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
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finite_type {f : A →ₐ[R] B} (hf : f.finite) : finite_type f | ring_hom.finite.finite_type hf | lemma | alg_hom.finite.finite_type | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"ring_hom.finite.finite_type"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : finite_type (alg_hom.id R A) | ring_hom.finite_type.id A | lemma | alg_hom.finite_type.id | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"alg_hom.id",
"ring_hom.finite_type.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {g : B →ₐ[R] C} {f : A →ₐ[R] B} (hg : g.finite_type) (hf : f.finite_type) :
(g.comp f).finite_type | ring_hom.finite_type.comp hg hf | lemma | alg_hom.finite_type.comp | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"ring_hom.finite_type.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_surjective {f : A →ₐ[R] B} {g : B →ₐ[R] C} (hf : f.finite_type) (hg : surjective g) :
(g.comp f).finite_type | ring_hom.finite_type.comp_surjective hf hg | lemma | alg_hom.finite_type.comp_surjective | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"ring_hom.finite_type.comp_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_surjective (f : A →ₐ[R] B) (hf : surjective f) : f.finite_type | ring_hom.finite_type.of_surjective f hf | lemma | alg_hom.finite_type.of_surjective | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"ring_hom.finite_type.of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_comp_finite_type {f : A →ₐ[R] B} {g : B →ₐ[R] C} (h : (g.comp f).finite_type) :
g.finite_type | ring_hom.finite_type.of_comp_finite_type h | lemma | alg_hom.finite_type.of_comp_finite_type | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"ring_hom.finite_type.of_comp_finite_type"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_adjoin_support (f : add_monoid_algebra R M) : f ∈ adjoin R (of' R M '' f.support) | begin
suffices : span R (of' R M '' f.support) ≤ (adjoin R (of' R M '' f.support)).to_submodule,
{ exact this (mem_span_support f) },
rw submodule.span_le,
exact subset_adjoin
end | lemma | add_monoid_algebra.mem_adjoin_support | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"add_monoid_algebra",
"submodule.span_le"
] | An element of `add_monoid_algebra R M` is in the subalgebra generated by its support. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
support_gen_of_gen {S : set (add_monoid_algebra R M)} (hS : algebra.adjoin R S = ⊤) :
algebra.adjoin R (⋃ f ∈ S, (of' R M '' (f.support : set M))) = ⊤ | begin
refine le_antisymm le_top _,
rw [← hS, adjoin_le_iff],
intros f hf,
have hincl : of' R M '' f.support ⊆
⋃ (g : add_monoid_algebra R M) (H : g ∈ S), of' R M '' g.support,
{ intros s hs,
exact set.mem_Union₂.2 ⟨f, ⟨hf, hs⟩⟩ },
exact adjoin_mono hincl (mem_adjoin_support f)
end | lemma | add_monoid_algebra.support_gen_of_gen | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"add_monoid_algebra",
"algebra.adjoin",
"le_top"
] | If a set `S` generates, as algebra, `add_monoid_algebra R M`, then the set of supports of
elements of `S` generates `add_monoid_algebra R M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
support_gen_of_gen' {S : set (add_monoid_algebra R M)} (hS : algebra.adjoin R S = ⊤) :
algebra.adjoin R (of' R M '' (⋃ f ∈ S, (f.support : set M))) = ⊤ | begin
suffices : of' R M '' (⋃ f ∈ S, (f.support : set M)) = ⋃ f ∈ S, (of' R M '' (f.support : set M)),
{ rw this,
exact support_gen_of_gen hS },
simp only [set.image_Union]
end | lemma | add_monoid_algebra.support_gen_of_gen' | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"add_monoid_algebra",
"algebra.adjoin",
"set.image_Union"
] | If a set `S` generates, as algebra, `add_monoid_algebra R M`, then the image of the union of
the supports of elements of `S` generates `add_monoid_algebra R M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_finset_adjoin_eq_top [h : finite_type R (add_monoid_algebra R M)] :
∃ G : finset M, algebra.adjoin R (of' R M '' G) = ⊤ | begin
unfreezingI { obtain ⟨S, hS⟩ := h },
letI : decidable_eq M := classical.dec_eq M,
use finset.bUnion S (λ f, f.support),
have : (finset.bUnion S (λ f, f.support) : set M) = ⋃ f ∈ S, (f.support : set M),
{ simp only [finset.set_bUnion_coe, finset.coe_bUnion] },
rw [this],
exact support_gen_of_gen' hS
... | lemma | add_monoid_algebra.exists_finset_adjoin_eq_top | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"add_monoid_algebra",
"algebra.adjoin",
"classical.dec_eq",
"finset",
"finset.bUnion",
"finset.coe_bUnion",
"finset.set_bUnion_coe"
] | If `add_monoid_algebra R M` is of finite type, there there is a `G : finset M` such that its
image generates, as algera, `add_monoid_algebra R M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of'_mem_span [nontrivial R] {m : M} {S : set M} :
of' R M m ∈ span R (of' R M '' S) ↔ m ∈ S | begin
refine ⟨λ h, _, λ h, submodule.subset_span $ set.mem_image_of_mem (of R M) h⟩,
rw [of', ← finsupp.supported_eq_span_single, finsupp.mem_supported,
finsupp.support_single_ne_zero _ (one_ne_zero' R)] at h,
simpa using h
end | lemma | add_monoid_algebra.of'_mem_span | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"finsupp.mem_supported",
"finsupp.support_single_ne_zero",
"finsupp.supported_eq_span_single",
"nontrivial",
"one_ne_zero'",
"set.mem_image_of_mem",
"submodule.subset_span"
] | The image of an element `m : M` in `add_monoid_algebra R M` belongs the submodule generated by
`S : set M` if and only if `m ∈ S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_closure_of_mem_span_closure [nontrivial R] {m : M} {S : set M}
(h : of' R M m ∈ span R (submonoid.closure (of' R M '' S) : set (add_monoid_algebra R M))) :
m ∈ closure S | begin
suffices : multiplicative.of_add m ∈ submonoid.closure (multiplicative.to_add ⁻¹' S),
{ simpa [← to_submonoid_closure] },
let S' := @submonoid.closure M multiplicative.mul_one_class S,
have h' : submonoid.map (of R M) S' = submonoid.closure ((λ (x : M), (of R M) x) '' S) :=
monoid_hom.map_mclosure _ _... | lemma | add_monoid_algebra.mem_closure_of_mem_span_closure | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"add_monoid_algebra",
"closure",
"monoid_hom.map_mclosure",
"multiplicative.of_add",
"multiplicative.to_add",
"nontrivial",
"set.image_congr'",
"submonoid.closure",
"submonoid.map"
] | If the image of an element `m : M` in `add_monoid_algebra R M` belongs the submodule generated by
the closure of some `S : set M` then `m ∈ closure S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mv_polynomial_aeval_of_surjective_of_closure [comm_semiring R] {S : set M}
(hS : closure S = ⊤) : function.surjective (mv_polynomial.aeval
(λ (s : S), of' R M ↑s) : mv_polynomial S R → add_monoid_algebra R M) | begin
refine λ f, induction_on f (λ m, _) _ _,
{ have : m ∈ closure S := hS.symm ▸ mem_top _,
refine closure_induction this (λ m hm, _) _ _,
{ exact ⟨mv_polynomial.X ⟨m, hm⟩, mv_polynomial.aeval_X _ _⟩ },
{ exact ⟨1, alg_hom.map_one _⟩ },
{ rintro m₁ m₂ ⟨P₁, hP₁⟩ ⟨P₂, hP₂⟩,
exact ⟨P₁ * P₂, by ... | lemma | add_monoid_algebra.mv_polynomial_aeval_of_surjective_of_closure | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"add_monoid_algebra",
"alg_hom.map_add",
"alg_hom.map_mul",
"alg_hom.map_one",
"alg_hom.map_smul",
"closure",
"comm_semiring",
"mv_polynomial",
"mv_polynomial.aeval",
"mv_polynomial.aeval_X",
"one_mul"
] | If a set `S` generates an additive monoid `M`, then the image of `M` generates, as algebra,
`add_monoid_algebra R M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_type_of_fg [comm_ring R] [h : add_monoid.fg M] :
finite_type R (add_monoid_algebra R M) | begin
obtain ⟨S, hS⟩ := h.out,
exact (finite_type.mv_polynomial R (S : set M)).of_surjective (mv_polynomial.aeval
(λ (s : (S : set M)), of' R M ↑s)) (mv_polynomial_aeval_of_surjective_of_closure hS)
end | instance | add_monoid_algebra.finite_type_of_fg | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"add_monoid.fg",
"add_monoid_algebra",
"comm_ring",
"mv_polynomial.aeval"
] | If an additive monoid `M` is finitely generated then `add_monoid_algebra R M` is of finite
type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_type_iff_fg [comm_ring R] [nontrivial R] :
finite_type R (add_monoid_algebra R M) ↔ add_monoid.fg M | begin
refine ⟨λ h, _, λ h, @add_monoid_algebra.finite_type_of_fg _ _ _ _ h⟩,
obtain ⟨S, hS⟩ := @exists_finset_adjoin_eq_top R M _ _ h,
refine add_monoid.fg_def.2 ⟨S, (eq_top_iff' _).2 (λ m, _)⟩,
have hm : of' R M m ∈ (adjoin R (of' R M '' ↑S)).to_submodule,
{ simp only [hS, top_to_submodule, submodule.mem_top... | lemma | add_monoid_algebra.finite_type_iff_fg | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"add_monoid.fg",
"add_monoid_algebra",
"add_monoid_algebra.finite_type_of_fg",
"comm_ring",
"nontrivial",
"submodule.mem_top"
] | An additive monoid `M` is finitely generated if and only if `add_monoid_algebra R M` is of
finite type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fg_of_finite_type [comm_ring R] [nontrivial R] [h : finite_type R (add_monoid_algebra R M)] :
add_monoid.fg M | finite_type_iff_fg.1 h | lemma | add_monoid_algebra.fg_of_finite_type | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"add_monoid.fg",
"add_monoid_algebra",
"comm_ring",
"nontrivial"
] | If `add_monoid_algebra R M` is of finite type then `M` is finitely generated. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_type_iff_group_fg {G : Type*} [add_comm_group G] [comm_ring R] [nontrivial R] :
finite_type R (add_monoid_algebra R G) ↔ add_group.fg G | by simpa [add_group.fg_iff_add_monoid.fg] using finite_type_iff_fg | lemma | add_monoid_algebra.finite_type_iff_group_fg | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"add_comm_group",
"add_group.fg",
"add_monoid_algebra",
"comm_ring",
"nontrivial"
] | An additive group `G` is finitely generated if and only if `add_monoid_algebra R G` is of
finite type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_adjoin_support (f : monoid_algebra R M) : f ∈ adjoin R (of R M '' f.support) | begin
suffices : span R (of R M '' f.support) ≤ (adjoin R (of R M '' f.support)).to_submodule,
{ exact this (mem_span_support f) },
rw submodule.span_le,
exact subset_adjoin
end | lemma | monoid_algebra.mem_adjoin_support | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"monoid_algebra",
"submodule.span_le"
] | An element of `monoid_algebra R M` is in the subalgebra generated by its support. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
support_gen_of_gen {S : set (monoid_algebra R M)} (hS : algebra.adjoin R S = ⊤) :
algebra.adjoin R (⋃ f ∈ S, (of R M '' (f.support : set M))) = ⊤ | begin
refine le_antisymm le_top _,
rw [← hS, adjoin_le_iff],
intros f hf,
have hincl : (of R M) '' f.support ⊆
⋃ (g : monoid_algebra R M) (H : g ∈ S), of R M '' g.support,
{ intros s hs,
exact set.mem_Union₂.2 ⟨f, ⟨hf, hs⟩⟩ },
exact adjoin_mono hincl (mem_adjoin_support f)
end | lemma | monoid_algebra.support_gen_of_gen | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"algebra.adjoin",
"le_top",
"monoid_algebra"
] | If a set `S` generates, as algebra, `monoid_algebra R M`, then the set of supports of elements
of `S` generates `monoid_algebra R M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
support_gen_of_gen' {S : set (monoid_algebra R M)} (hS : algebra.adjoin R S = ⊤) :
algebra.adjoin R (of R M '' (⋃ f ∈ S, (f.support : set M))) = ⊤ | begin
suffices : of R M '' (⋃ f ∈ S, (f.support : set M)) = ⋃ f ∈ S, (of R M '' (f.support : set M)),
{ rw this,
exact support_gen_of_gen hS },
simp only [set.image_Union]
end | lemma | monoid_algebra.support_gen_of_gen' | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"algebra.adjoin",
"monoid_algebra",
"set.image_Union"
] | If a set `S` generates, as algebra, `monoid_algebra R M`, then the image of the union of the
supports of elements of `S` generates `monoid_algebra R M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_finset_adjoin_eq_top [h :finite_type R (monoid_algebra R M)] :
∃ G : finset M, algebra.adjoin R (of R M '' G) = ⊤ | begin
unfreezingI { obtain ⟨S, hS⟩ := h },
letI : decidable_eq M := classical.dec_eq M,
use finset.bUnion S (λ f, f.support),
have : (finset.bUnion S (λ f, f.support) : set M) = ⋃ f ∈ S, (f.support : set M),
{ simp only [finset.set_bUnion_coe, finset.coe_bUnion] },
rw [this],
exact support_gen_of_gen' hS
... | lemma | monoid_algebra.exists_finset_adjoin_eq_top | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"algebra.adjoin",
"classical.dec_eq",
"finset",
"finset.bUnion",
"finset.coe_bUnion",
"finset.set_bUnion_coe",
"monoid_algebra"
] | If `monoid_algebra R M` is of finite type, there there is a `G : finset M` such that its image
generates, as algera, `monoid_algebra R M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_mem_span_of_iff [nontrivial R] {m : M} {S : set M} :
of R M m ∈ span R (of R M '' S) ↔ m ∈ S | begin
refine ⟨λ h, _, λ h, submodule.subset_span $ set.mem_image_of_mem (of R M) h⟩,
rw [of, monoid_hom.coe_mk, ← finsupp.supported_eq_span_single, finsupp.mem_supported,
finsupp.support_single_ne_zero _ (one_ne_zero' R)] at h,
simpa using h
end | lemma | monoid_algebra.of_mem_span_of_iff | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"finsupp.mem_supported",
"finsupp.support_single_ne_zero",
"finsupp.supported_eq_span_single",
"monoid_hom.coe_mk",
"nontrivial",
"one_ne_zero'",
"set.mem_image_of_mem",
"submodule.subset_span"
] | The image of an element `m : M` in `monoid_algebra R M` belongs the submodule generated by
`S : set M` if and only if `m ∈ S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_closure_of_mem_span_closure [nontrivial R] {m : M} {S : set M}
(h : of R M m ∈ span R (submonoid.closure (of R M '' S) : set (monoid_algebra R M))) :
m ∈ closure S | begin
rw ← monoid_hom.map_mclosure at h,
simpa using of_mem_span_of_iff.1 h
end | lemma | monoid_algebra.mem_closure_of_mem_span_closure | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"closure",
"monoid_algebra",
"monoid_hom.map_mclosure",
"nontrivial",
"submonoid.closure"
] | If the image of an element `m : M` in `monoid_algebra R M` belongs the submodule generated by the
closure of some `S : set M` then `m ∈ closure S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mv_polynomial_aeval_of_surjective_of_closure [comm_semiring R] {S : set M}
(hS : closure S = ⊤) : function.surjective (mv_polynomial.aeval
(λ (s : S), of R M ↑s) : mv_polynomial S R → monoid_algebra R M) | begin
refine λ f, induction_on f (λ m, _) _ _,
{ have : m ∈ closure S := hS.symm ▸ mem_top _,
refine closure_induction this (λ m hm, _) _ _,
{ exact ⟨mv_polynomial.X ⟨m, hm⟩, mv_polynomial.aeval_X _ _⟩ },
{ exact ⟨1, alg_hom.map_one _⟩ },
{ rintro m₁ m₂ ⟨P₁, hP₁⟩ ⟨P₂, hP₂⟩,
exact ⟨P₁ * P₂, by ... | lemma | monoid_algebra.mv_polynomial_aeval_of_surjective_of_closure | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"alg_hom.map_add",
"alg_hom.map_mul",
"alg_hom.map_one",
"alg_hom.map_smul",
"closure",
"comm_semiring",
"monoid_algebra",
"mv_polynomial",
"mv_polynomial.aeval",
"mv_polynomial.aeval_X",
"one_mul"
] | If a set `S` generates a monoid `M`, then the image of `M` generates, as algebra,
`monoid_algebra R M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_type_of_fg [comm_ring R] [monoid.fg M] : finite_type R (monoid_algebra R M) | (add_monoid_algebra.finite_type_of_fg R (additive M)).equiv (to_additive_alg_equiv R M).symm | instance | monoid_algebra.finite_type_of_fg | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"add_monoid_algebra.finite_type_of_fg",
"additive",
"comm_ring",
"equiv",
"monoid.fg",
"monoid_algebra"
] | If a monoid `M` is finitely generated then `monoid_algebra R M` is of finite type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_type_iff_fg [comm_ring R] [nontrivial R] :
finite_type R (monoid_algebra R M) ↔ monoid.fg M | ⟨λ h, monoid.fg_iff_add_fg.2 $ add_monoid_algebra.finite_type_iff_fg.1 $ h.equiv $
to_additive_alg_equiv R M, λ h, @monoid_algebra.finite_type_of_fg _ _ _ _ h⟩ | lemma | monoid_algebra.finite_type_iff_fg | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"comm_ring",
"monoid.fg",
"monoid_algebra",
"monoid_algebra.finite_type_of_fg",
"nontrivial"
] | A monoid `M` is finitely generated if and only if `monoid_algebra R M` is of finite type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fg_of_finite_type [comm_ring R] [nontrivial R] [h : finite_type R (monoid_algebra R M)] :
monoid.fg M | finite_type_iff_fg.1 h | lemma | monoid_algebra.fg_of_finite_type | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"comm_ring",
"monoid.fg",
"monoid_algebra",
"nontrivial"
] | If `monoid_algebra R M` is of finite type then `M` is finitely generated. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_type_iff_group_fg {G : Type*} [comm_group G] [comm_ring R] [nontrivial R] :
finite_type R (monoid_algebra R G) ↔ group.fg G | by simpa [group.fg_iff_monoid.fg] using finite_type_iff_fg | lemma | monoid_algebra.finite_type_iff_group_fg | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"comm_group",
"comm_ring",
"group.fg",
"group.fg_iff_monoid.fg",
"monoid_algebra",
"nontrivial"
] | A group `G` is finitely generated if and only if `add_monoid_algebra R G` is of finite type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module_polynomial_of_endo : module R[X] M | module.comp_hom M (polynomial.aeval f).to_ring_hom | def | module_polynomial_of_endo | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"module",
"module.comp_hom",
"polynomial.aeval"
] | The structure of a module `M` over a ring `R` as a module over `R[X]` when given a
choice of how `X` acts by choosing a linear map `f : M →ₗ[R] M` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
module_polynomial_of_endo_smul_def (n : R[X]) (a : M) :
@@has_smul.smul (module_polynomial_of_endo f).to_has_smul n a = polynomial.aeval f n a | rfl | lemma | module_polynomial_of_endo_smul_def | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"module_polynomial_of_endo",
"polynomial.aeval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module_polynomial_of_endo.is_scalar_tower : @is_scalar_tower R R[X] M _
(by { letI := module_polynomial_of_endo f, apply_instance }) _ | begin
letI := module_polynomial_of_endo f,
constructor,
intros x y z,
simp,
end | lemma | module_polynomial_of_endo.is_scalar_tower | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"is_scalar_tower",
"module_polynomial_of_endo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module.finite.injective_of_surjective_endomorphism [hfg : finite R M]
(f_surj : function.surjective f) : function.injective f | begin
letI := module_polynomial_of_endo f,
haveI : is_scalar_tower R R[X] M := module_polynomial_of_endo.is_scalar_tower f,
have hfgpoly : finite R[X] M, from finite.of_restrict_scalars_finite R _ _,
have X_mul : ∀ o, (X : R[X]) • o = f o,
{ intro,
simp, },
have : (⊤ : submodule R[X] M) ≤ ideal.span {X}... | theorem | module.finite.injective_of_surjective_endomorphism | ring_theory | src/ring_theory/finite_type.lean | [
"group_theory.finiteness",
"ring_theory.adjoin.tower",
"ring_theory.finiteness",
"ring_theory.noetherian"
] | [
"add_smul",
"dvd_refl",
"finite",
"ideal.mem_span_singleton'",
"ideal.span",
"is_scalar_tower",
"linear_map.ker_eq_bot",
"linear_map.ker_eq_bot'",
"module_polynomial_of_endo",
"module_polynomial_of_endo.is_scalar_tower",
"one_smul",
"smul_zero",
"submodule",
"submodule.exists_sub_one_mem_a... | A theorem/proof by Vasconcelos, given a finite module `M` over a commutative ring, any
surjective endomorphism of `M` is also injective. Based on,
https://math.stackexchange.com/a/239419/31917,
https://www.ams.org/journals/tran/1969-138-00/S0002-9947-1969-0238839-5/.
This is similar to `is_noetherian.injective_of_surje... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_units_lt (M₀ : Type*) [monoid_with_zero M₀] [nontrivial M₀] [fintype M₀] :
fintype.card M₀ˣ < fintype.card M₀ | fintype.card_lt_of_injective_of_not_mem (coe : M₀ˣ → M₀) units.ext not_is_unit_zero | lemma | card_units_lt | ring_theory | src/ring_theory/fintype.lean | [
"data.fintype.units"
] | [
"fintype",
"fintype.card",
"fintype.card_lt_of_injective_of_not_mem",
"monoid_with_zero",
"nontrivial",
"not_is_unit_zero",
"units.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flat (R : Type u) (M : Type v) [comm_ring R] [add_comm_group M] [module R M] : Prop | (out : ∀ ⦃I : ideal R⦄ (hI : I.fg), injective (tensor_product.lift ((lsmul R M).comp I.subtype))) | class | module.flat | ring_theory | src/ring_theory/flat.lean | [
"ring_theory.noetherian"
] | [
"add_comm_group",
"comm_ring",
"ideal",
"module",
"tensor_product.lift"
] | An `R`-module `M` is flat if for all finitely generated ideals `I` of `R`,
the canonical map `I ⊗ M →ₗ M` is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self (R : Type u) [comm_ring R] : flat R R | ⟨begin
intros I hI,
rw ← equiv.injective_comp (tensor_product.rid R I).symm.to_equiv,
convert subtype.coe_injective using 1,
ext x,
simp only [function.comp_app, linear_equiv.coe_to_equiv, rid_symm_apply, comp_apply,
mul_one, lift.tmul, subtype_apply, algebra.id.smul_eq_mul, lsmul_apply]
end⟩ | instance | module.flat.self | ring_theory | src/ring_theory/flat.lean | [
"ring_theory.noetherian"
] | [
"algebra.id.smul_eq_mul",
"comm_ring",
"equiv.injective_comp",
"linear_equiv.coe_to_equiv",
"mul_one",
"subtype.coe_injective",
"tensor_product.rid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_fractional (I : submodule R P) | ∃ a ∈ S, ∀ b ∈ I, is_integer R (a • b) | def | is_fractional | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"submodule"
] | A submodule `I` is a fractional ideal if `a I ⊆ R` for some `a ≠ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fractional_ideal | {I : submodule R P // is_fractional S I} | def | fractional_ideal | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"is_fractional",
"submodule"
] | The fractional ideals of a domain `R` are ideals of `R` divided by some `a ∈ R`.
More precisely, let `P` be a localization of `R` at some submonoid `S`,
then a fractional ideal `I ⊆ P` is an `R`-submodule of `P`,
such that there is a nonzero `a : R` with `a I ⊆ R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_fractional (I : fractional_ideal S P) :
is_fractional S (I : submodule R P) | I.prop | lemma | fractional_ideal.is_fractional | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"is_fractional",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coe {I : fractional_ideal S P} {x : P} :
x ∈ (I : submodule R P) ↔ x ∈ I | iff.rfl | lemma | fractional_ideal.mem_coe | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {I J : fractional_ideal S P} : (∀ x, x ∈ I ↔ x ∈ J) → I = J | set_like.ext | lemma | fractional_ideal.ext | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"set_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (p : fractional_ideal S P) (s : set P) (hs : s = ↑p) : fractional_ideal S P | ⟨submodule.copy p s hs, by { convert p.is_fractional, ext, simp only [hs], refl }⟩ | def | fractional_ideal.copy | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | Copy of a `fractional_ideal` with a new underlying set equal to the old one.
Useful to fix definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (p : fractional_ideal S P) (s : set P) (hs : s = ↑p) :
↑(p.copy s hs) = s | rfl | lemma | fractional_ideal.coe_copy | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq (p : fractional_ideal S P) (s : set P) (hs : s = ↑p) : p.copy s hs = p | set_like.coe_injective hs | lemma | fractional_ideal.coe_eq | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
val_eq_coe (I : fractional_ideal S P) : I.val = I | rfl | lemma | fractional_ideal.val_eq_coe | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (I : submodule R P) (hI : is_fractional S I) :
(subtype.mk I hI : submodule R P) = I | rfl | lemma | fractional_ideal.coe_mk | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"is_fractional",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_submodule_injective :
function.injective (coe : fractional_ideal S P → submodule R P) | subtype.coe_injective | lemma | fractional_ideal.coe_to_submodule_injective | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_submodule_inj {I J : fractional_ideal S P} : (I : submodule R P) = J ↔ I = J | coe_to_submodule_injective.eq_iff | lemma | fractional_ideal.coe_to_submodule_inj | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_fractional_of_le_one (I : submodule R P) (h : I ≤ 1) : is_fractional S I | begin
use [1, S.one_mem],
intros b hb,
rw one_smul,
obtain ⟨b', b'_mem, rfl⟩ := h hb,
exact set.mem_range_self b',
end | lemma | fractional_ideal.is_fractional_of_le_one | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"is_fractional",
"one_smul",
"set.mem_range_self",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_fractional_of_le {I : submodule R P} {J : fractional_ideal S P} (hIJ : I ≤ J) :
is_fractional S I | begin
obtain ⟨a, a_mem, ha⟩ := J.is_fractional,
use [a, a_mem],
intros b b_mem,
exact ha b (hIJ b_mem)
end | lemma | fractional_ideal.is_fractional_of_le | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"is_fractional",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe_ideal (I : ideal R) :
((I : fractional_ideal S P) : submodule R P) = coe_submodule P I | rfl | lemma | fractional_ideal.coe_coe_ideal | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coe_ideal {x : P} {I : ideal R} :
x ∈ (I : fractional_ideal S P) ↔ ∃ x', x' ∈ I ∧ algebra_map R P x' = x | mem_coe_submodule _ _ | lemma | fractional_ideal.mem_coe_ideal | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra_map",
"fractional_ideal",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coe_ideal_of_mem {x : R} {I : ideal R} (hx : x ∈ I) :
algebra_map R P x ∈ (I : fractional_ideal S P) | (mem_coe_ideal S).mpr ⟨x, hx, rfl⟩ | lemma | fractional_ideal.mem_coe_ideal_of_mem | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra_map",
"fractional_ideal",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_le_coe_ideal' [is_localization S P] (h : S ≤ non_zero_divisors R)
{I J : ideal R} : (I : fractional_ideal S P) ≤ J ↔ I ≤ J | coe_submodule_le_coe_submodule h | lemma | fractional_ideal.coe_ideal_le_coe_ideal' | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"is_localization",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_le_coe_ideal (K : Type*) [comm_ring K] [algebra R K] [is_fraction_ring R K]
{I J : ideal R} : (I : fractional_ideal R⁰ K) ≤ J ↔ I ≤ J | is_fraction_ring.coe_submodule_le_coe_submodule | lemma | fractional_ideal.coe_ideal_le_coe_ideal | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra",
"comm_ring",
"fractional_ideal",
"ideal",
"is_fraction_ring",
"is_fraction_ring.coe_submodule_le_coe_submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_zero_iff {x : P} : x ∈ (0 : fractional_ideal S P) ↔ x = 0 | ⟨(λ ⟨x', x'_mem_zero, x'_eq_x⟩,
have x'_eq_zero : x' = 0 := x'_mem_zero,
by simp [x'_eq_x.symm, x'_eq_zero]),
(λ hx, ⟨0, rfl, by simp [hx]⟩)⟩ | lemma | fractional_ideal.mem_zero_iff | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ↑(0 : fractional_ideal S P) = (⊥ : submodule R P) | submodule.ext $ λ _, mem_zero_iff S | lemma | fractional_ideal.coe_zero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule",
"submodule.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_bot : ((⊥ : ideal R) : fractional_ideal S P) = 0 | rfl | lemma | fractional_ideal.coe_ideal_bot | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_to_map_eq {x : R} {I : ideal R} (h : S ≤ non_zero_divisors R) :
(∃ x', x' ∈ I ∧ algebra_map R P x' = algebra_map R P x) ↔ x ∈ I | ⟨λ ⟨x', hx', eq⟩, is_localization.injective _ h eq ▸ hx', λ h, ⟨x, h, rfl⟩⟩ | lemma | fractional_ideal.exists_mem_to_map_eq | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra_map",
"ideal",
"is_localization.injective",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_injective' (h : S ≤ non_zero_divisors R) :
function.injective (coe : ideal R → fractional_ideal S P) | λ _ _ h', ((coe_ideal_le_coe_ideal' S h).mp h'.le).antisymm ((coe_ideal_le_coe_ideal' S h).mp h'.ge) | lemma | fractional_ideal.coe_ideal_injective' | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_inj' (h : S ≤ non_zero_divisors R) {I J : ideal R} :
(I : fractional_ideal S P) = J ↔ I = J | (coe_ideal_injective' h).eq_iff | lemma | fractional_ideal.coe_ideal_inj' | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_eq_zero' {I : ideal R} (h : S ≤ non_zero_divisors R) :
(I : fractional_ideal S P) = 0 ↔ I = (⊥ : ideal R) | coe_ideal_inj' h | lemma | fractional_ideal.coe_ideal_eq_zero' | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_ne_zero' {I : ideal R} (h : S ≤ non_zero_divisors R) :
(I : fractional_ideal S P) ≠ 0 ↔ I ≠ (⊥ : ideal R) | not_iff_not.mpr $ coe_ideal_eq_zero' h | lemma | fractional_ideal.coe_ideal_ne_zero' | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_submodule_eq_bot {I : fractional_ideal S P} :
(I : submodule R P) = ⊥ ↔ I = 0 | ⟨λ h, coe_to_submodule_injective (by simp [h]),
λ h, by simp [h]⟩ | lemma | fractional_ideal.coe_to_submodule_eq_bot | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_submodule_ne_bot {I : fractional_ideal S P} :
↑I ≠ (⊥ : submodule R P) ↔ I ≠ 0 | not_iff_not.mpr coe_to_submodule_eq_bot | lemma | fractional_ideal.coe_to_submodule_ne_bot | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_top : ((⊤ : ideal R) : fractional_ideal S P) = 1 | rfl | lemma | fractional_ideal.coe_ideal_top | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_one_iff {x : P} : x ∈ (1 : fractional_ideal S P) ↔ ∃ x' : R, algebra_map R P x' = x | iff.intro (λ ⟨x', _, h⟩, ⟨x', h⟩) (λ ⟨x', h⟩, ⟨x', ⟨⟩, h⟩) | lemma | fractional_ideal.mem_one_iff | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra_map",
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mem_one (x : R) : algebra_map R P x ∈ (1 : fractional_ideal S P) | (mem_one_iff S).mpr ⟨x, rfl⟩ | lemma | fractional_ideal.coe_mem_one | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra_map",
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mem_one : (1 : P) ∈ (1 : fractional_ideal S P) | (mem_one_iff S).mpr ⟨1, ring_hom.map_one _⟩ | lemma | fractional_ideal.one_mem_one | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ring_hom.map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one_eq_coe_submodule_top :
↑(1 : fractional_ideal S P) = coe_submodule P (⊤ : ideal R) | rfl | lemma | fractional_ideal.coe_one_eq_coe_submodule_top | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal"
] | `(1 : fractional_ideal S P)` is defined as the R-submodule `f(R) ≤ P`.
However, this is not definitionally equal to `1 : submodule R P`,
which is proved in the actual `simp` lemma `coe_one`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_one :
(↑(1 : fractional_ideal S P) : submodule R P) = 1 | by rw [coe_one_eq_coe_submodule_top, coe_submodule_top] | lemma | fractional_ideal.coe_one | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_le_coe {I J : fractional_ideal S P} :
(I : submodule R P) ≤ (J : submodule R P) ↔ I ≤ J | iff.rfl | lemma | fractional_ideal.coe_le_coe | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le (I : fractional_ideal S P) : 0 ≤ I | begin
intros x hx,
convert submodule.zero_mem _,
simpa using hx
end | lemma | fractional_ideal.zero_le | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule.zero_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_bot : order_bot (fractional_ideal S P) | { bot := 0,
bot_le := zero_le } | instance | fractional_ideal.order_bot | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"bot_le",
"fractional_ideal",
"order_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_eq_zero : (⊥ : fractional_ideal S P) = 0 | rfl | lemma | fractional_ideal.bot_eq_zero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_zero_iff {I : fractional_ideal S P} : I ≤ 0 ↔ I = 0 | le_bot_iff | lemma | fractional_ideal.le_zero_iff | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"le_bot_iff",
"le_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_iff {I : fractional_ideal S P} : I = 0 ↔ (∀ x ∈ I, x = (0 : P)) | ⟨ (λ h x hx, by simpa [h, mem_zero_iff] using hx),
(λ h, le_bot_iff.mp (λ x hx, (mem_zero_iff S).mpr (h x hx))) ⟩ | lemma | fractional_ideal.eq_zero_iff | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"eq_zero_iff",
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_fractional.sup {I J : submodule R P} :
is_fractional S I → is_fractional S J → is_fractional S (I ⊔ J) | | ⟨aI, haI, hI⟩ ⟨aJ, haJ, hJ⟩ := ⟨aI * aJ, S.mul_mem haI haJ, λ b hb, begin
rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩,
rw smul_add,
apply is_integer_add,
{ rw [mul_smul, smul_comm],
exact is_integer_smul (hI bI hbI), },
{ rw mul_smul,
exact is_integer_smul (hJ bJ hbJ) }
end⟩ | lemma | is_fractional.sup | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"is_fractional",
"smul_add",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_fractional.inf_right {I : submodule R P} :
is_fractional S I → ∀ J, is_fractional S (I ⊓ J) | | ⟨aI, haI, hI⟩ J := ⟨aI, haI, λ b hb, begin
rcases mem_inf.mp hb with ⟨hbI, hbJ⟩,
exact hI b hbI
end⟩ | lemma | is_fractional.inf_right | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"is_fractional",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (I J : fractional_ideal S P) : ↑(I ⊓ J) = (I ⊓ J : submodule R P) | rfl | lemma | fractional_ideal.coe_inf | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sup (I J : fractional_ideal S P) : ↑(I ⊔ J) = (I ⊔ J : submodule R P) | rfl | lemma | fractional_ideal.coe_sup | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lattice : lattice (fractional_ideal S P) | function.injective.lattice _ subtype.coe_injective coe_sup coe_inf | instance | fractional_ideal.lattice | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"function.injective.lattice",
"lattice",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_eq_add (I J : fractional_ideal S P) : I ⊔ J = I + J | rfl | lemma | fractional_ideal.sup_eq_add | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (I J : fractional_ideal S P) : (↑(I + J) : submodule R P) = I + J | rfl | lemma | fractional_ideal.coe_add | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_sup (I J : ideal R) : ↑(I ⊔ J) = (I + J : fractional_ideal S P) | coe_to_submodule_injective $ coe_submodule_sup _ _ _ | lemma | fractional_ideal.coe_ideal_sup | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_fractional.nsmul {I : submodule R P} :
Π n : ℕ, is_fractional S I → is_fractional S (n • I : submodule R P) | | 0 _ := begin
rw [zero_smul],
convert ((0 : ideal R) : fractional_ideal S P).is_fractional,
simp,
end
| (n + 1) h := begin
rw succ_nsmul,
exact h.sup (_root_.is_fractional.nsmul n h)
end | lemma | is_fractional.nsmul | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"is_fractional",
"submodule",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nsmul (n : ℕ) (I : fractional_ideal S P) : (↑(n • I) : submodule R P) = n • I | rfl | lemma | fractional_ideal.coe_nsmul | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_fractional.mul {I J : submodule R P} :
is_fractional S I → is_fractional S J → is_fractional S (I * J : submodule R P) | | ⟨aI, haI, hI⟩ ⟨aJ, haJ, hJ⟩ := ⟨aI * aJ, S.mul_mem haI haJ, λ b hb, begin
apply submodule.mul_induction_on hb,
{ intros m hm n hn,
obtain ⟨n', hn'⟩ := hJ n hn,
rw [mul_smul, mul_comm m, ← smul_mul_assoc, ← hn', ← algebra.smul_def],
apply hI,
exact submodule.smul_mem _ _ hm },
{ intros x y hx hy,... | lemma | is_fractional.mul | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra.smul_def",
"is_fractional",
"mul_comm",
"smul_add",
"smul_mul_assoc",
"submodule",
"submodule.mul_induction_on",
"submodule.smul_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_fractional.pow {I : submodule R P} (h : is_fractional S I) :
∀ n : ℕ, is_fractional S (I ^ n : submodule R P) | | 0 := is_fractional_of_le_one _ (pow_zero _).le
| (n + 1) := (pow_succ I n).symm ▸ h.mul (_root_.is_fractional.pow n) | lemma | is_fractional.pow | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"is_fractional",
"pow_succ",
"pow_zero",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul (I J : fractional_ideal S P) : fractional_ideal S P | ⟨I * J, I.is_fractional.mul J.is_fractional⟩ | def | fractional_ideal.mul | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | `fractional_ideal.mul` is the product of two fractional ideals,
used to define the `has_mul` instance.
This is only an auxiliary definition: the preferred way of writing `I.mul J` is `I * J`.
Elaborated terms involving `fractional_ideal` tend to grow quite large,
so by making definitions irreducible, we hope to avoid... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_eq_mul (I J : fractional_ideal S P) : mul I J = I * J | rfl | lemma | fractional_ideal.mul_eq_mul | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_def (I J : fractional_ideal S P) : I * J = ⟨I * J, I.is_fractional.mul J.is_fractional⟩ | by simp only [← mul_eq_mul, mul] | lemma | fractional_ideal.mul_def | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (I J : fractional_ideal S P) : (↑(I * J) : submodule R P) = I * J | by { simp only [mul_def], refl } | lemma | fractional_ideal.coe_mul | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_mul (I J : ideal R) : (↑(I * J) : fractional_ideal S P) = I * J | begin
simp only [mul_def],
exact coe_to_submodule_injective (coe_submodule_mul _ _ _)
end | lemma | fractional_ideal.coe_ideal_mul | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_mono (I : fractional_ideal S P) : monotone ((*) I) | begin
intros J J' h,
simp only [mul_def],
exact mul_le.mpr (λ x hx y hy, mul_mem_mul hx (h hy))
end | lemma | fractional_ideal.mul_left_mono | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_mono (I : fractional_ideal S P) : monotone (λ J, J * I) | begin
intros J J' h,
simp only [mul_def],
exact mul_le.mpr (λ x hx y hy, mul_mem_mul (h hx) hy)
end | lemma | fractional_ideal.mul_right_mono | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_mul {I J : fractional_ideal S P} {i j : P} (hi : i ∈ I) (hj : j ∈ J) :
i * j ∈ I * J | by { simp only [mul_def], exact submodule.mul_mem_mul hi hj } | lemma | fractional_ideal.mul_mem_mul | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule.mul_mem_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le {I J K : fractional_ideal S P} :
I * J ≤ K ↔ (∀ (i ∈ I) (j ∈ J), i * j ∈ K) | by { simp only [mul_def], exact submodule.mul_le } | lemma | fractional_ideal.mul_le | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule.mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow (I : fractional_ideal S P) (n : ℕ) : ↑(I ^ n) = (I ^ n : submodule R P) | rfl | lemma | fractional_ideal.coe_pow | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_induction_on
{I J : fractional_ideal S P}
{C : P → Prop} {r : P} (hr : r ∈ I * J)
(hm : ∀ (i ∈ I) (j ∈ J), C (i * j))
(ha : ∀ x y, C x → C y → C (x + y)) : C r | begin
simp only [mul_def] at hr,
exact submodule.mul_induction_on hr hm ha
end | theorem | fractional_ideal.mul_induction_on | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule.mul_induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nat_cast (n : ℕ) : ((n : fractional_ideal S P) : submodule R P) = n | show ↑n.unary_cast = ↑n, by induction n; simp [*, nat.unary_cast] | lemma | fractional_ideal.coe_nat_cast | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"nat.unary_cast",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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