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 |
|---|---|---|---|---|---|---|---|---|---|---|
top_smul : (⊤ : ideal R) • N = N | le_antisymm smul_le_right $ λ r hri, one_smul R r ▸ smul_mem_smul mem_top hri | theorem | submodule.top_smul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_sup : I • (N ⊔ P) = I • N ⊔ I • P | map₂_sup_right _ _ _ _ | theorem | submodule.smul_sup | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_smul : (I ⊔ J) • N = I • N ⊔ J • N | map₂_sup_left _ _ _ _ | theorem | submodule.sup_smul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_assoc : (I • J) • N = I • (J • N) | le_antisymm (smul_le.2 $ λ rs hrsij t htn,
smul_induction_on hrsij
(λ r hr s hs,
(@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn))
(λ x y, (add_smul x y t).symm ▸ submodule.add_mem _))
(smul_le.2 $ λ r hr sn hsn,
suffices J • N ≤ submodule.comap (r • (linear_map.id : M... | theorem | submodule.smul_assoc | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"add_smul",
"linear_map.id",
"smul_assoc",
"smul_eq_mul",
"smul_smul",
"submodule.add_mem",
"submodule.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_inf_le (M₁ M₂ : submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ | le_inf (submodule.smul_mono_right inf_le_left) (submodule.smul_mono_right inf_le_right) | lemma | submodule.smul_inf_le | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"inf_le_left",
"inf_le_right",
"le_inf",
"submodule",
"submodule.smul_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_supr {ι : Sort*} {I : ideal R} {t : ι → submodule R M} :
I • supr t = ⨆ i, I • t i | map₂_supr_right _ _ _ | lemma | submodule.smul_supr | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"submodule",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_infi_le {ι : Sort*} {I : ideal R} {t : ι → submodule R M} :
I • infi t ≤ ⨅ i, I • t i | le_infi (λ i, smul_mono_right (infi_le _ _)) | lemma | submodule.smul_infi_le | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"infi",
"infi_le",
"le_infi",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_smul_span : (ideal.span S) • (span R T) =
span R (⋃ (s ∈ S) (t ∈ T), {s • t}) | (map₂_span_span _ _ _ _).trans $ congr_arg _ $ set.image2_eq_Union _ _ _ | theorem | submodule.span_smul_span | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.span",
"set.image2_eq_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_span_singleton_smul (r : R) (N : submodule R M) :
(ideal.span {r} : ideal R) • N = r • N | begin
have : span R (⋃ (t : M) (x : t ∈ N), {r • t}) = r • N,
{ convert span_eq _, exact (set.image_eq_Union _ (N : set M)).symm },
conv_lhs { rw [← span_eq N, span_smul_span] },
simpa
end | lemma | submodule.ideal_span_singleton_smul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ideal.span",
"set.image_eq_Union",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_of_span_top_of_smul_mem (M' : submodule R M)
(s : set R) (hs : ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' | begin
suffices : (⊤ : ideal R) • (span R ({x} : set M)) ≤ M',
{ rw top_smul at this, exact this (subset_span (set.mem_singleton x)) },
rw [← hs, span_smul_span, span_le],
simpa using H
end | lemma | submodule.mem_of_span_top_of_smul_mem | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ideal.span",
"set.mem_singleton",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_of_span_eq_top_of_smul_pow_mem (M' : submodule R M)
(s : set R) (hs : ideal.span s = ⊤) (x : M)
(H : ∀ r : s, ∃ (n : ℕ), (r ^ n : R) • x ∈ M') : x ∈ M' | begin
obtain ⟨s', hs₁, hs₂⟩ := (ideal.span_eq_top_iff_finite _).mp hs,
replace H : ∀ r : s', ∃ (n : ℕ), (r ^ n : R) • x ∈ M' := λ r, H ⟨_, hs₁ r.prop⟩,
choose n₁ n₂ using H,
let N := s'.attach.sup n₁,
have hs' := ideal.span_pow_eq_top (s' : set R) hs₂ N,
apply M'.mem_of_span_top_of_smul_mem _ hs',
rintro ... | lemma | submodule.mem_of_span_eq_top_of_smul_pow_mem | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset.le_sup",
"ideal.span",
"ideal.span_eq_top_iff_finite",
"ideal.span_pow_eq_top",
"pow_add",
"smul_smul",
"submodule",
"subtype.coe_mk",
"tsub_add_cancel_of_le"
] | Given `s`, a generating set of `R`, to check that an `x : M` falls in a
submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f | le_antisymm (map_le_iff_le_comap.2 $ smul_le.2 $ λ r hr n hn, show f (r • n) ∈ I • N.map f,
from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) $
smul_le.2 $ λ r hr n hn, let ⟨p, hp, hfp⟩ := mem_map.1 hn in
hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) | theorem | submodule.map_smul'' | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_smul_span {s : set M} {x : M} :
x ∈ I • submodule.span R s ↔ x ∈ submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : set M)) | by rw [← I.span_eq, submodule.span_smul_span, I.span_eq]; refl | lemma | submodule.mem_smul_span | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"submodule.span",
"submodule.span_smul_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
x ∈ I • span R (set.range f) ↔
∃ (a : ι →₀ R) (ha : ∀ i, a i ∈ I), a.sum (λ i c, c • f i) = x | begin
split, swap,
{ rintro ⟨a, ha, rfl⟩,
exact submodule.sum_mem _ (λ c _, smul_mem_smul (ha c) $ subset_span $ set.mem_range_self _) },
refine λ hx, span_induction (mem_smul_span.mp hx) _ _ _ _,
{ simp only [set.mem_Union, set.mem_range, set.mem_singleton_iff],
rintros x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩,
... | lemma | submodule.mem_ideal_smul_span_iff_exists_sum | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"add_smul",
"classical.dec_eq",
"finsupp.single_apply",
"finsupp.smul_sum",
"finsupp.sum_smul_index",
"set.mem_Union",
"set.mem_range",
"set.mem_range_self",
"set.mem_singleton_iff",
"set.range",
"submodule.sum_mem",
"zero_smul"
] | If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : set ι) (f : ι → M) (x : M) :
x ∈ I • span R (f '' s) ↔
∃ (a : s →₀ R) (ha : ∀ i, a i ∈ I), a.sum (λ i c, c • f i) = x | by rw [← submodule.mem_ideal_smul_span_iff_exists_sum, ← set.image_eq_range] | theorem | submodule.mem_ideal_smul_span_iff_exists_sum' | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"set.image_eq_range",
"submodule.mem_ideal_smul_span_iff_exists_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_smul_top_iff (N : submodule R M) (x : N) :
x ∈ I • (⊤ : submodule R N) ↔ (x : M) ∈ I • N | begin
change _ ↔ N.subtype x ∈ I • N,
have : submodule.map N.subtype (I • ⊤) = I • N,
{ rw [submodule.map_smul'', submodule.map_top, submodule.range_subtype] },
rw ← this,
convert (function.injective.mem_set_image N.injective_subtype).symm using 1,
refl,
end | lemma | submodule.mem_smul_top_iff | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"function.injective.mem_set_image",
"submodule",
"submodule.map",
"submodule.map_smul''",
"submodule.map_top",
"submodule.range_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : submodule R M') (I : ideal R) :
I • S.comap f ≤ (I • S).comap f | begin
refine (submodule.smul_le.mpr (λ r hr x hx, _)),
rw [submodule.mem_comap] at ⊢ hx,
rw f.map_smul,
exact submodule.smul_mem_smul hr hx
end | lemma | submodule.smul_comap_le_comap_smul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"submodule",
"submodule.mem_comap",
"submodule.smul_mem_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colon (N P : submodule R M) : ideal R | annihilator (P.map N.mkq) | def | submodule.colon | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"submodule"
] | `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N | mem_annihilator.trans ⟨λ H p hp, (quotient.mk_eq_zero N).1 (H (quotient.mk p) (mem_map_of_mem hp)),
λ H m ⟨p, hp, hpm⟩, hpm ▸ (N.mkq).map_smul r p ▸ (quotient.mk_eq_zero N).2 $ H p hp⟩ | theorem | submodule.mem_colon | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (linear_map.id : M →ₗ[R] M)) N | mem_colon | theorem | submodule.mem_colon' | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ | λ r hrnp, mem_colon.2 $ λ p₁ hp₁, hn $ mem_colon.1 hrnp p₁ $ hp hp₁ | theorem | submodule.colon_mono | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_colon_supr (ι₁ : Sort w) (f : ι₁ → submodule R M)
(ι₂ : Sort x) (g : ι₂ → submodule R M) :
(⨅ i, f i).colon (⨆ j, g j) = ⨅ i j, (f i).colon (g j) | le_antisymm (le_infi $ λ i, le_infi $ λ j, colon_mono (infi_le _ _) (le_supr _ _))
(λ r H, mem_colon'.2 $ supr_le $ λ j, map_le_iff_le_comap.1 $ le_infi $ λ i,
map_le_iff_le_comap.2 $ mem_colon'.1 $ have _ := ((mem_infi _).1 H i),
have _ := ((mem_infi _).1 this j), this) | theorem | submodule.infi_colon_supr | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"infi_le",
"le_infi",
"le_supr",
"submodule",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_colon_singleton {N : submodule R M} {x : M} {r : R} :
r ∈ N.colon (submodule.span R {x}) ↔ r • x ∈ N | calc r ∈ N.colon (submodule.span R {x}) ↔ ∀ (a : R), r • (a • x) ∈ N :
by simp [submodule.mem_colon, submodule.mem_span_singleton]
... ↔ r • x ∈ N :
by { simp_rw [smul_comm r]; exact set_like.forall_smul_mem_iff } | lemma | submodule.mem_colon_singleton | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"set_like.forall_smul_mem_iff",
"submodule",
"submodule.mem_colon",
"submodule.mem_span_singleton",
"submodule.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.ideal.mem_colon_singleton {I : ideal R} {x r : R} :
r ∈ I.colon (ideal.span {x}) ↔ r * x ∈ I | by simp [← ideal.submodule_span_eq, submodule.mem_colon_singleton, smul_eq_mul] | lemma | ideal.mem_colon_singleton | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ideal.span",
"ideal.submodule_span_eq",
"smul_eq_mul",
"submodule.mem_colon_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_sup {I J : ideal R} : I + J = I ⊔ J | rfl | lemma | ideal.add_eq_sup | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"add_eq_sup",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_eq_bot : (0 : ideal R) = ⊥ | rfl | lemma | ideal.zero_eq_bot | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_eq_sup {ι : Type*} (s : finset ι) (f : ι → ideal R) : s.sum f = s.sup f | rfl | lemma | ideal.sum_eq_sup | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_eq_top : (1 : ideal R) = ⊤ | by erw [submodule.one_eq_range, linear_map.range_id] | lemma | ideal.one_eq_top | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"linear_map.range_id",
"submodule.one_eq_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J | submodule.smul_mem_smul hr hs | theorem | ideal.mul_mem_mul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"submodule.smul_mem_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J | mul_comm r s ▸ mul_mem_mul hr hs | theorem | ideal.mul_mem_mul_rev | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n | submodule.pow_mem_pow _ hx _ | lemma | ideal.pow_mem_pow | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"submodule.pow_mem_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mem_prod {ι : Type*} {s : finset ι} {I : ι → ideal R} {x : ι → R} :
(∀ i ∈ s, x i ∈ I i) → ∏ i in s, x i ∈ ∏ i in s, I i | begin
classical,
apply finset.induction_on s,
{ intro _, rw [finset.prod_empty, finset.prod_empty, one_eq_top], exact submodule.mem_top },
{ intros a s ha IH h,
rw [finset.prod_insert ha, finset.prod_insert ha],
exact mul_mem_mul (h a $ finset.mem_insert_self a s)
(IH $ λ i hi, h i $ finset.mem_in... | lemma | ideal.prod_mem_prod | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset",
"finset.induction_on",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.prod_empty",
"finset.prod_insert",
"ideal",
"submodule.mem_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le : I * J ≤ K ↔ ∀ (r ∈ I) (s ∈ J), r * s ∈ K | submodule.smul_le | theorem | ideal.mul_le | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"submodule.smul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_left : I * J ≤ J | ideal.mul_le.2 (λ r hr s, J.mul_mem_left _) | lemma | ideal.mul_le_left | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_right : I * J ≤ I | ideal.mul_le.2 (λ r hr s hs, I.mul_mem_right _ hr) | lemma | ideal.mul_le_right | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_mul_right_self : I ⊔ (I * J) = I | sup_eq_left.2 ideal.mul_le_right | lemma | ideal.sup_mul_right_self | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.mul_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_mul_left_self : I ⊔ (J * I) = I | sup_eq_left.2 ideal.mul_le_left | lemma | ideal.sup_mul_left_self | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.mul_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_self_sup : (I * J) ⊔ I = I | sup_eq_right.2 ideal.mul_le_right | lemma | ideal.mul_right_self_sup | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.mul_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_self_sup : (J * I) ⊔ I = I | sup_eq_right.2 ideal.mul_le_left | lemma | ideal.mul_left_self_sup | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.mul_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_comm : I * J = J * I | le_antisymm (mul_le.2 $ λ r hrI s hsJ, mul_mem_mul_rev hsJ hrI)
(mul_le.2 $ λ r hrJ s hsI, mul_mem_mul_rev hsI hrJ) | theorem | ideal.mul_comm | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_assoc : (I * J) * K = I * (J * K) | submodule.smul_assoc I J K | theorem | ideal.mul_assoc | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"mul_assoc",
"submodule.smul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_mul_span (S T : set R) : span S * span T =
span ⋃ (s ∈ S) (t ∈ T), {s * t} | submodule.span_smul_span S T | theorem | ideal.span_mul_span | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"submodule.span_smul_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_mul_span' (S T : set R) : span S * span T = span (S*T) | by { unfold span, rw submodule.span_mul_span, } | lemma | ideal.span_mul_span' | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"submodule.span_mul_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_span_singleton (r s : R) :
span {r} * span {s} = (span {r * s} : ideal R) | by { unfold span, rw [submodule.span_mul_span, set.singleton_mul_singleton], } | lemma | ideal.span_singleton_mul_span_singleton | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"set.singleton_mul_singleton",
"submodule.span_mul_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_pow (s : R) (n : ℕ):
span {s} ^ n = (span {s ^ n} : ideal R) | begin
induction n with n ih, { simp [set.singleton_one], },
simp only [pow_succ, ih, span_singleton_mul_span_singleton],
end | lemma | ideal.span_singleton_pow | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ih",
"pow_succ",
"set.singleton_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mul_span_singleton {x y : R} {I : ideal R} :
x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x | submodule.mem_smul_span_singleton | lemma | ideal.mem_mul_span_singleton | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"submodule.mem_smul_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_singleton_mul {x y : R} {I : ideal R} :
x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x | by simp only [mul_comm, mem_mul_span_singleton] | lemma | ideal.mem_span_singleton_mul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_span_singleton_mul_iff {x : R} {I J : ideal R} :
I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI | show (∀ {zI} (hzI : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI,
by simp only [mem_span_singleton_mul] | lemma | ideal.le_span_singleton_mul_iff | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_le_iff {x : R} {I J : ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J | begin
simp only [mul_le, mem_span_singleton_mul, mem_span_singleton],
split,
{ intros h zI hzI,
exact h x (dvd_refl x) zI hzI },
{ rintros h _ ⟨z, rfl⟩ zI hzI,
rw [mul_comm x z, mul_assoc],
exact J.mul_mem_left _ (h zI hzI) },
end | lemma | ideal.span_singleton_mul_le_iff | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"dvd_refl",
"ideal",
"mul_assoc",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_le_span_singleton_mul {x y : R} {I J : ideal R} :
span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ | by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] | lemma | ideal.span_singleton_mul_le_span_singleton_mul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_right_mono [is_domain R] {x : R} (hx : x ≠ 0) :
span {x} * I ≤ span {x} * J ↔ I ≤ J | by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_prop,
exists_eq_right', set_like.le_def] | lemma | ideal.span_singleton_mul_right_mono | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"exists_eq_right'",
"exists_prop",
"is_domain",
"mul_right_inj'",
"set_like.le_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_left_mono [is_domain R] {x : R} (hx : x ≠ 0) :
I * span {x} ≤ J * span {x} ↔ I ≤ J | by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx | lemma | ideal.span_singleton_mul_left_mono | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"is_domain",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_right_inj [is_domain R] {x : R} (hx : x ≠ 0) :
span {x} * I = span {x} * J ↔ I = J | by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] | lemma | ideal.span_singleton_mul_right_inj | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_left_inj [is_domain R] {x : R} (hx : x ≠ 0) :
I * span {x} = J * span {x} ↔ I = J | by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] | lemma | ideal.span_singleton_mul_left_inj | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_right_injective [is_domain R] {x : R} (hx : x ≠ 0) :
function.injective ((*) (span {x} : ideal R)) | λ _ _, (span_singleton_mul_right_inj hx).mp | lemma | ideal.span_singleton_mul_right_injective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_left_injective [is_domain R] {x : R} (hx : x ≠ 0) :
function.injective (λ I : ideal R, I * span {x}) | λ _ _, (span_singleton_mul_left_inj hx).mp | lemma | ideal.span_singleton_mul_left_injective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_span_singleton_mul {x : R} (I J : ideal R) :
I = span {x} * J ↔ ((∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ (∀ z ∈ J, x * z ∈ I)) | by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] | lemma | ideal.eq_span_singleton_mul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : ideal R) :
span {x} * I = span {y} * J ↔
((∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧
(∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ)) | by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] | lemma | ideal.span_singleton_mul_eq_span_singleton_mul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_span {ι : Type*} (s : finset ι) (I : ι → set R) :
(∏ i in s, ideal.span (I i)) = ideal.span (∏ i in s, I i) | submodule.prod_span s I | lemma | ideal.prod_span | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset",
"ideal.span",
"submodule.prod_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_span_singleton {ι : Type*} (s : finset ι) (I : ι → R) :
(∏ i in s, ideal.span ({I i} : set R)) = ideal.span {∏ i in s, I i} | submodule.prod_span_singleton s I | lemma | ideal.prod_span_singleton | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset",
"ideal.span",
"submodule.prod_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset_prod_span_singleton (m : multiset R) :
(m.map (λ x, ideal.span {x})).prod = ideal.span ({multiset.prod m} : set R) | multiset.induction_on m (by simp)
(λ a m ih, by simp only [multiset.map_cons, multiset.prod_cons, ih,
← ideal.span_singleton_mul_span_singleton]) | lemma | ideal.multiset_prod_span_singleton | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.span",
"ideal.span_singleton_mul_span_singleton",
"ih",
"multiset",
"multiset.induction_on",
"multiset.map_cons",
"multiset.prod",
"multiset.prod_cons"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_inf_span_singleton {ι : Type*} (s : finset ι) (I : ι → R)
(hI : set.pairwise ↑s (is_coprime on I)) :
(s.inf $ λ i, ideal.span ({I i} : set R)) = ideal.span {∏ i in s, I i} | begin
ext x,
simp only [submodule.mem_finset_inf, ideal.mem_span_singleton],
exact ⟨finset.prod_dvd_of_coprime hI,
λ h i hi, (finset.dvd_prod_of_mem _ hi).trans h⟩
end | lemma | ideal.finset_inf_span_singleton | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset",
"finset.dvd_prod_of_mem",
"ideal.mem_span_singleton",
"ideal.span",
"is_coprime",
"set.pairwise",
"submodule.mem_finset_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_span_singleton {ι : Type*} [fintype ι] (I : ι → R)
(hI : ∀ i j (hij : i ≠ j), is_coprime (I i) (I j)):
(⨅ i, ideal.span ({I i} : set R)) = ideal.span {∏ i, I i} | begin
rw [← finset.inf_univ_eq_infi, finset_inf_span_singleton],
rwa [finset.coe_univ, set.pairwise_univ]
end | lemma | ideal.infi_span_singleton | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset.coe_univ",
"finset.inf_univ_eq_infi",
"fintype",
"ideal.span",
"is_coprime",
"set.pairwise_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_eq_top_iff_is_coprime {R : Type*} [comm_semiring R] (x y : R) :
span ({x} : set R) ⊔ span {y} = ⊤ ↔ is_coprime x y | begin
rw [eq_top_iff_one, submodule.mem_sup],
split,
{ rintro ⟨u, hu, v, hv, h1⟩,
rw mem_span_singleton' at hu hv,
rw [← hu.some_spec, ← hv.some_spec] at h1,
exact ⟨_, _, h1⟩ },
{ exact λ ⟨u, v, h1⟩,
⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ },
end | lemma | ideal.sup_eq_top_iff_is_coprime | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"comm_semiring",
"is_coprime",
"submodule.mem_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_inf : I * J ≤ I ⊓ J | mul_le.2 $ λ r hri s hsj, ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ | theorem | ideal.mul_le_inf | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset_prod_le_inf {s : multiset (ideal R)} :
s.prod ≤ s.inf | begin
classical, refine s.induction_on _ _,
{ rw [multiset.inf_zero], exact le_top },
intros a s ih,
rw [multiset.prod_cons, multiset.inf_cons],
exact le_trans mul_le_inf (inf_le_inf le_rfl ih)
end | theorem | ideal.multiset_prod_le_inf | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ih",
"inf_le_inf",
"le_rfl",
"le_top",
"multiset",
"multiset.inf_cons",
"multiset.inf_zero",
"multiset.prod_cons"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_le_inf {s : finset ι} {f : ι → ideal R} : s.prod f ≤ s.inf f | multiset_prod_le_inf | theorem | ideal.prod_le_inf | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J | le_antisymm mul_le_inf $ λ r ⟨hri, hrj⟩,
let ⟨s, hsi, t, htj, hst⟩ := submodule.mem_sup.1 ((eq_top_iff_one _).1 h) in
mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj)
(mul_mem_mul hri htj) | theorem | ideal.mul_eq_inf_of_coprime | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.add_mem",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ (J * K) = I ⊔ K | le_antisymm (sup_le_sup_left mul_le_left _) $ λ i hi,
begin
rw eq_top_iff_one at h, rw submodule.mem_sup at h hi ⊢,
obtain ⟨i1, hi1, j, hj, h⟩ := h, obtain ⟨i', hi', k, hk, hi⟩ := hi,
refine ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩,
rw [add_assoc, ← add_mul, h, one_mul, hi]
end | lemma | ideal.sup_mul_eq_of_coprime_left | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"one_mul",
"submodule.mem_sup",
"sup_le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ (J * K) = I ⊔ J | by { rw mul_comm, exact sup_mul_eq_of_coprime_left h } | lemma | ideal.sup_mul_eq_of_coprime_right | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : (I * K) ⊔ J = K ⊔ J | by { rw sup_comm at h, rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] } | lemma | ideal.mul_sup_eq_of_coprime_left | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : (I * K) ⊔ J = I ⊔ J | by { rw sup_comm at h, rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] } | lemma | ideal.mul_sup_eq_of_coprime_right | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_prod_eq_top {s : finset ι} {J : ι → ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) :
I ⊔ ∏ i in s, J i = ⊤ | finset.prod_induction _ (λ J, I ⊔ J = ⊤) (λ J K hJ hK, (sup_mul_eq_of_coprime_left hJ).trans hK)
(by rw [one_eq_top, sup_top_eq]) h | lemma | ideal.sup_prod_eq_top | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset",
"finset.prod_induction",
"ideal",
"sup_top_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_infi_eq_top {s : finset ι} {J : ι → ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) :
I ⊔ (⨅ i ∈ s, J i) = ⊤ | eq_top_iff.mpr $ le_of_eq_of_le (sup_prod_eq_top h).symm $ sup_le_sup_left
(le_of_le_of_eq prod_le_inf $ finset.inf_eq_infi _ _) _ | lemma | ideal.sup_infi_eq_top | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset",
"finset.inf_eq_infi",
"ideal",
"le_of_eq_of_le",
"le_of_le_of_eq",
"sup_le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_sup_eq_top {s : finset ι} {J : ι → ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) :
(∏ i in s, J i) ⊔ I = ⊤ | sup_comm.trans (sup_prod_eq_top $ λ i hi, sup_comm.trans $ h i hi) | lemma | ideal.prod_sup_eq_top | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_sup_eq_top {s : finset ι} {J : ι → ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) :
(⨅ i ∈ s, J i) ⊔ I = ⊤ | sup_comm.trans (sup_infi_eq_top $ λ i hi, sup_comm.trans $ h i hi) | lemma | ideal.infi_sup_eq_top | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ (J ^ n) = ⊤ | by { rw [← finset.card_range n, ← finset.prod_const], exact sup_prod_eq_top (λ _ _, h) } | lemma | ideal.sup_pow_eq_top | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset.card_range",
"finset.prod_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : (I ^ n) ⊔ J = ⊤ | by { rw [← finset.card_range n, ← finset.prod_const], exact prod_sup_eq_top (λ _ _, h) } | lemma | ideal.pow_sup_eq_top | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset.card_range",
"finset.prod_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : (I ^ m) ⊔ (J ^ n) = ⊤ | sup_pow_eq_top (pow_sup_eq_top h) | lemma | ideal.pow_sup_pow_eq_top | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_bot : I * ⊥ = ⊥ | submodule.smul_bot I | theorem | ideal.mul_bot | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"submodule.smul_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_mul : ⊥ * I = ⊥ | submodule.bot_smul I | theorem | ideal.bot_mul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"submodule.bot_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_top : I * ⊤ = I | ideal.mul_comm ⊤ I ▸ submodule.top_smul I | theorem | ideal.mul_top | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.mul_comm",
"submodule.top_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_mul : ⊤ * I = I | submodule.top_smul I | theorem | ideal.top_mul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"submodule.top_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L | submodule.smul_mono hik hjl | theorem | ideal.mul_mono | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"submodule.smul_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mono_left (h : I ≤ J) : I * K ≤ J * K | submodule.smul_mono_left h | theorem | ideal.mul_mono_left | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"submodule.smul_mono_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mono_right (h : J ≤ K) : I * J ≤ I * K | submodule.smul_mono_right h | theorem | ideal.mul_mono_right | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"submodule.smul_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_sup : I * (J ⊔ K) = I * J ⊔ I * K | submodule.smul_sup I J K | theorem | ideal.mul_sup | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"mul_sup",
"submodule.smul_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_mul : (I ⊔ J) * K = I * K ⊔ J * K | submodule.sup_smul I J K | theorem | ideal.sup_mul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"submodule.sup_smul",
"sup_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_pow {m n : ℕ} (h : m ≤ n) :
I^n ≤ I^m | begin
cases nat.exists_eq_add_of_le h with k hk,
rw [hk, pow_add],
exact le_trans (mul_le_inf) (inf_le_left)
end | lemma | ideal.pow_le_pow | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"inf_le_left",
"nat.exists_eq_add_of_le",
"pow_add",
"pow_le_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_self {n : ℕ} (hn : n ≠ 0) : I^n ≤ I | calc I^n ≤ I ^ 1 : pow_le_pow (nat.pos_of_ne_zero hn)
... = I : pow_one _ | lemma | ideal.pow_le_self | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"pow_le_pow",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mono {I J : ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n | begin
induction n,
{ rw [pow_zero, pow_zero], exact rfl.le },
{ rw [pow_succ, pow_succ], exact ideal.mul_mono e n_ih }
end | lemma | ideal.pow_mono | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ideal.mul_mono",
"pow_mono",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_bot {R : Type*} [comm_semiring R] [no_zero_divisors R] {I J : ideal R} :
I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ | ⟨λ hij, or_iff_not_imp_left.mpr (λ I_ne_bot, J.eq_bot_iff.mpr (λ j hj,
let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot in
or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0)),
λ h, by cases h; rw [← ideal.mul_bot, h, ideal.mul_comm]⟩ | lemma | ideal.mul_eq_bot | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"comm_semiring",
"ideal",
"ideal.mul_bot",
"ideal.mul_comm",
"no_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_bot {R : Type*} [comm_ring R] [is_domain R]
{s : multiset (ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ | prod_zero_iff_exists_zero | lemma | ideal.prod_eq_bot | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"comm_ring",
"ideal",
"is_domain",
"multiset",
"prod_zero_iff_exists_zero"
] | A product of ideals in an integral domain is zero if and only if one of the terms is zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_pair_mul_span_pair (w x y z : R) :
(span {w, x} : ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} | by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] | lemma | ideal.span_pair_mul_span_pair | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"mul_sup",
"sup_assoc",
"sup_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical (I : ideal R) : ideal R | { carrier := { r | ∃ n : ℕ, r ^ n ∈ I },
zero_mem' := ⟨1, (pow_one (0:R)).symm ▸ I.zero_mem⟩,
add_mem' := λ x y ⟨m, hxmi⟩ ⟨n, hyni⟩, ⟨m + n,
(add_pow x y (m + n)).symm ▸ I.sum_mem $
show ∀ c ∈ finset.range (nat.succ (m + n)),
x ^ c * y ^ (m + n - c) * (nat.choose (m + n) c) ∈ I,
from λ c hc, or.ca... | def | ideal.radical | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"add_pow",
"add_tsub_assoc_of_le",
"add_tsub_cancel_of_le",
"finset.range",
"ideal",
"mul_pow",
"nat.choose",
"pow_add",
"pow_one"
] | The radical of an ideal `I` consists of the elements `r` such that `r^n ∈ I` for some `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_radical (I : ideal R) : Prop | I.radical ≤ I | def | ideal.is_radical | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"is_radical"
] | An ideal is radical if it contains its radical. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_radical : I ≤ radical I | λ r hri, ⟨1, (pow_one r).symm ▸ hri⟩ | theorem | ideal.le_radical | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical_eq_iff : I.radical = I ↔ I.is_radical | by rw [le_antisymm_iff, and_iff_left le_radical, is_radical] | theorem | ideal.radical_eq_iff | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"is_radical"
] | An ideal is radical iff it is equal to its radical. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
radical_top : (radical ⊤ : ideal R) = ⊤ | (eq_top_iff_one _).2 ⟨0, submodule.mem_top⟩ | theorem | ideal.radical_top | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical_mono (H : I ≤ J) : radical I ≤ radical J | λ r ⟨n, hrni⟩, ⟨n, H hrni⟩ | theorem | ideal.radical_mono | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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