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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tuple_succ : (fin E.order → α) →ₗ[α] (fin E.order → α) | { to_fun := λ X i, if h : (i : ℕ) + 1 < E.order then X ⟨i+1, h⟩ else (∑ i, E.coeffs i * X i),
map_add' := λ x y,
begin
ext i,
split_ifs ; simp [h, mul_add, sum_add_distrib],
end,
map_smul' := λ x y,
begin
ext i,
split_ifs ; simp [h, mul_sum],
exact sum_congr rfl (λ x _, by ... | def | linear_recurrence.tuple_succ | algebra | src/algebra/linear_recurrence.lean | [
"data.polynomial.eval",
"linear_algebra.dimension"
] | [] | `E.tuple_succ` maps `![s₀, s₁, ..., sₙ]` to `![s₁, ..., sₙ, ∑ (E.coeffs i) * sᵢ]`,
where `n := E.order`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sol_space_rank : module.rank α E.sol_space = E.order | begin
letI := nontrivial_of_invariant_basis_number α,
exact @rank_fin_fun α _ _ E.order ▸ E.to_init.rank_eq
end | lemma | linear_recurrence.sol_space_rank | algebra | src/algebra/linear_recurrence.lean | [
"data.polynomial.eval",
"linear_algebra.dimension"
] | [
"module.rank",
"nontrivial_of_invariant_basis_number",
"rank_fin_fun"
] | The dimension of `E.sol_space` is `E.order`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
char_poly : α[X] | polynomial.monomial E.order 1 - (∑ i : fin E.order, polynomial.monomial i (E.coeffs i)) | def | linear_recurrence.char_poly | algebra | src/algebra/linear_recurrence.lean | [
"data.polynomial.eval",
"linear_algebra.dimension"
] | [
"polynomial.monomial"
] | The characteristic polynomial of `E` is
`X ^ E.order - ∑ i : fin E.order, (E.coeffs i) * X ^ i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
geom_sol_iff_root_char_poly (q : α) : E.is_solution (λ n, q^n) ↔ E.char_poly.is_root q | begin
rw [char_poly, polynomial.is_root.def, polynomial.eval],
simp only [polynomial.eval₂_finset_sum, one_mul,
ring_hom.id_apply, polynomial.eval₂_monomial, polynomial.eval₂_sub],
split,
{ intro h,
simpa [sub_eq_zero] using h 0 },
{ intros h n,
simp only [pow_add, sub_eq_zero.mp h, mul_... | lemma | linear_recurrence.geom_sol_iff_root_char_poly | algebra | src/algebra/linear_recurrence.lean | [
"data.polynomial.eval",
"linear_algebra.dimension"
] | [
"one_mul",
"polynomial.eval",
"polynomial.eval₂_finset_sum",
"polynomial.eval₂_monomial",
"polynomial.eval₂_sub",
"polynomial.is_root.def",
"pow_add",
"ring",
"ring_hom.id_apply"
] | The geometric sequence `q^n` is a solution of `E` iff
`q` is a root of `E`'s characteristic polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
modeq (p a b : α) : Prop | ∃ z : ℤ, b - a = z • p | def | add_comm_group.modeq | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | `a ≡ b [PMOD p]` means that `b` is congruent to `a` modulo `p`.
Equivalently (as shown in `algebra.order.to_interval_mod`), `b` does not lie in the open interval
`(a, a + p)` modulo `p`, or `to_Ico_mod hp a` disagrees with `to_Ioc_mod hp a` at `b`, or
`to_Ico_div hp a` disagrees with `to_Ioc_div hp a` at `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
modeq_refl (a : α) : a ≡ a [PMOD p] | ⟨0, by simp⟩ | lemma | add_comm_group.modeq_refl | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_rfl : a ≡ a [PMOD p] | modeq_refl _ | lemma | add_comm_group.modeq_rfl | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_comm : a ≡ b [PMOD p] ↔ b ≡ a [PMOD p] | (equiv.neg _).exists_congr_left.trans $ by simp [modeq, ←neg_eq_iff_eq_neg] | lemma | add_comm_group.modeq_comm | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq.trans : a ≡ b [PMOD p] → b ≡ c [PMOD p] → a ≡ c [PMOD p] | λ ⟨m, hm⟩ ⟨n, hn⟩, ⟨m + n, by simp [add_smul, ←hm, ←hn]⟩ | lemma | add_comm_group.modeq.trans | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"add_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_modeq_neg : -a ≡ -b [PMOD p] ↔ a ≡ b [PMOD p] | modeq_comm.trans $ by simp [modeq] | lemma | add_comm_group.neg_modeq_neg | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_neg : a ≡ b [PMOD -p] ↔ a ≡ b [PMOD p] | modeq_comm.trans $ by simp [modeq, ←neg_eq_iff_eq_neg] | lemma | add_comm_group.modeq_neg | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_sub (a b : α) : a ≡ b [PMOD b - a] | ⟨1, (one_smul _ _).symm⟩ | lemma | add_comm_group.modeq_sub | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_zero : a ≡ b [PMOD 0] ↔ a = b | by simp [modeq, sub_eq_zero, eq_comm] | lemma | add_comm_group.modeq_zero | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_modeq_zero : p ≡ 0 [PMOD p] | ⟨-1, by simp⟩ | lemma | add_comm_group.self_modeq_zero | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_modeq_zero (z : ℤ) : z • p ≡ 0 [PMOD p] | ⟨-z, by simp⟩ | lemma | add_comm_group.zsmul_modeq_zero | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_zsmul_modeq (z : ℤ) : a + z • p ≡ a [PMOD p] | ⟨-z, by simp⟩ | lemma | add_comm_group.add_zsmul_modeq | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_add_modeq (z : ℤ) : z • p + a ≡ a [PMOD p] | ⟨-z, by simp⟩ | lemma | add_comm_group.zsmul_add_modeq | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_nsmul_modeq (n : ℕ) : a + n • p ≡ a [PMOD p] | ⟨-n, by simp⟩ | lemma | add_comm_group.add_nsmul_modeq | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_add_modeq (n : ℕ) : n • p + a ≡ a [PMOD p] | ⟨-n, by simp⟩ | lemma | add_comm_group.nsmul_add_modeq | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_zsmul (z : ℤ) : a ≡ b [PMOD p] → a + z • p ≡ b [PMOD p] | (add_zsmul_modeq _).trans | lemma | add_comm_group.modeq.add_zsmul | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_add (z : ℤ) : a ≡ b [PMOD p] → z • p + a ≡ b [PMOD p] | (zsmul_add_modeq _).trans | lemma | add_comm_group.modeq.zsmul_add | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_nsmul (n : ℕ) : a ≡ b [PMOD p] → a + n • p ≡ b [PMOD p] | (add_nsmul_modeq _).trans | lemma | add_comm_group.modeq.add_nsmul | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_add (n : ℕ) : a ≡ b [PMOD p] → n • p + a ≡ b [PMOD p] | (nsmul_add_modeq _).trans | lemma | add_comm_group.modeq.nsmul_add | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_zsmul : a ≡ b [PMOD (z • p)] → a ≡ b [PMOD p] | λ ⟨m, hm⟩, ⟨m * z, by rwa [mul_smul]⟩ | lemma | add_comm_group.modeq.of_zsmul | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_nsmul : a ≡ b [PMOD (n • p)] → a ≡ b [PMOD p] | λ ⟨m, hm⟩, ⟨m * n, by rwa [mul_smul, coe_nat_zsmul]⟩ | lemma | add_comm_group.modeq.of_nsmul | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul : a ≡ b [PMOD p] → z • a ≡ z • b [PMOD (z • p)] | Exists.imp $ λ m hm, by rw [←smul_sub, hm, smul_comm] | lemma | add_comm_group.modeq.zsmul | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"Exists.imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul : a ≡ b [PMOD p] → n • a ≡ n • b [PMOD (n • p)] | Exists.imp $ λ m hm, by rw [←smul_sub, hm, smul_comm] | lemma | add_comm_group.modeq.nsmul | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"Exists.imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_modeq_zsmul [no_zero_smul_divisors ℤ α] (hn : z ≠ 0) :
z • a ≡ z • b [PMOD (z • p)] ↔ a ≡ b [PMOD p] | exists_congr $ λ m, by rw [←smul_sub, smul_comm, smul_right_inj hn]; apply_instance | lemma | add_comm_group.zsmul_modeq_zsmul | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"no_zero_smul_divisors",
"smul_right_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_modeq_nsmul [no_zero_smul_divisors ℕ α] (hn : n ≠ 0) :
n • a ≡ n • b [PMOD (n • p)] ↔ a ≡ b [PMOD p] | exists_congr $ λ m, by rw [←smul_sub, smul_comm, smul_right_inj hn]; apply_instance | lemma | add_comm_group.nsmul_modeq_nsmul | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"no_zero_smul_divisors",
"smul_right_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_iff_left :
a₁ ≡ b₁ [PMOD p] → (a₁ + a₂ ≡ b₁ + b₂ [PMOD p] ↔ a₂ ≡ b₂ [PMOD p]) | λ ⟨m, hm⟩, (equiv.add_left m).symm.exists_congr_left.trans $
by simpa [add_sub_add_comm, hm, add_smul] | lemma | add_comm_group.modeq.add_iff_left | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"add_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_iff_right :
a₂ ≡ b₂ [PMOD p] → (a₁ + a₂ ≡ b₁ + b₂ [PMOD p] ↔ a₁ ≡ b₁ [PMOD p]) | λ ⟨m, hm⟩, (equiv.add_right m).symm.exists_congr_left.trans $
by simpa [add_sub_add_comm, hm, add_smul] | lemma | add_comm_group.modeq.add_iff_right | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"add_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_iff_left :
a₁ ≡ b₁ [PMOD p] → (a₁ - a₂ ≡ b₁ - b₂ [PMOD p] ↔ a₂ ≡ b₂ [PMOD p]) | λ ⟨m, hm⟩, (equiv.sub_left m).symm.exists_congr_left.trans $
by simpa [sub_sub_sub_comm, hm, sub_smul] | lemma | add_comm_group.modeq.sub_iff_left | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"sub_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_iff_right :
a₂ ≡ b₂ [PMOD p] → (a₁ - a₂ ≡ b₁ - b₂ [PMOD p] ↔ a₁ ≡ b₁ [PMOD p]) | λ ⟨m, hm⟩, (equiv.sub_right m).symm.exists_congr_left.trans $
by simpa [sub_sub_sub_comm, hm, sub_smul] | lemma | add_comm_group.modeq.sub_iff_right | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"sub_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left (c : α) (h : a ≡ b [PMOD p]) : c + a ≡ c + b [PMOD p] | modeq_rfl.add h | lemma | add_comm_group.modeq.add_left | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_left (c : α) (h : a ≡ b [PMOD p]) : c - a ≡ c - b [PMOD p] | modeq_rfl.sub h | lemma | add_comm_group.modeq.sub_left | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_right (c : α) (h : a ≡ b [PMOD p]) : a + c ≡ b + c [PMOD p] | h.add modeq_rfl | lemma | add_comm_group.modeq.add_right | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_right (c : α) (h : a ≡ b [PMOD p]) : a - c ≡ b - c [PMOD p] | h.sub modeq_rfl | lemma | add_comm_group.modeq.sub_right | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_cancel' (c : α) : c + a ≡ c + b [PMOD p] → a ≡ b [PMOD p] | modeq_rfl.add_left_cancel | lemma | add_comm_group.modeq.add_left_cancel' | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_right_cancel' (c : α) : a + c ≡ b + c [PMOD p] → a ≡ b [PMOD p] | modeq_rfl.add_right_cancel | lemma | add_comm_group.modeq.add_right_cancel' | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_left_cancel' (c : α) : c - a ≡ c - b [PMOD p] → a ≡ b [PMOD p] | modeq_rfl.sub_left_cancel | lemma | add_comm_group.modeq.sub_left_cancel' | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_right_cancel' (c : α) : a - c ≡ b - c [PMOD p] → a ≡ b [PMOD p] | modeq_rfl.sub_right_cancel | lemma | add_comm_group.modeq.sub_right_cancel' | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_sub_iff_add_modeq' : a ≡ b - c [PMOD p] ↔ c + a ≡ b [PMOD p] | by simp [modeq, sub_sub] | lemma | add_comm_group.modeq_sub_iff_add_modeq' | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_sub_iff_add_modeq : a ≡ b - c [PMOD p] ↔ a + c ≡ b [PMOD p] | modeq_sub_iff_add_modeq'.trans $ by rw add_comm | lemma | add_comm_group.modeq_sub_iff_add_modeq | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_modeq_iff_modeq_add' : a - b ≡ c [PMOD p] ↔ a ≡ b + c [PMOD p] | modeq_comm.trans $ modeq_sub_iff_add_modeq'.trans modeq_comm | lemma | add_comm_group.sub_modeq_iff_modeq_add' | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_modeq_iff_modeq_add : a - b ≡ c [PMOD p] ↔ a ≡ c + b [PMOD p] | modeq_comm.trans $ modeq_sub_iff_add_modeq.trans modeq_comm | lemma | add_comm_group.sub_modeq_iff_modeq_add | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_modeq_zero : a - b ≡ 0 [PMOD p] ↔ a ≡ b [PMOD p] | by simp [sub_modeq_iff_modeq_add] | lemma | add_comm_group.sub_modeq_zero | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_modeq_left : a + b ≡ a [PMOD p] ↔ b ≡ 0 [PMOD p] | by simp [←modeq_sub_iff_add_modeq'] | lemma | add_comm_group.add_modeq_left | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_modeq_right : a + b ≡ b [PMOD p] ↔ a ≡ 0 [PMOD p] | by simp [←modeq_sub_iff_add_modeq] | lemma | add_comm_group.add_modeq_right | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_iff_eq_add_zsmul : a ≡ b [PMOD p] ↔ ∃ z : ℤ, b = a + z • p | by simp_rw [modeq, sub_eq_iff_eq_add'] | lemma | add_comm_group.modeq_iff_eq_add_zsmul | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_modeq_iff_ne_add_zsmul : ¬a ≡ b [PMOD p] ↔ ∀ z : ℤ, b ≠ a + z • p | by rw [modeq_iff_eq_add_zsmul, not_exists] | lemma | add_comm_group.not_modeq_iff_ne_add_zsmul | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"not_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_iff_eq_mod_zmultiples : a ≡ b [PMOD p] ↔ (b : α ⧸ add_subgroup.zmultiples p) = a | by simp_rw [modeq_iff_eq_add_zsmul, quotient_add_group.eq_iff_sub_mem,
add_subgroup.mem_zmultiples_iff, eq_sub_iff_add_eq', eq_comm] | lemma | add_comm_group.modeq_iff_eq_mod_zmultiples | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"add_subgroup.zmultiples"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_modeq_iff_ne_mod_zmultiples :
¬a ≡ b [PMOD p] ↔ (b : α ⧸ add_subgroup.zmultiples p) ≠ a | modeq_iff_eq_mod_zmultiples.not | lemma | add_comm_group.not_modeq_iff_ne_mod_zmultiples | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"add_subgroup.zmultiples"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
modeq_iff_int_modeq {a b z : ℤ} : a ≡ b [PMOD z] ↔ a ≡ b [ZMOD z] | by simp [modeq, dvd_iff_exists_eq_mul_left, int.modeq_iff_dvd] | lemma | add_comm_group.modeq_iff_int_modeq | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"dvd_iff_exists_eq_mul_left",
"int.modeq_iff_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_modeq_int_cast {a b z : ℤ} : a ≡ b [PMOD (z : α)] ↔ a ≡ b [PMOD z] | by simp_rw [modeq, ←int.cast_mul_eq_zsmul_cast]; norm_cast | lemma | add_comm_group.int_cast_modeq_int_cast | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_modeq_nat_cast {a b n : ℕ} : a ≡ b [PMOD (n : α)] ↔ a ≡ b [MOD n] | by simp_rw [←int.coe_nat_modeq_iff, ←modeq_iff_int_modeq, ←@int_cast_modeq_int_cast α,
int.cast_coe_nat] | lemma | add_comm_group.nat_cast_modeq_nat_cast | algebra | src/algebra/modeq.lean | [
"data.int.modeq",
"group_theory.quotient_group"
] | [
"int.cast_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero {R} [has_zero R] (n : R) : Prop | (out : n ≠ 0) | class | ne_zero | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [] | A type-class version of `n ≠ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ne_zero.ne {R} [has_zero R] (n : R) [h : ne_zero n] : n ≠ 0 | h.out | lemma | ne_zero.ne | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero.ne' {R} [has_zero R] (n : R) [h : ne_zero n] : 0 ≠ n | h.out.symm | lemma | ne_zero.ne' | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_iff {R : Type*} [has_zero R] {n : R} : ne_zero n ↔ n ≠ 0 | ⟨λ h, h.out, ne_zero.mk⟩ | lemma | ne_zero_iff | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_ne_zero {R : Type*} [has_zero R] {n : R} : ¬ ne_zero n ↔ n = 0 | by simp [ne_zero_iff] | lemma | not_ne_zero | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_or_ne_zero {α} [has_zero α] (a : α) : a = 0 ∨ ne_zero a | (eq_or_ne a 0).imp_right ne_zero.mk | lemma | eq_zero_or_ne_zero | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"eq_or_ne",
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_ne_one [ne_zero (1 : α)] : (0 : α) ≠ 1 | ne_zero.ne' (1 : α) | lemma | zero_ne_one | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_ne_zero [ne_zero (1 : α)] : (1 : α) ≠ 0 | ne_zero.ne (1 : α) | lemma | one_ne_zero | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_ne_zero [has_add α] [ne_zero (2 : α)] : (2 : α) ≠ 0 | ne_zero.ne (2 : α) | lemma | two_ne_zero | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
three_ne_zero [has_add α] [ne_zero (3 : α)] : (3 : α) ≠ 0 | ne_zero.ne (3 : α) | lemma | three_ne_zero | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
four_ne_zero [has_add α] [ne_zero (4 : α)] : (4 : α) ≠ 0 | ne_zero.ne (4 : α) | lemma | four_ne_zero | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_eq_one [ne_zero (1 : α)] {a : α} (h : a = 1) : a ≠ 0 | calc a = 1 : h
... ≠ 0 : one_ne_zero | lemma | ne_zero_of_eq_one | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_ne_one' [ne_zero (1 : α)] : (0 : α) ≠ 1 | ne_zero.ne' (1 : α) | lemma | zero_ne_one' | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_ne_zero' [ne_zero (1 : α)] : (1 : α) ≠ 0 | ne_zero.ne (1 : α) | lemma | one_ne_zero' | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_ne_zero' [has_add α] [ne_zero (2 : α)] : (2 : α) ≠ 0 | ne_zero.ne (2 : α) | lemma | two_ne_zero' | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
three_ne_zero' [has_add α] [ne_zero (3 : α)] : (3 : α) ≠ 0 | ne_zero.ne (3 : α) | lemma | three_ne_zero' | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
four_ne_zero' [has_add α] [ne_zero (4 : α)] : (4 : α) ≠ 0 | ne_zero.ne (4 : α) | lemma | four_ne_zero' | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero",
"ne_zero.ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ : ne_zero (n + 1) | ⟨n.succ_ne_zero⟩ | instance | ne_zero.succ | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_pos [preorder M] [has_zero M] (h : 0 < x) : ne_zero x | ⟨ne_of_gt h⟩ | lemma | ne_zero.of_pos | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_trans [has_zero M] [has_coe R S] [has_coe_t S M] [h : ne_zero (r : M)] :
ne_zero ((r : S) : M) | ⟨h.out⟩ | instance | ne_zero.coe_trans | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans [has_zero M] [has_coe R S] [has_coe_t S M] (h : ne_zero ((r : S) : M)) :
ne_zero (r : M) | ⟨h.out⟩ | lemma | ne_zero.trans | algebra | src/algebra/ne_zero.lean | [
"logic.basic"
] | [
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_opposite (α : Type u) : Type u | α | def | mul_opposite | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | Multiplicative opposite of a type. This type inherits all additive structures on `α` and
reverses left and right in multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op : α → αᵐᵒᵖ | id | def | mul_opposite.op | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | The element of `mul_opposite α` that represents `x : α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop : αᵐᵒᵖ → α | id | def | mul_opposite.unop | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | The element of `α` represented by `x : αᵐᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unop_op (x : α) : unop (op x) = x | rfl | lemma | mul_opposite.unop_op | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_unop (x : αᵐᵒᵖ) : op (unop x) = x | rfl | lemma | mul_opposite.op_unop | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_comp_unop : (op : α → αᵐᵒᵖ) ∘ unop = id | rfl | lemma | mul_opposite.op_comp_unop | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_comp_op : (unop : αᵐᵒᵖ → α) ∘ op = id | rfl | lemma | mul_opposite.unop_comp_op | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rec {F : Π (X : αᵐᵒᵖ), Sort v} (h : Π X, F (op X)) : Π X, F X | λ X, h (unop X) | def | mul_opposite.rec | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | A recursor for `mul_opposite`. Use as `induction x using mul_opposite.rec`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_equiv : α ≃ αᵐᵒᵖ | ⟨op, unop, unop_op, op_unop⟩ | def | mul_opposite.op_equiv | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | The canonical bijection between `α` and `αᵐᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_bijective : bijective (op : α → αᵐᵒᵖ) | op_equiv.bijective | lemma | mul_opposite.op_bijective | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_bijective : bijective (unop : αᵐᵒᵖ → α) | op_equiv.symm.bijective | lemma | mul_opposite.unop_bijective | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_injective : injective (op : α → αᵐᵒᵖ) | op_bijective.injective | lemma | mul_opposite.op_injective | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_surjective : surjective (op : α → αᵐᵒᵖ) | op_bijective.surjective | lemma | mul_opposite.op_surjective | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_injective : injective (unop : αᵐᵒᵖ → α) | unop_bijective.injective | lemma | mul_opposite.unop_injective | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_surjective : surjective (unop : αᵐᵒᵖ → α) | unop_bijective.surjective | lemma | mul_opposite.unop_surjective | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_inj {x y : α} : op x = op y ↔ x = y | op_injective.eq_iff | lemma | mul_opposite.op_inj | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_inj {x y : αᵐᵒᵖ} : unop x = unop y ↔ x = y | unop_injective.eq_iff | lemma | mul_opposite.unop_inj | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_zero [has_zero α] : op (0 : α) = 0 | rfl | lemma | mul_opposite.op_zero | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_zero [has_zero α] : unop (0 : αᵐᵒᵖ) = 0 | rfl | lemma | mul_opposite.unop_zero | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_one [has_one α] : op (1 : α) = 1 | rfl | lemma | mul_opposite.op_one | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_one [has_one α] : unop (1 : αᵐᵒᵖ) = 1 | rfl | lemma | mul_opposite.unop_one | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_add [has_add α] (x y : α) : op (x + y) = op x + op y | rfl | lemma | mul_opposite.op_add | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_add [has_add α] (x y : αᵐᵒᵖ) : unop (x + y) = unop x + unop y | rfl | lemma | mul_opposite.unop_add | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_neg [has_neg α] (x : α) : op (-x) = -op x | rfl | lemma | mul_opposite.op_neg | algebra | src/algebra/opposites.lean | [
"algebra.group.defs",
"logic.equiv.defs",
"logic.nontrivial"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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