_of_not_mem_inter, ih _],
{ assume i hi j hj hij,
exact hI i (finset.mem_insert_of_mem hi) j (finset.mem_insert_of_mem hj... | lemma | set.mul_indicator_finset_bUnion | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint",
"exists_prop",
"finset",
"finset.induction_on",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.prod_insert",
"finset.set_bUnion_insert",
"ih",
"ne_of_mem_of_not_mem",
"not_and",
"not_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_finset_bUnion_apply {ι} (I : finset ι)
(s : ι → set α) {f : α → M} (h : ∀ (i ∈ I) (j ∈ I), i ≠ j → disjoint (s i) (s j)) (x : α) :
mul_indicator (⋃ i ∈ I, s i) f x = ∏ i in I, mul_indicator (s i) f x | by rw set.mul_indicator_finset_bUnion I s h | lemma | set.mul_indicator_finset_bUnion_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint",
"finset",
"set.mul_indicator_finset_bUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_mul (s : set α) (f g : α → M) :
indicator s (λa, f a * g a) = λa, indicator s f a * indicator s g a | by { funext, simp only [indicator], split_ifs, { refl }, rw mul_zero } | lemma | set.indicator_mul | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_mul_left (s : set α) (f g : α → M) :
indicator s (λa, f a * g a) a = indicator s f a * g a | by { simp only [indicator], split_ifs, { refl }, rw [zero_mul] } | lemma | set.indicator_mul_left | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_mul_right (s : set α) (f g : α → M) :
indicator s (λa, f a * g a) a = f a * indicator s g a | by { simp only [indicator], split_ifs, { refl }, rw [mul_zero] } | lemma | set.indicator_mul_right | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inter_indicator_mul {t1 t2 : set α} (f g : α → M) (x : α) :
(t1 ∩ t2).indicator (λ x, f x * g x) x = t1.indicator f x * t2.indicator g x | by { rw [← set.indicator_indicator], simp [indicator] } | lemma | set.inter_indicator_mul | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inter_indicator_one {s t : set α} :
(s ∩ t).indicator (1 : _ → M) = s.indicator 1 * t.indicator 1 | funext (λ _, by simpa only [← inter_indicator_mul, pi.mul_apply, pi.one_apply, one_mul]) | lemma | set.inter_indicator_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"one_mul",
"pi.mul_apply",
"pi.one_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_prod_one {s : set α} {t : set β} {x : α} {y : β} :
(s ×ˢ t).indicator (1 : _ → M) (x, y) = s.indicator 1 x * t.indicator 1 y | by { classical, simp [indicator_apply, ←ite_and] } | lemma | set.indicator_prod_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_eq_zero_iff_not_mem {U : set α} {x : α} :
indicator U 1 x = (0 : M) ↔ x ∉ U | by { classical, simp [indicator_apply, imp_false] } | lemma | set.indicator_eq_zero_iff_not_mem | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"imp_false"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_eq_one_iff_mem {U : set α} {x : α} :
indicator U 1 x = (1 : M) ↔ x ∈ U | by { classical, simp [indicator_apply, imp_false] } | lemma | set.indicator_eq_one_iff_mem | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"imp_false"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
indicator_one_inj {U V : set α} (h : indicator U (1 : α → M) = indicator V 1) : U = V | by { ext, simp_rw [← indicator_eq_one_iff_mem M, h] } | lemma | set.indicator_one_inj | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_apply_le' (hfg : a ∈ s → f a ≤ y) (hg : a ∉ s → 1 ≤ y) :
mul_indicator s f a ≤ y | begin
by_cases ha : a ∈ s,
{ simpa [ha] using hfg ha },
{ simpa [ha] using hg ha },
end | lemma | set.mul_indicator_apply_le' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_indicator_le' (hfg : ∀ a ∈ s, f a ≤ g a) (hg : ∀ a ∉ s, 1 ≤ g a) :
mul_indicator s f ≤ g | λ a, mul_indicator_apply_le' (hfg _) (hg _) | lemma | set.mul_indicator_le' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_indicator_apply {y} (hfg : a ∈ s → y ≤ g a) (hf : a ∉ s → y ≤ 1) :
y ≤ mul_indicator s g a | @mul_indicator_apply_le' α Mᵒᵈ ‹_› _ _ _ _ _ hfg hf | lemma | set.le_mul_indicator_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_indicator (hfg : ∀ a ∈ s, f a ≤ g a) (hf : ∀ a ∉ s, f a ≤ 1) :
f ≤ mul_indicator s g | λ a, le_mul_indicator_apply (hfg _) (hf _) | lemma | set.le_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_mul_indicator_apply (h : a ∈ s → 1 ≤ f a) : 1 ≤ mul_indicator s f a | le_mul_indicator_apply h (λ _, le_rfl) | lemma | set.one_le_mul_indicator_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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