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
value |
|---|---|---|---|---|---|---|---|---|---|---|
adjoin.power_basis_aux {x : S} (hx : is_integral K x) :
basis (fin (minpoly K x).nat_degree) K (adjoin K ({x} : set S)) | begin
have hST : function.injective (algebra_map (adjoin K ({x} : set S)) S) := subtype.coe_injective,
have hx' : is_integral K
(show adjoin K ({x} : set S), from ⟨x, subset_adjoin (set.mem_singleton x)⟩),
{ apply (is_integral_algebra_map_iff hST).mp,
convert hx,
apply_instance },
have minpoly_eq :=... | def | algebra.adjoin.power_basis_aux | ring_theory.adjoin | src/ring_theory/adjoin/power_basis.lean | [
"ring_theory.adjoin.basic",
"ring_theory.power_basis",
"linear_algebra.matrix.basis"
] | [
"algebra_map",
"basis",
"basis.mk",
"is_integral",
"is_integral_algebra_map_iff",
"linear_independent_pow",
"minpoly",
"minpoly.eq_of_algebra_map_eq",
"set.mem_singleton",
"subtype.coe_injective"
] | The elements `1, x, ..., x ^ (d - 1)` for a basis for the `K`-module `K[x]`,
where `d` is the degree of the minimal polynomial of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoin.power_basis {x : S} (hx : is_integral K x) :
power_basis K (adjoin K ({x} : set S)) | { gen := ⟨x, subset_adjoin (set.mem_singleton x)⟩,
dim := (minpoly K x).nat_degree,
basis := adjoin.power_basis_aux hx,
basis_eq_pow := basis.mk_apply _ _ } | def | algebra.adjoin.power_basis | ring_theory.adjoin | src/ring_theory/adjoin/power_basis.lean | [
"ring_theory.adjoin.basic",
"ring_theory.power_basis",
"linear_algebra.matrix.basis"
] | [
"basis",
"basis.mk_apply",
"is_integral",
"minpoly",
"power_basis",
"set.mem_singleton"
] | The power basis `1, x, ..., x ^ (d - 1)` for `K[x]`,
where `d` is the degree of the minimal polynomial of `x`. See `algebra.adjoin.power_basis'` for
a version over a more general base ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
power_basis.of_gen_mem_adjoin {x : S} (B : power_basis K S)
(hint : is_integral K x) (hx : B.gen ∈ adjoin K ({x} : set S)) : power_basis K S | (algebra.adjoin.power_basis hint).map $
(subalgebra.equiv_of_eq _ _ $ power_basis.adjoin_eq_top_of_gen_mem_adjoin hx).trans
subalgebra.top_equiv | def | power_basis.of_gen_mem_adjoin | ring_theory.adjoin | src/ring_theory/adjoin/power_basis.lean | [
"ring_theory.adjoin.basic",
"ring_theory.power_basis",
"linear_algebra.matrix.basis"
] | [
"algebra.adjoin.power_basis",
"is_integral",
"power_basis",
"power_basis.adjoin_eq_top_of_gen_mem_adjoin",
"subalgebra.equiv_of_eq",
"subalgebra.top_equiv"
] | The power basis given by `x` if `B.gen ∈ adjoin K {x}`. See `power_basis.of_gen_mem_adjoin'`
for a version over a more general base ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
repr_gen_pow_is_integral [is_domain S]
(hmin : minpoly S B.gen = (minpoly R B.gen).map (algebra_map R S)) (n : ℕ) :
∀ i, is_integral R (B.basis.repr (B.gen ^ n) i) | begin
intro i,
let Q := (X ^ n) %ₘ (minpoly R B.gen),
have : B.gen ^ n = aeval B.gen Q,
{ rw [← @aeval_X_pow R _ _ _ _ B.gen, ← mod_by_monic_add_div (X ^ n) (minpoly.monic hB)],
simp },
by_cases hQ : Q = 0,
{ simp [this, hQ, is_integral_zero] },
have hlt : Q.nat_degree < B.dim,
{ rw [← B.nat_degree_... | lemma | power_basis.repr_gen_pow_is_integral | ring_theory.adjoin | src/ring_theory/adjoin/power_basis.lean | [
"ring_theory.adjoin.basic",
"ring_theory.power_basis",
"linear_algebra.matrix.basis"
] | [
"algebra.smul_def",
"algebra_map",
"algebra_map_smul",
"fin.coe_mk",
"finset.sum_apply'",
"finsupp.coe_smul",
"is_domain",
"is_integral",
"is_integral.sum",
"is_integral_algebra_map",
"is_integral_zero",
"is_scalar_tower.algebra_map_apply",
"linear_equiv.map_smul",
"linear_equiv.map_smulₛₗ... | If `B : power_basis S A` is such that `is_integral R B.gen`, then
`is_integral R (B.basis.repr (B.gen ^ n) i)` for all `i` if
`minpoly S B.gen = (minpoly R B.gen).map (algebra_map R S)`. This is the case if `R` is a GCD domain
and `S` is its fraction ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
repr_mul_is_integral [is_domain S] {x y : A} (hx : ∀ i, is_integral R (B.basis.repr x i))
(hy : ∀ i, is_integral R (B.basis.repr y i))
(hmin : minpoly S B.gen = (minpoly R B.gen).map (algebra_map R S)) :
∀ i, is_integral R ((B.basis.repr (x * y) i)) | begin
intro i,
rw [← B.basis.sum_repr x, ← B.basis.sum_repr y, finset.sum_mul_sum, linear_equiv.map_sum,
finset.sum_apply'],
refine is_integral.sum _ (λ I hI, _),
simp only [algebra.smul_mul_assoc, algebra.mul_smul_comm, linear_equiv.map_smulₛₗ,
ring_hom.id_apply, finsupp.coe_smul, pi.smul_apply, id.smu... | lemma | power_basis.repr_mul_is_integral | ring_theory.adjoin | src/ring_theory/adjoin/power_basis.lean | [
"ring_theory.adjoin.basic",
"ring_theory.power_basis",
"linear_algebra.matrix.basis"
] | [
"algebra.mul_smul_comm",
"algebra.smul_mul_assoc",
"algebra_map",
"finset.sum_apply'",
"finset.sum_mul_sum",
"finsupp.coe_smul",
"is_domain",
"is_integral",
"is_integral.sum",
"is_integral_mul",
"linear_equiv.map_smulₛₗ",
"linear_equiv.map_sum",
"minpoly",
"pi.smul_apply",
"pow_add",
"... | Let `B : power_basis S A` be such that `is_integral R B.gen`, and let `x y : A` be elements with
integral coordinates in the base `B.basis`. Then `is_integral R ((B.basis.repr (x * y) i)` for all
`i` if `minpoly S B.gen = (minpoly R B.gen).map (algebra_map R S)`. This is the case if `R` is a GCD
domain and `S` is its f... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
repr_pow_is_integral [is_domain S] {x : A} (hx : ∀ i, is_integral R (B.basis.repr x i))
(hmin : minpoly S B.gen = (minpoly R B.gen).map (algebra_map R S)) (n : ℕ) :
∀ i, is_integral R ((B.basis.repr (x ^ n) i)) | begin
nontriviality A using [subsingleton.elim (x ^ n) 0, is_integral_zero],
revert hx,
refine nat.case_strong_induction_on n _ (λ n hn, _),
{ intros hx i,
rw [pow_zero, ← pow_zero B.gen, ← fin.coe_mk B.dim_pos, ← B.basis_eq_pow,
B.basis.repr_self_apply],
split_ifs,
{ exact is_integral_one },
... | lemma | power_basis.repr_pow_is_integral | ring_theory.adjoin | src/ring_theory/adjoin/power_basis.lean | [
"ring_theory.adjoin.basic",
"ring_theory.power_basis",
"linear_algebra.matrix.basis"
] | [
"algebra_map",
"fin.coe_mk",
"is_domain",
"is_integral",
"is_integral_one",
"is_integral_zero",
"le_rfl",
"minpoly",
"pow_succ",
"pow_zero"
] | Let `B : power_basis S A` be such that `is_integral R B.gen`, and let `x : A` be and element
with integral coordinates in the base `B.basis`. Then `is_integral R ((B.basis.repr (x ^ n) i)` for
all `i` and all `n` if `minpoly S B.gen = (minpoly R B.gen).map (algebra_map R S)`. This is the case
if `R` is a GCD domain and... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_matrix_is_integral {B B' : power_basis K S} {P : R[X]} (h : aeval B.gen P = B'.gen)
(hB : is_integral R B.gen) (hmin : minpoly K B.gen = (minpoly R B.gen).map (algebra_map R K)) :
∀ i j, is_integral R (B.basis.to_matrix B'.basis i j) | begin
intros i j,
rw [B.basis.to_matrix_apply, B'.coe_basis],
refine repr_pow_is_integral hB (λ i, _) hmin _ _,
rw [← h, aeval_eq_sum_range, linear_equiv.map_sum, finset.sum_apply'],
refine is_integral.sum _ (λ n hn, _),
rw [algebra.smul_def, is_scalar_tower.algebra_map_apply R K S, ← algebra.smul_def,
... | lemma | power_basis.to_matrix_is_integral | ring_theory.adjoin | src/ring_theory/adjoin/power_basis.lean | [
"ring_theory.adjoin.basic",
"ring_theory.power_basis",
"linear_algebra.matrix.basis"
] | [
"algebra.smul_def",
"algebra_map",
"algebra_map_smul",
"finset.sum_apply'",
"is_integral",
"is_integral.sum",
"is_integral_smul",
"is_scalar_tower.algebra_map_apply",
"linear_equiv.map_smul",
"linear_equiv.map_sum",
"minpoly",
"power_basis"
] | Let `B B' : power_basis K S` be such that `is_integral R B.gen`, and let `P : R[X]` be such that
`aeval B.gen P = B'.gen`. Then `is_integral R (B.basis.to_matrix B'.basis i j)` for all `i` and `j`
if `minpoly K B.gen = (minpoly R B.gen).map (algebra_map R L)`. This is the case
if `R` is a GCD domain and `K` is its frac... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoin_algebra_map (R : Type u) (S : Type v) (A : Type w)
[comm_semiring R] [comm_semiring S] [semiring A] [algebra R S] [algebra S A] [algebra R A]
[is_scalar_tower R S A] (s : set S) :
adjoin R (algebra_map S A '' s) = (adjoin R s).map (is_scalar_tower.to_alg_hom R S A) | le_antisymm (adjoin_le $ set.image_subset_iff.2 $ λ y hy, ⟨y, subset_adjoin hy, rfl⟩)
(subalgebra.map_le.2 $ adjoin_le $ λ y hy, subset_adjoin ⟨y, hy, rfl⟩) | theorem | algebra.adjoin_algebra_map | ring_theory.adjoin | src/ring_theory/adjoin/tower.lean | [
"ring_theory.adjoin.fg"
] | [
"algebra",
"algebra_map",
"comm_semiring",
"is_scalar_tower",
"is_scalar_tower.to_alg_hom",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoin_restrict_scalars (C D E : Type*) [comm_semiring C] [comm_semiring D] [comm_semiring E]
[algebra C D] [algebra C E] [algebra D E] [is_scalar_tower C D E] (S : set E) :
(algebra.adjoin D S).restrict_scalars C =
(algebra.adjoin
((⊤ : subalgebra C D).map (is_scalar_tower.to_alg_hom C D E)) S).restrict_scalar... | begin
suffices : set.range (algebra_map D E) =
set.range (algebra_map ((⊤ : subalgebra C D).map (is_scalar_tower.to_alg_hom C D E)) E),
{ ext x, change x ∈ subsemiring.closure (_ ∪ S) ↔ x ∈ subsemiring.closure (_ ∪ S), rw this },
ext x,
split,
{ rintros ⟨y, hy⟩,
exact ⟨⟨algebra_map D E y, ⟨y, ⟨algebra... | lemma | algebra.adjoin_restrict_scalars | ring_theory.adjoin | src/ring_theory/adjoin/tower.lean | [
"ring_theory.adjoin.fg"
] | [
"algebra",
"algebra.adjoin",
"algebra_map",
"comm_semiring",
"is_scalar_tower",
"is_scalar_tower.to_alg_hom",
"restrict_scalars",
"set.range",
"subalgebra",
"subsemiring.closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoin_res_eq_adjoin_res (C D E F : Type*) [comm_semiring C] [comm_semiring D]
[comm_semiring E] [comm_semiring F] [algebra C D] [algebra C E] [algebra C F] [algebra D F]
[algebra E F] [is_scalar_tower C D F] [is_scalar_tower C E F] {S : set D} {T : set E}
(hS : algebra.adjoin C S = ⊤) (hT : algebra.adjoin C T = ... | by rw [adjoin_restrict_scalars C E, adjoin_restrict_scalars C D, ←hS, ←hT, ←algebra.adjoin_image,
←algebra.adjoin_image, ←alg_hom.coe_to_ring_hom, ←alg_hom.coe_to_ring_hom,
is_scalar_tower.coe_to_alg_hom, is_scalar_tower.coe_to_alg_hom, ←adjoin_union_eq_adjoin_adjoin,
←adjoin_union_eq_adjoin_adjoin, set.union_com... | lemma | algebra.adjoin_res_eq_adjoin_res | ring_theory.adjoin | src/ring_theory/adjoin/tower.lean | [
"ring_theory.adjoin.fg"
] | [
"algebra",
"algebra.adjoin",
"algebra_map",
"comm_semiring",
"is_scalar_tower",
"is_scalar_tower.coe_to_alg_hom",
"restrict_scalars",
"set.union_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.fg_trans' {R S A : Type*} [comm_semiring R] [comm_semiring S] [comm_semiring A]
[algebra R S] [algebra S A] [algebra R A] [is_scalar_tower R S A]
(hRS : (⊤ : subalgebra R S).fg) (hSA : (⊤ : subalgebra S A).fg) :
(⊤ : subalgebra R A).fg | let ⟨s, hs⟩ := hRS, ⟨t, ht⟩ := hSA in ⟨s.image (algebra_map S A) ∪ t,
by rw [finset.coe_union, finset.coe_image, algebra.adjoin_union_eq_adjoin_adjoin,
algebra.adjoin_algebra_map, hs, algebra.map_top, is_scalar_tower.adjoin_range_to_alg_hom, ht,
subalgebra.restrict_scalars_top]⟩ | lemma | algebra.fg_trans' | ring_theory.adjoin | src/ring_theory/adjoin/tower.lean | [
"ring_theory.adjoin.fg"
] | [
"algebra",
"algebra.adjoin_algebra_map",
"algebra.adjoin_union_eq_adjoin_adjoin",
"algebra.map_top",
"algebra_map",
"comm_semiring",
"finset.coe_image",
"finset.coe_union",
"is_scalar_tower",
"is_scalar_tower.adjoin_range_to_alg_hom",
"subalgebra",
"subalgebra.restrict_scalars_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_subalgebra_of_fg (hAC : (⊤ : subalgebra A C).fg) (hBC : (⊤ : submodule B C).fg) :
∃ B₀ : subalgebra A B, B₀.fg ∧ (⊤ : submodule B₀ C).fg | begin
cases hAC with x hx,
cases hBC with y hy, have := hy,
simp_rw [eq_top_iff', mem_span_finset] at this, choose f hf,
let s : finset B := finset.image₂ f (x ∪ (y * y)) y,
have hxy : ∀ xi ∈ x, xi ∈ span (algebra.adjoin A (↑s : set B))
(↑(insert 1 y : finset C) : set C) :=
λ xi hxi, hf xi ... | lemma | exists_subalgebra_of_fg | ring_theory.adjoin | src/ring_theory/adjoin/tower.lean | [
"ring_theory.adjoin.fg"
] | [
"algebra.adjoin",
"algebra.adjoin_eq_span",
"algebra.subset_adjoin",
"algebra.top_to_submodule",
"eq_top_iff",
"finset",
"finset.image₂",
"mem_span_finset",
"mul_one",
"one_mul",
"set.mem_insert",
"set.mem_insert_of_mem",
"subalgebra",
"subalgebra.fg_adjoin_finset",
"submodule",
"submo... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_of_fg_of_fg [is_noetherian_ring A]
(hAC : (⊤ : subalgebra A C).fg) (hBC : (⊤ : submodule B C).fg)
(hBCi : function.injective (algebra_map B C)) :
(⊤ : subalgebra A B).fg | let ⟨B₀, hAB₀, hB₀C⟩ := exists_subalgebra_of_fg A B C hAC hBC in
algebra.fg_trans' (B₀.fg_top.2 hAB₀) $ subalgebra.fg_of_submodule_fg $
have is_noetherian_ring B₀, from is_noetherian_ring_of_fg hAB₀,
have is_noetherian B₀ C, by exactI is_noetherian_of_fg_of_noetherian' hB₀C,
by exactI fg_of_injective (is_scalar_tower.t... | theorem | fg_of_fg_of_fg | ring_theory.adjoin | src/ring_theory/adjoin/tower.lean | [
"ring_theory.adjoin.fg"
] | [
"algebra.fg_trans'",
"algebra_map",
"exists_subalgebra_of_fg",
"fg_of_injective",
"is_noetherian",
"is_noetherian_of_fg_of_noetherian'",
"is_noetherian_ring",
"is_noetherian_ring_of_fg",
"is_scalar_tower.to_alg_hom",
"subalgebra",
"subalgebra.fg_of_submodule_fg",
"submodule"
] | **Artin--Tate lemma**: if A ⊆ B ⊆ C is a chain of subrings of commutative rings, and
A is noetherian, and C is algebra-finite over A, and C is module-finite over B,
then B is algebra-finite over A.
References: Atiyah--Macdonald Proposition 7.8; Stacks 00IS; Altman--Kleiman 16.17. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_coprime : Prop | ∃ a b, a * x + b * y = 1 | def | is_coprime | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [] | The proposition that `x` and `y` are coprime, defined to be the existence of `a` and `b` such
that `a * x + b * y = 1`. Note that elements with no common divisors are not necessarily coprime,
e.g., the multivariate polynomials `x₁` and `x₂` are not coprime. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_coprime.symm (H : is_coprime x y) : is_coprime y x | let ⟨a, b, H⟩ := H in ⟨b, a, by rw [add_comm, H]⟩ | theorem | is_coprime.symm | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_comm : is_coprime x y ↔ is_coprime y x | ⟨is_coprime.symm, is_coprime.symm⟩ | theorem | is_coprime_comm | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_self : is_coprime x x ↔ is_unit x | ⟨λ ⟨a, b, h⟩, is_unit_of_mul_eq_one x (a + b) $ by rwa [mul_comm, add_mul],
λ h, let ⟨b, hb⟩ := is_unit_iff_exists_inv'.1 h in ⟨b, 0, by rwa [zero_mul, add_zero]⟩⟩ | theorem | is_coprime_self | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_unit",
"is_unit_of_mul_eq_one",
"mul_comm",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_zero_left : is_coprime 0 x ↔ is_unit x | ⟨λ ⟨a, b, H⟩, is_unit_of_mul_eq_one x b $ by rwa [mul_zero, zero_add, mul_comm] at H,
λ H, let ⟨b, hb⟩ := is_unit_iff_exists_inv'.1 H in ⟨1, b, by rwa [one_mul, zero_add]⟩⟩ | theorem | is_coprime_zero_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_unit",
"is_unit_of_mul_eq_one",
"mul_comm",
"mul_zero",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_zero_right : is_coprime x 0 ↔ is_unit x | is_coprime_comm.trans is_coprime_zero_left | theorem | is_coprime_zero_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime_zero_left",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_coprime_zero_zero [nontrivial R] : ¬ is_coprime (0 : R) 0 | mt is_coprime_zero_right.mp not_is_unit_zero | lemma | not_coprime_zero_zero | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"nontrivial",
"not_is_unit_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.ne_zero [nontrivial R] {p : fin 2 → R} (h : is_coprime (p 0) (p 1)) : p ≠ 0 | by { rintro rfl, exact not_coprime_zero_zero h } | lemma | is_coprime.ne_zero | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"nontrivial",
"not_coprime_zero_zero"
] | If a 2-vector `p` satisfies `is_coprime (p 0) (p 1)`, then `p ≠ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_coprime_one_left : is_coprime 1 x | ⟨1, 0, by rw [one_mul, zero_mul, add_zero]⟩ | theorem | is_coprime_one_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"one_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_one_right : is_coprime x 1 | ⟨0, 1, by rw [one_mul, zero_mul, zero_add]⟩ | theorem | is_coprime_one_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"one_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.dvd_of_dvd_mul_right (H1 : is_coprime x z) (H2 : x ∣ y * z) : x ∣ y | let ⟨a, b, H⟩ := H1 in by { rw [← mul_one y, ← H, mul_add, ← mul_assoc, mul_left_comm],
exact dvd_add (dvd_mul_left _ _) (H2.mul_left _) } | theorem | is_coprime.dvd_of_dvd_mul_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"dvd_add",
"dvd_mul_left",
"is_coprime",
"mul_assoc",
"mul_left_comm",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.dvd_of_dvd_mul_left (H1 : is_coprime x y) (H2 : x ∣ y * z) : x ∣ z | let ⟨a, b, H⟩ := H1 in by { rw [← one_mul z, ← H, add_mul, mul_right_comm, mul_assoc b],
exact dvd_add (dvd_mul_left _ _) (H2.mul_left _) } | theorem | is_coprime.dvd_of_dvd_mul_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"dvd_add",
"dvd_mul_left",
"is_coprime",
"mul_assoc",
"mul_right_comm",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.mul_left (H1 : is_coprime x z) (H2 : is_coprime y z) : is_coprime (x * y) z | let ⟨a, b, h1⟩ := H1, ⟨c, d, h2⟩ := H2 in
⟨a * c, a * x * d + b * c * y + b * d * z,
calc a * c * (x * y) + (a * x * d + b * c * y + b * d * z) * z
= (a * x + b * z) * (c * y + d * z) : by ring
... = 1 : by rw [h1, h2, mul_one]⟩ | theorem | is_coprime.mul_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"mul_one",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.mul_right (H1 : is_coprime x y) (H2 : is_coprime x z) : is_coprime x (y * z) | by { rw is_coprime_comm at H1 H2 ⊢, exact H1.mul_left H2 } | theorem | is_coprime.mul_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.mul_dvd (H : is_coprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z | begin
obtain ⟨a, b, h⟩ := H,
rw [← mul_one z, ← h, mul_add],
apply dvd_add,
{ rw [mul_comm z, mul_assoc],
exact (mul_dvd_mul_left _ H2).mul_left _ },
{ rw [mul_comm b, ← mul_assoc],
exact (mul_dvd_mul_right H1 _).mul_right _ }
end | theorem | is_coprime.mul_dvd | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"dvd_add",
"is_coprime",
"mul_assoc",
"mul_comm",
"mul_dvd_mul_left",
"mul_dvd_mul_right",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_mul_left_left (H : is_coprime (x * y) z) : is_coprime x z | let ⟨a, b, h⟩ := H in ⟨a * y, b, by rwa [mul_right_comm, mul_assoc]⟩ | theorem | is_coprime.of_mul_left_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"mul_assoc",
"mul_right_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_mul_left_right (H : is_coprime (x * y) z) : is_coprime y z | by { rw mul_comm at H, exact H.of_mul_left_left } | theorem | is_coprime.of_mul_left_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_mul_right_left (H : is_coprime x (y * z)) : is_coprime x y | by { rw is_coprime_comm at H ⊢, exact H.of_mul_left_left } | theorem | is_coprime.of_mul_right_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_mul_right_right (H : is_coprime x (y * z)) : is_coprime x z | by { rw mul_comm at H, exact H.of_mul_right_left } | theorem | is_coprime.of_mul_right_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.mul_left_iff : is_coprime (x * y) z ↔ is_coprime x z ∧ is_coprime y z | ⟨λ H, ⟨H.of_mul_left_left, H.of_mul_left_right⟩, λ ⟨H1, H2⟩, H1.mul_left H2⟩ | theorem | is_coprime.mul_left_iff | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.mul_right_iff : is_coprime x (y * z) ↔ is_coprime x y ∧ is_coprime x z | by rw [is_coprime_comm, is_coprime.mul_left_iff, is_coprime_comm, @is_coprime_comm _ _ z] | theorem | is_coprime.mul_right_iff | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime.mul_left_iff",
"is_coprime_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_coprime_of_dvd_left (h : is_coprime y z) (hdvd : x ∣ y) : is_coprime x z | begin
obtain ⟨d, rfl⟩ := hdvd,
exact is_coprime.of_mul_left_left h
end | theorem | is_coprime.of_coprime_of_dvd_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime.of_mul_left_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_coprime_of_dvd_right (h : is_coprime z y) (hdvd : x ∣ y) : is_coprime z x | (h.symm.of_coprime_of_dvd_left hdvd).symm | theorem | is_coprime.of_coprime_of_dvd_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.is_unit_of_dvd (H : is_coprime x y) (d : x ∣ y) : is_unit x | let ⟨k, hk⟩ := d in is_coprime_self.1 $ is_coprime.of_mul_right_left $
show is_coprime x (x * k), from hk ▸ H | theorem | is_coprime.is_unit_of_dvd | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime.of_mul_right_left",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.is_unit_of_dvd' {a b x : R} (h : is_coprime a b) (ha : x ∣ a) (hb : x ∣ b) :
is_unit x | (h.of_coprime_of_dvd_left ha).is_unit_of_dvd hb | theorem | is_coprime.is_unit_of_dvd' | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.map (H : is_coprime x y) {S : Type v} [comm_semiring S] (f : R →+* S) :
is_coprime (f x) (f y) | let ⟨a, b, h⟩ := H in ⟨f a, f b, by rw [← f.map_mul, ← f.map_mul, ← f.map_add, h, f.map_one]⟩ | theorem | is_coprime.map | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"comm_semiring",
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_add_mul_left_left (h : is_coprime (x + y * z) y) : is_coprime x y | let ⟨a, b, H⟩ := h in ⟨a, a * z + b, by simpa only [add_mul, mul_add,
add_assoc, add_comm, add_left_comm, mul_assoc, mul_comm, mul_left_comm] using H⟩ | lemma | is_coprime.of_add_mul_left_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"mul_assoc",
"mul_comm",
"mul_left_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_add_mul_right_left (h : is_coprime (x + z * y) y) : is_coprime x y | by { rw mul_comm at h, exact h.of_add_mul_left_left } | lemma | is_coprime.of_add_mul_right_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_add_mul_left_right (h : is_coprime x (y + x * z)) : is_coprime x y | by { rw is_coprime_comm at h ⊢, exact h.of_add_mul_left_left } | lemma | is_coprime.of_add_mul_left_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_add_mul_right_right (h : is_coprime x (y + z * x)) : is_coprime x y | by { rw mul_comm at h, exact h.of_add_mul_left_right } | lemma | is_coprime.of_add_mul_right_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_mul_add_left_left (h : is_coprime (y * z + x) y) : is_coprime x y | by { rw add_comm at h, exact h.of_add_mul_left_left } | lemma | is_coprime.of_mul_add_left_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_mul_add_right_left (h : is_coprime (z * y + x) y) : is_coprime x y | by { rw add_comm at h, exact h.of_add_mul_right_left } | lemma | is_coprime.of_mul_add_right_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_mul_add_left_right (h : is_coprime x (x * z + y)) : is_coprime x y | by { rw add_comm at h, exact h.of_add_mul_left_right } | lemma | is_coprime.of_mul_add_left_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_mul_add_right_right (h : is_coprime x (z * x + y)) : is_coprime x y | by { rw add_comm at h, exact h.of_add_mul_right_right } | lemma | is_coprime.of_mul_add_right_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_group_smul_left : is_coprime (x • y) z ↔ is_coprime y z | ⟨λ ⟨a, b, h⟩, ⟨x • a, b, by rwa [smul_mul_assoc, ←mul_smul_comm]⟩,
λ ⟨a, b, h⟩, ⟨x⁻¹ • a, b, by rwa [smul_mul_smul, inv_mul_self, one_smul]⟩⟩ | lemma | is_coprime_group_smul_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"inv_mul_self",
"is_coprime",
"one_smul",
"smul_mul_assoc",
"smul_mul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_group_smul_right : is_coprime y (x • z) ↔ is_coprime y z | is_coprime_comm.trans $ (is_coprime_group_smul_left x z y).trans is_coprime_comm | lemma | is_coprime_group_smul_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime_comm",
"is_coprime_group_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_group_smul : is_coprime (x • y) (x • z) ↔ is_coprime y z | (is_coprime_group_smul_left x y (x • z)).trans (is_coprime_group_smul_right x y z) | lemma | is_coprime_group_smul | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime_group_smul_left",
"is_coprime_group_smul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_mul_unit_left_left : is_coprime (x * y) z ↔ is_coprime y z | let ⟨u, hu⟩ := hu in hu ▸ is_coprime_group_smul_left u y z | lemma | is_coprime_mul_unit_left_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime_group_smul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_mul_unit_left_right : is_coprime y (x * z) ↔ is_coprime y z | let ⟨u, hu⟩ := hu in hu ▸ is_coprime_group_smul_right u y z | lemma | is_coprime_mul_unit_left_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime_group_smul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_mul_unit_left : is_coprime (x * y) (x * z) ↔ is_coprime y z | (is_coprime_mul_unit_left_left hu y (x * z)).trans (is_coprime_mul_unit_left_right hu y z) | lemma | is_coprime_mul_unit_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime_mul_unit_left_left",
"is_coprime_mul_unit_left_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_mul_unit_right_left : is_coprime (y * x) z ↔ is_coprime y z | mul_comm x y ▸ is_coprime_mul_unit_left_left hu y z | lemma | is_coprime_mul_unit_right_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime_mul_unit_left_left",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_mul_unit_right_right : is_coprime y (z * x) ↔ is_coprime y z | mul_comm x z ▸ is_coprime_mul_unit_left_right hu y z | lemma | is_coprime_mul_unit_right_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime_mul_unit_left_right",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_mul_unit_right : is_coprime (y * x) (z * x) ↔ is_coprime y z | (is_coprime_mul_unit_right_left hu y (z * x)).trans (is_coprime_mul_unit_right_right hu y z) | lemma | is_coprime_mul_unit_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime_mul_unit_right_left",
"is_coprime_mul_unit_right_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mul_left_left {x y : R} (h : is_coprime x y) (z : R) : is_coprime (x + y * z) y | @of_add_mul_left_left R _ _ _ (-z) $
by simpa only [mul_neg, add_neg_cancel_right] using h | lemma | is_coprime.add_mul_left_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mul_right_left {x y : R} (h : is_coprime x y) (z : R) : is_coprime (x + z * y) y | by { rw mul_comm, exact h.add_mul_left_left z } | lemma | is_coprime.add_mul_right_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mul_left_right {x y : R} (h : is_coprime x y) (z : R) : is_coprime x (y + x * z) | by { rw is_coprime_comm, exact h.symm.add_mul_left_left z } | lemma | is_coprime.add_mul_left_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mul_right_right {x y : R} (h : is_coprime x y) (z : R) : is_coprime x (y + z * x) | by { rw is_coprime_comm, exact h.symm.add_mul_right_left z } | lemma | is_coprime.add_mul_right_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"is_coprime_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_add_left_left {x y : R} (h : is_coprime x y) (z : R) : is_coprime (y * z + x) y | by { rw add_comm, exact h.add_mul_left_left z } | lemma | is_coprime.mul_add_left_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_add_right_left {x y : R} (h : is_coprime x y) (z : R) : is_coprime (z * y + x) y | by { rw add_comm, exact h.add_mul_right_left z } | lemma | is_coprime.mul_add_right_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_add_left_right {x y : R} (h : is_coprime x y) (z : R) : is_coprime x (x * z + y) | by { rw add_comm, exact h.add_mul_left_right z } | lemma | is_coprime.mul_add_left_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_add_right_right {x y : R} (h : is_coprime x y) (z : R) : is_coprime x (z * x + y) | by { rw add_comm, exact h.add_mul_right_right z } | lemma | is_coprime.mul_add_right_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mul_left_left_iff {x y z : R} : is_coprime (x + y * z) y ↔ is_coprime x y | ⟨of_add_mul_left_left, λ h, h.add_mul_left_left z⟩ | lemma | is_coprime.add_mul_left_left_iff | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mul_right_left_iff {x y z : R} : is_coprime (x + z * y) y ↔ is_coprime x y | ⟨of_add_mul_right_left, λ h, h.add_mul_right_left z⟩ | lemma | is_coprime.add_mul_right_left_iff | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mul_left_right_iff {x y z : R} : is_coprime x (y + x * z) ↔ is_coprime x y | ⟨of_add_mul_left_right, λ h, h.add_mul_left_right z⟩ | lemma | is_coprime.add_mul_left_right_iff | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mul_right_right_iff {x y z : R} : is_coprime x (y + z * x) ↔ is_coprime x y | ⟨of_add_mul_right_right, λ h, h.add_mul_right_right z⟩ | lemma | is_coprime.add_mul_right_right_iff | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_add_left_left_iff {x y z : R} : is_coprime (y * z + x) y ↔ is_coprime x y | ⟨of_mul_add_left_left, λ h, h.mul_add_left_left z⟩ | lemma | is_coprime.mul_add_left_left_iff | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_add_right_left_iff {x y z : R} : is_coprime (z * y + x) y ↔ is_coprime x y | ⟨of_mul_add_right_left, λ h, h.mul_add_right_left z⟩ | lemma | is_coprime.mul_add_right_left_iff | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_add_left_right_iff {x y z : R} : is_coprime x (x * z + y) ↔ is_coprime x y | ⟨of_mul_add_left_right, λ h, h.mul_add_left_right z⟩ | lemma | is_coprime.mul_add_left_right_iff | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_add_right_right_iff {x y z : R} : is_coprime x (z * x + y) ↔ is_coprime x y | ⟨of_mul_add_right_right, λ h, h.mul_add_right_right z⟩ | lemma | is_coprime.mul_add_right_right_iff | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_left {x y : R} (h : is_coprime x y) : is_coprime (-x) y | begin
obtain ⟨a, b, h⟩ := h,
use [-a, b],
rwa neg_mul_neg,
end | lemma | is_coprime.neg_left | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"neg_mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_left_iff (x y : R) : is_coprime (-x) y ↔ is_coprime x y | ⟨λ h, neg_neg x ▸ h.neg_left, neg_left⟩ | lemma | is_coprime.neg_left_iff | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_right {x y : R} (h : is_coprime x y) : is_coprime x (-y) | h.symm.neg_left.symm | lemma | is_coprime.neg_right | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_right_iff (x y : R) : is_coprime x (-y) ↔ is_coprime x y | ⟨λ h, neg_neg y ▸ h.neg_right, neg_right⟩ | lemma | is_coprime.neg_right_iff | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_neg {x y : R} (h : is_coprime x y) : is_coprime (-x) (-y) | h.neg_left.neg_right | lemma | is_coprime.neg_neg | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_neg_iff (x y : R) : is_coprime (-x) (-y) ↔ is_coprime x y | (neg_left_iff _ _).trans (neg_right_iff _ _) | lemma | is_coprime.neg_neg_iff | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_add_sq_ne_zero {R : Type*} [linear_ordered_comm_ring R] {a b : R} (h : is_coprime a b) :
a ^ 2 + b ^ 2 ≠ 0 | begin
intros h',
obtain ⟨ha, hb⟩ := (add_eq_zero_iff' (sq_nonneg a) (sq_nonneg b)).mp h',
obtain rfl := pow_eq_zero ha,
obtain rfl := pow_eq_zero hb,
exact not_coprime_zero_zero h
end | lemma | is_coprime.sq_add_sq_ne_zero | ring_theory.coprime | src/ring_theory/coprime/basic.lean | [
"tactic.ring",
"group_theory.group_action.units"
] | [
"is_coprime",
"linear_ordered_comm_ring",
"not_coprime_zero_zero",
"pow_eq_zero",
"sq_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_infi_eq_top_iff_pairwise {t : finset ι} (h : t.nonempty) (I : ι → ideal R) :
(⨆ i ∈ t, ⨅ j (hj : j ∈ t) (ij : j ≠ i), I j) = ⊤ ↔
(t : set ι).pairwise (λ i j, I i ⊔ I j = ⊤) | begin
haveI : decidable_eq ι := classical.dec_eq ι,
rw [eq_top_iff_one, submodule.mem_supr_finset_iff_exists_sum],
refine h.cons_induction _ _; clear' t h,
{ simp only [finset.sum_singleton, finset.coe_singleton, set.pairwise_singleton, iff_true],
refine λ a, ⟨λ i, if h : i = a then ⟨1, _⟩ else 0, _⟩,
{... | lemma | ideal.supr_infi_eq_top_iff_pairwise | ring_theory.coprime | src/ring_theory/coprime/ideal.lean | [
"linear_algebra.dfinsupp",
"ring_theory.ideal.operations"
] | [
"classical.dec_eq",
"finset",
"finset.coe_cons",
"finset.coe_singleton",
"finset.mem_cons",
"finset.mem_cons_self",
"finset.mem_singleton",
"finset.mul_sum",
"finset.subset_cons",
"ideal",
"ih",
"infi_false",
"infi_infi_eq_left",
"mul_one",
"ne_of_mem_of_not_mem",
"pairwise",
"set.pa... | A finite family of ideals is pairwise coprime (that is, any two of them generate the whole ring)
iff when taking all the possible intersections of all but one of these ideals, the resulting family
of ideals still generate the whole ring.
For example with three ideals : `I ⊔ J = I ⊔ K = J ⊔ K = ⊤ ↔ (I ⊓ J) ⊔ (I ⊓ K) ⊔ ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat.is_coprime_iff_coprime {m n : ℕ} : is_coprime (m : ℤ) n ↔ nat.coprime m n | ⟨λ ⟨a, b, H⟩, nat.eq_one_of_dvd_one $ int.coe_nat_dvd.1 $ by { rw [int.coe_nat_one, ← H],
exact dvd_add (dvd_mul_of_dvd_right (int.coe_nat_dvd.2 $ nat.gcd_dvd_left m n) _)
(dvd_mul_of_dvd_right (int.coe_nat_dvd.2 $ nat.gcd_dvd_right m n) _) },
λ H, ⟨nat.gcd_a m n, nat.gcd_b m n, by rw [mul_comm _ (m : ℤ), mul_com... | theorem | nat.is_coprime_iff_coprime | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"dvd_add",
"dvd_mul_of_dvd_right",
"is_coprime",
"mul_comm",
"nat.gcd_b",
"nat.gcd_dvd_left",
"nat.gcd_dvd_right",
"nat.gcd_eq_gcd_ab"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.prod_left : (∀ i ∈ t, is_coprime (s i) x) → is_coprime (∏ i in t, s i) x | finset.induction_on t (λ _, is_coprime_one_left) $ λ b t hbt ih H,
by { rw finset.prod_insert hbt, rw finset.forall_mem_insert at H, exact H.1.mul_left (ih H.2) } | theorem | is_coprime.prod_left | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"finset.forall_mem_insert",
"finset.induction_on",
"finset.prod_insert",
"ih",
"is_coprime",
"is_coprime_one_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.prod_right : (∀ i ∈ t, is_coprime x (s i)) → is_coprime x (∏ i in t, s i) | by simpa only [is_coprime_comm] using is_coprime.prod_left | theorem | is_coprime.prod_right | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"is_coprime",
"is_coprime.prod_left",
"is_coprime_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.prod_left_iff : is_coprime (∏ i in t, s i) x ↔ ∀ i ∈ t, is_coprime (s i) x | finset.induction_on t (iff_of_true is_coprime_one_left $ λ _, false.elim) $ λ b t hbt ih,
by rw [finset.prod_insert hbt, is_coprime.mul_left_iff, ih, finset.forall_mem_insert] | theorem | is_coprime.prod_left_iff | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"finset.forall_mem_insert",
"finset.induction_on",
"finset.prod_insert",
"iff_of_true",
"ih",
"is_coprime",
"is_coprime.mul_left_iff",
"is_coprime_one_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.prod_right_iff : is_coprime x (∏ i in t, s i) ↔ ∀ i ∈ t, is_coprime x (s i) | by simpa only [is_coprime_comm] using is_coprime.prod_left_iff | theorem | is_coprime.prod_right_iff | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"is_coprime",
"is_coprime.prod_left_iff",
"is_coprime_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_prod_left (H1 : is_coprime (∏ i in t, s i) x) (i : I) (hit : i ∈ t) :
is_coprime (s i) x | is_coprime.prod_left_iff.1 H1 i hit | theorem | is_coprime.of_prod_left | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.of_prod_right (H1 : is_coprime x (∏ i in t, s i)) (i : I) (hit : i ∈ t) :
is_coprime x (s i) | is_coprime.prod_right_iff.1 H1 i hit | theorem | is_coprime.of_prod_right | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.prod_dvd_of_coprime :
∀ (Hs : (t : set I).pairwise (is_coprime on s)) (Hs1 : ∀ i ∈ t, s i ∣ z),
∏ x in t, s x ∣ z | finset.induction_on t (λ _ _, one_dvd z)
begin
intros a r har ih Hs Hs1,
rw finset.prod_insert har,
have aux1 : a ∈ (↑(insert a r) : set I) := finset.mem_insert_self a r,
refine (is_coprime.prod_right $ λ i hir, Hs aux1 (finset.mem_insert_of_mem hir)
$ by { rintro rfl, exact har hir }).mul_dvd
(Hs1 a au... | theorem | finset.prod_dvd_of_coprime | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"finset.coe_insert",
"finset.induction_on",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.prod_insert",
"ih",
"is_coprime",
"is_coprime.prod_right",
"one_dvd",
"pairwise",
"set.subset_insert"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fintype.prod_dvd_of_coprime [fintype I] (Hs : pairwise (is_coprime on s))
(Hs1 : ∀ i, s i ∣ z) : ∏ x, s x ∣ z | finset.prod_dvd_of_coprime (Hs.set_pairwise _) (λ i _, Hs1 i) | theorem | fintype.prod_dvd_of_coprime | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"finset.prod_dvd_of_coprime",
"fintype",
"is_coprime",
"pairwise"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_sum_eq_one_iff_pairwise_coprime [decidable_eq I] (h : t.nonempty) :
(∃ μ : I → R, ∑ i in t, μ i * ∏ j in t \ {i}, s j = 1) ↔ pairwise (is_coprime on λ i : t, s i) | begin
refine h.cons_induction _ _; clear' t h,
{ simp only [pairwise, sum_singleton, finset.sdiff_self, prod_empty, mul_one,
exists_apply_eq_apply, ne.def, true_iff],
rintro a ⟨i, hi⟩ ⟨j, hj⟩ h,
rw finset.mem_singleton at hi hj,
simpa [hi, hj] using h },
intros a t hat h ih,
rw pairwise_cons',... | lemma | exists_sum_eq_one_iff_pairwise_coprime | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"exists_apply_eq_apply",
"finset.mem_singleton",
"finset.sdiff_self",
"ih",
"is_coprime",
"is_coprime.prod_left",
"mul_assoc",
"mul_one",
"one_mul",
"pairwise",
"sdiff_sdiff_comm",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_sum_eq_one_iff_pairwise_coprime' [fintype I] [nonempty I] [decidable_eq I] :
(∃ μ : I → R, ∑ (i : I), μ i * ∏ j in {i}ᶜ, s j = 1) ↔ pairwise (is_coprime on s) | begin
convert exists_sum_eq_one_iff_pairwise_coprime finset.univ_nonempty using 1,
simp only [function.on_fun, pairwise_subtype_iff_pairwise_finset', coe_univ, set.pairwise_univ],
assumption
end | lemma | exists_sum_eq_one_iff_pairwise_coprime' | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"exists_sum_eq_one_iff_pairwise_coprime",
"finset.univ_nonempty",
"fintype",
"is_coprime",
"pairwise",
"set.pairwise_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pairwise_coprime_iff_coprime_prod [decidable_eq I] :
pairwise (is_coprime on λ i : t, s i) ↔ ∀ i ∈ t, is_coprime (s i) (∏ j in t \ {i}, (s j)) | begin
refine ⟨λ hp i hi, is_coprime.prod_right_iff.mpr (λ j hj, _), λ hp, _⟩,
{ rw [finset.mem_sdiff, finset.mem_singleton] at hj,
obtain ⟨hj, ji⟩ := hj,
exact @hp ⟨i, hi⟩ ⟨j, hj⟩ (λ h, ji (congr_arg coe h).symm) },
{ rintro ⟨i, hi⟩ ⟨j, hj⟩ h,
apply is_coprime.prod_right_iff.mp (hp i hi),
exact fi... | lemma | pairwise_coprime_iff_coprime_prod | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"finset.mem_sdiff",
"finset.mem_singleton",
"is_coprime",
"pairwise",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.pow_left (H : is_coprime x y) : is_coprime (x ^ m) y | by { rw [← finset.card_range m, ← finset.prod_const], exact is_coprime.prod_left (λ _ _, H) } | theorem | is_coprime.pow_left | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"finset.card_range",
"finset.prod_const",
"is_coprime",
"is_coprime.prod_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.pow_right (H : is_coprime x y) : is_coprime x (y ^ n) | by { rw [← finset.card_range n, ← finset.prod_const], exact is_coprime.prod_right (λ _ _, H) } | theorem | is_coprime.pow_right | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"finset.card_range",
"finset.prod_const",
"is_coprime",
"is_coprime.prod_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.pow (H : is_coprime x y) : is_coprime (x ^ m) (y ^ n) | H.pow_left.pow_right | theorem | is_coprime.pow | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.pow_left_iff (hm : 0 < m) : is_coprime (x ^ m) y ↔ is_coprime x y | begin
refine ⟨λ h, _, is_coprime.pow_left⟩,
rw [← finset.card_range m, ← finset.prod_const] at h,
exact h.of_prod_left 0 (finset.mem_range.mpr hm),
end | theorem | is_coprime.pow_left_iff | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"finset.card_range",
"finset.prod_const",
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.pow_right_iff (hm : 0 < m) : is_coprime x (y ^ m) ↔ is_coprime x y | is_coprime_comm.trans $ (is_coprime.pow_left_iff hm).trans $ is_coprime_comm | theorem | is_coprime.pow_right_iff | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"is_coprime",
"is_coprime.pow_left_iff",
"is_coprime_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime.pow_iff (hm : 0 < m) (hn : 0 < n) :
is_coprime (x ^ m) (y ^ n) ↔ is_coprime x y | (is_coprime.pow_left_iff hm).trans $ is_coprime.pow_right_iff hn | theorem | is_coprime.pow_iff | ring_theory.coprime | src/ring_theory/coprime/lemmas.lean | [
"algebra.big_operators.ring",
"data.fintype.basic",
"data.int.gcd",
"ring_theory.coprime.basic"
] | [
"is_coprime",
"is_coprime.pow_left_iff",
"is_coprime.pow_right_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_valuation_def (r : R) : ℤₘ₀ | if r = 0 then 0 else multiplicative.of_add
(-(associates.mk v.as_ideal).count (associates.mk (ideal.span {r} : ideal R)).factors : ℤ) | def | is_dedekind_domain.height_one_spectrum.int_valuation_def | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [
"associates.mk",
"ideal",
"ideal.span",
"multiplicative.of_add"
] | The additive `v`-adic valuation of `r ∈ R` is the exponent of `v` in the factorization of the
ideal `(r)`, if `r` is nonzero, or infinity, if `r = 0`. `int_valuation_def` is the corresponding
multiplicative valuation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int_valuation_def_if_pos {r : R} (hr : r = 0) : v.int_valuation_def r = 0 | if_pos hr | lemma | is_dedekind_domain.height_one_spectrum.int_valuation_def_if_pos | ring_theory.dedekind_domain | src/ring_theory/dedekind_domain/adic_valuation.lean | [
"ring_theory.dedekind_domain.ideal",
"ring_theory.valuation.extend_to_localization",
"ring_theory.valuation.valuation_subring",
"topology.algebra.valued_field",
"algebra.order.group.type_tags"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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