statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
map_equiv_trans (e₁ : R ≃+* S) (e₂ : S ≃+* T) :
map_equiv (e₁.trans e₂) = (map_equiv e₁).trans (map_equiv e₂) | ring_equiv.to_ring_hom_injective $ map_comp (e₁ : R →+* S) (e₂ : S →+* T) | lemma | local_ring.residue_field.map_equiv_trans | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"map_comp",
"ring_equiv.to_ring_hom_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_equiv_refl : map_equiv (ring_equiv.refl R) = ring_equiv.refl _ | ring_equiv.to_ring_hom_injective map_id | lemma | local_ring.residue_field.map_equiv_refl | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"map_id",
"ring_equiv.refl",
"ring_equiv.to_ring_hom_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_aut : ring_aut R →* ring_aut (local_ring.residue_field R) | { to_fun := map_equiv,
map_mul' := λ e₁ e₂, map_equiv_trans e₂ e₁,
map_one' := map_equiv_refl } | def | local_ring.residue_field.map_aut | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"local_ring.residue_field",
"ring_aut"
] | The group homomorphism from `ring_aut R` to `ring_aut k` where `k`
is the residue field of `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
residue_smul (g : G) (r : R) : residue R (g • r) = g • residue R r | rfl | lemma | local_ring.residue_field.residue_smul | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ker_eq_maximal_ideal [field K] (φ : R →+* K) (hφ : function.surjective φ) :
φ.ker = maximal_ideal R | local_ring.eq_maximal_ideal $ (ring_hom.ker_is_maximal_of_surjective φ) hφ | lemma | local_ring.ker_eq_maximal_ideal | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"field",
"local_ring.eq_maximal_ideal",
"ring_hom.ker_is_maximal_of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_ring_hom_residue :
is_local_ring_hom (local_ring.residue R) | begin
constructor,
intros a ha,
by_contra,
erw ideal.quotient.eq_zero_iff_mem.mpr ((local_ring.mem_maximal_ideal _).mpr h) at ha,
exact ha.ne_zero rfl,
end | lemma | local_ring.is_local_ring_hom_residue | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"by_contra",
"is_local_ring_hom",
"local_ring.mem_maximal_ideal",
"local_ring.residue"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_ring.maximal_ideal_eq_bot {R : Type*} [field R] :
local_ring.maximal_ideal R = ⊥ | local_ring.is_field_iff_maximal_ideal_eq.mp (field.to_is_field R) | lemma | local_ring.maximal_ideal_eq_bot | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"field",
"field.to_is_field",
"local_ring.maximal_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_ring {A B : Type*} [comm_semiring A] [local_ring A]
[comm_semiring B] (e : A ≃+* B) : local_ring B | begin
haveI := e.symm.to_equiv.nontrivial,
exact local_ring.of_surjective (e : A →+* B) e.surjective
end | lemma | ring_equiv.local_ring | ring_theory.ideal | src/ring_theory/ideal/local_ring.lean | [
"algebra.algebra.basic",
"ring_theory.ideal.operations",
"ring_theory.jacobson_ideal",
"logic.equiv.transfer_instance"
] | [
"comm_semiring",
"local_ring",
"local_ring.of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.minimal_primes : set (ideal R) | minimals (≤) { p | p.is_prime ∧ I ≤ p } | def | ideal.minimal_primes | ring_theory.ideal | src/ring_theory/ideal/minimal_prime.lean | [
"ring_theory.localization.at_prime",
"order.minimal"
] | [
"ideal",
"minimals"
] | `I.minimal_primes` is the set of ideals that are minimal primes over `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
minimal_primes (R : Type*) [comm_ring R] : set (ideal R) | ideal.minimal_primes ⊥ | def | minimal_primes | ring_theory.ideal | src/ring_theory/ideal/minimal_prime.lean | [
"ring_theory.localization.at_prime",
"order.minimal"
] | [
"comm_ring",
"ideal",
"ideal.minimal_primes"
] | `minimal_primes R` is the set of minimal primes of `R`.
This is defined as `ideal.minimal_primes ⊥`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.exists_minimal_primes_le [J.is_prime] (e : I ≤ J) :
∃ p ∈ I.minimal_primes, p ≤ J | begin
suffices : ∃ m ∈ { p : (ideal R)ᵒᵈ | ideal.is_prime p ∧ I ≤ order_dual.of_dual p },
(order_dual.to_dual J) ≤ m ∧
∀ z ∈ { p : (ideal R)ᵒᵈ | ideal.is_prime p ∧ I ≤ p }, m ≤ z → z = m,
{ obtain ⟨p, h₁, h₂, h₃⟩ := this,
simp_rw ← @eq_comm _ p at h₃,
exact ⟨p, ⟨h₁, λ a b c, (h₃ a b c).le⟩, h₂⟩ },... | lemma | ideal.exists_minimal_primes_le | ring_theory.ideal | src/ring_theory/ideal/minimal_prime.lean | [
"ring_theory.localization.at_prime",
"order.minimal"
] | [
"Inf_le",
"ideal",
"ideal.is_prime",
"le_Inf",
"order_dual.le_to_dual",
"order_dual.of_dual",
"order_dual.of_dual_to_dual",
"order_dual.to_dual",
"zorn_nonempty_partial_order₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.radical_minimal_primes : I.radical.minimal_primes = I.minimal_primes | begin
rw [ideal.minimal_primes, ideal.minimal_primes],
congr,
ext p,
exact ⟨λ ⟨a, b⟩, ⟨a, ideal.le_radical.trans b⟩, λ ⟨a, b⟩, ⟨a, a.radical_le_iff.mpr b⟩⟩,
end | lemma | ideal.radical_minimal_primes | ring_theory.ideal | src/ring_theory/ideal/minimal_prime.lean | [
"ring_theory.localization.at_prime",
"order.minimal"
] | [
"ideal.minimal_primes"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.Inf_minimal_primes :
Inf I.minimal_primes = I.radical | begin
rw I.radical_eq_Inf,
apply le_antisymm,
{ intros x hx,
rw ideal.mem_Inf at hx ⊢,
rintros J ⟨e, hJ⟩,
resetI,
obtain ⟨p, hp, hp'⟩ := ideal.exists_minimal_primes_le e,
exact hp' (hx hp) },
{ apply Inf_le_Inf _,
intros I hI,
exact hI.1.symm },
end | lemma | ideal.Inf_minimal_primes | ring_theory.ideal | src/ring_theory/ideal/minimal_prime.lean | [
"ring_theory.localization.at_prime",
"order.minimal"
] | [
"Inf_le_Inf",
"ideal.exists_minimal_primes_le",
"ideal.mem_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.exists_comap_eq_of_mem_minimal_primes_of_injective {f : R →+* S}
(hf : function.injective f) (p ∈ minimal_primes R) :
∃ p' : ideal S, p'.is_prime ∧ p'.comap f = p | begin
haveI := H.1.1,
haveI : nontrivial (localization (submonoid.map f p.prime_compl)),
{ refine ⟨⟨1, 0, _⟩⟩,
convert (is_localization.map_injective_of_injective p.prime_compl (localization.at_prime p)
(localization $ p.prime_compl.map f) hf).ne one_ne_zero,
{ rw map_one }, { rw map_zero } },
obt... | lemma | ideal.exists_comap_eq_of_mem_minimal_primes_of_injective | ring_theory.ideal | src/ring_theory/ideal/minimal_prime.lean | [
"ring_theory.localization.at_prime",
"order.minimal"
] | [
"algebra_map",
"by_contra",
"ideal",
"ideal.comap_comap",
"ideal.exists_maximal",
"is_localization.map_comp",
"is_localization.map_injective_of_injective",
"is_unit.map",
"localization",
"localization.at_prime",
"map_one",
"minimal_primes",
"nontrivial",
"one_ne_zero",
"submonoid.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.exists_comap_eq_of_mem_minimal_primes {I : ideal S}
(f : R →+* S) (p ∈ (I.comap f).minimal_primes) :
∃ p' : ideal S, p'.is_prime ∧ I ≤ p' ∧ p'.comap f = p | begin
haveI := H.1.1,
let f' := I^.quotient.mk^.comp f,
have e : (I^.quotient.mk^.comp f).ker = I.comap f,
{ ext1, exact (submodule.quotient.mk_eq_zero _) },
have : (I^.quotient.mk^.comp f).ker^.quotient.mk^.ker ≤ p,
{ rw [ideal.mk_ker, e], exact H.1.2 },
obtain ⟨p', hp₁, hp₂⟩ := ideal.exists_comap_eq_of_... | lemma | ideal.exists_comap_eq_of_mem_minimal_primes | ring_theory.ideal | src/ring_theory/ideal/minimal_prime.lean | [
"ring_theory.localization.at_prime",
"order.minimal"
] | [
"bot_le",
"ideal",
"ideal.comap",
"ideal.comap_map_of_surjective",
"ideal.comap_mono",
"ideal.exists_comap_eq_of_mem_minimal_primes_of_injective",
"ideal.is_prime.comap",
"ideal.map_comap_of_surjective",
"ideal.map_is_prime_of_surjective",
"ideal.map_mono",
"ideal.mk_ker",
"ideal.quotient.mk_s... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.exists_minimal_primes_comap_eq {I : ideal S}
(f : R →+* S) (p ∈ (I.comap f).minimal_primes) :
∃ p' ∈ I.minimal_primes, ideal.comap f p' = p | begin
obtain ⟨p', h₁, h₂, h₃⟩ := ideal.exists_comap_eq_of_mem_minimal_primes f p H,
resetI,
obtain ⟨q, hq, hq'⟩ := ideal.exists_minimal_primes_le h₂,
refine ⟨q, hq, eq.symm _⟩,
haveI := hq.1.1,
have := (ideal.comap_mono hq').trans_eq h₃,
exact (H.2 ⟨infer_instance, ideal.comap_mono hq.1.2⟩ this).antisymm ... | lemma | ideal.exists_minimal_primes_comap_eq | ring_theory.ideal | src/ring_theory/ideal/minimal_prime.lean | [
"ring_theory.localization.at_prime",
"order.minimal"
] | [
"ideal",
"ideal.comap",
"ideal.comap_mono",
"ideal.exists_comap_eq_of_mem_minimal_primes",
"ideal.exists_minimal_primes_le",
"minimal_primes"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.mimimal_primes_comap_of_surjective {f : R →+* S} (hf : function.surjective f)
{I J : ideal S} (h : J ∈ I.minimal_primes) :
J.comap f ∈ (I.comap f).minimal_primes | begin
haveI := h.1.1,
refine ⟨⟨infer_instance, ideal.comap_mono h.1.2⟩, _⟩,
rintros K ⟨hK, e₁⟩ e₂,
have : f.ker ≤ K := (ideal.comap_mono bot_le).trans e₁,
rw [← sup_eq_left.mpr this, ring_hom.ker_eq_comap_bot, ← ideal.comap_map_of_surjective f hf],
apply ideal.comap_mono _,
apply h.2 _ _,
{ exactI ⟨idea... | lemma | ideal.mimimal_primes_comap_of_surjective | ring_theory.ideal | src/ring_theory/ideal/minimal_prime.lean | [
"ring_theory.localization.at_prime",
"order.minimal"
] | [
"bot_le",
"ideal",
"ideal.comap_map_of_surjective",
"ideal.comap_mono",
"ideal.le_map_of_comap_le_of_surjective",
"ideal.map_le_of_le_comap",
"minimal_primes",
"ring_hom.ker_eq_comap_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.comap_minimal_primes_eq_of_surjective {f : R →+* S} (hf : function.surjective f)
(I : ideal S) :
(I.comap f).minimal_primes = ideal.comap f '' I.minimal_primes | begin
ext J,
split,
{ intro H, obtain ⟨p, h, rfl⟩ := ideal.exists_minimal_primes_comap_eq f J H, exact ⟨p, h, rfl⟩ },
{ rintros ⟨J, hJ, rfl⟩, exact ideal.mimimal_primes_comap_of_surjective hf hJ }
end | lemma | ideal.comap_minimal_primes_eq_of_surjective | ring_theory.ideal | src/ring_theory/ideal/minimal_prime.lean | [
"ring_theory.localization.at_prime",
"order.minimal"
] | [
"ideal",
"ideal.comap",
"ideal.exists_minimal_primes_comap_eq",
"ideal.mimimal_primes_comap_of_surjective",
"minimal_primes"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.minimal_primes_eq_comap :
I.minimal_primes = ideal.comap I^.quotient.mk '' minimal_primes (R ⧸ I) | begin
rw [minimal_primes, ← ideal.comap_minimal_primes_eq_of_surjective ideal.quotient.mk_surjective,
← ring_hom.ker_eq_comap_bot, ideal.mk_ker],
end | lemma | ideal.minimal_primes_eq_comap | ring_theory.ideal | src/ring_theory/ideal/minimal_prime.lean | [
"ring_theory.localization.at_prime",
"order.minimal"
] | [
"ideal.comap",
"ideal.comap_minimal_primes_eq_of_surjective",
"ideal.mk_ker",
"ideal.quotient.mk_surjective",
"minimal_primes",
"ring_hom.ker_eq_comap_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.minimal_primes_eq_subsingleton (hI : I.is_primary) :
I.minimal_primes = {I.radical} | begin
ext J,
split,
{ exact λ H, let e := H.1.1.radical_le_iff.mpr H.1.2 in
(H.2 ⟨ideal.is_prime_radical hI, ideal.le_radical⟩ e).antisymm e },
{ rintro (rfl : J = I.radical),
exact ⟨⟨ideal.is_prime_radical hI, ideal.le_radical⟩, λ _ H _, H.1.radical_le_iff.mpr H.2⟩ }
end | lemma | ideal.minimal_primes_eq_subsingleton | ring_theory.ideal | src/ring_theory/ideal/minimal_prime.lean | [
"ring_theory.localization.at_prime",
"order.minimal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.minimal_primes_eq_subsingleton_self [I.is_prime] :
I.minimal_primes = {I} | begin
ext J,
split,
{ exact λ H, (H.2 ⟨infer_instance, rfl.le⟩ H.1.2).antisymm H.1.2 },
{ unfreezingI { rintro (rfl : J = I) }, refine ⟨⟨infer_instance, rfl.le⟩, λ _ h _, h.2⟩ },
end | lemma | ideal.minimal_primes_eq_subsingleton_self | ring_theory.ideal | src/ring_theory/ideal/minimal_prime.lean | [
"ring_theory.localization.at_prime",
"order.minimal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_quot (S : submodule R M) : ℕ | add_subgroup.index S.to_add_subgroup | def | submodule.card_quot | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"submodule"
] | The cardinality of `(M ⧸ S)`, if `(M ⧸ S)` is finite, and `0` otherwise.
This is used to define the absolute ideal norm `ideal.abs_norm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_quot_apply (S : submodule R M) [fintype (M ⧸ S)] :
card_quot S = fintype.card (M ⧸ S) | add_subgroup.index_eq_card _ | lemma | submodule.card_quot_apply | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"fintype",
"fintype.card",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_quot_bot [infinite M] : card_quot (⊥ : submodule R M) = 0 | add_subgroup.index_bot.trans nat.card_eq_zero_of_infinite | lemma | submodule.card_quot_bot | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"infinite",
"nat.card_eq_zero_of_infinite",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_quot_top : card_quot (⊤ : submodule R M) = 1 | add_subgroup.index_top | lemma | submodule.card_quot_top | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_quot_eq_one_iff {P : submodule R M} : card_quot P = 1 ↔ P = ⊤ | add_subgroup.index_eq_one.trans (by simp [set_like.ext_iff]) | lemma | submodule.card_quot_eq_one_iff | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"set_like.ext_iff",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_quot_mul_of_coprime [is_dedekind_domain S] [module.free ℤ S] [module.finite ℤ S]
{I J : ideal S} (coprime : I ⊔ J = ⊤) : card_quot (I * J) = card_quot I * card_quot J | begin
let b := module.free.choose_basis ℤ S,
casesI is_empty_or_nonempty (module.free.choose_basis_index ℤ S),
{ haveI : subsingleton S := function.surjective.subsingleton b.repr.to_equiv.symm.surjective,
nontriviality S,
exfalso,
exact not_nontrivial_iff_subsingleton.mpr ‹subsingleton S› ‹nontrivial ... | lemma | card_quot_mul_of_coprime | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"classical.dec_eq",
"fintype.card_prod",
"function.surjective.subsingleton",
"ideal",
"ideal.quotient_mul_equiv_quotient_prod",
"infinite",
"infinite.of_surjective",
"is_dedekind_domain",
"is_empty_or_nonempty",
"module.finite",
"module.free",
"module.free.choose_basis",
"module.free.choose_... | Multiplicity of the ideal norm, for coprime ideals.
This is essentially just a repackaging of the Chinese Remainder Theorem. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.mul_add_mem_pow_succ_inj
(P : ideal S) {i : ℕ} (a d d' e e' : S) (a_mem : a ∈ P ^ i)
(e_mem : e ∈ P ^ (i + 1)) (e'_mem : e' ∈ P ^ (i + 1))
(h : d - d' ∈ P) : (a * d + e) - (a * d' + e') ∈ P ^ (i + 1) | begin
have : a * d - a * d' ∈ P ^ (i + 1),
{ convert ideal.mul_mem_mul a_mem h; simp [mul_sub, pow_succ, mul_comm] },
convert ideal.add_mem _ this (ideal.sub_mem _ e_mem e'_mem),
ring,
end | lemma | ideal.mul_add_mem_pow_succ_inj | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal",
"ideal.add_mem",
"ideal.mul_mem_mul",
"ideal.sub_mem",
"mul_comm",
"pow_succ",
"ring"
] | If the `d` from `ideal.exists_mul_add_mem_pow_succ` is unique, up to `P`,
then so are the `c`s, up to `P ^ (i + 1)`.
Inspired by [Neukirch], proposition 6.1 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.exists_mul_add_mem_pow_succ [is_dedekind_domain S] {i : ℕ}
(a c : S) (a_mem : a ∈ P ^ i) (a_not_mem : a ∉ P ^ (i + 1)) (c_mem : c ∈ P ^ i) :
∃ (d : S) (e ∈ P ^ (i + 1)), a * d + e = c | begin
suffices eq_b : P ^ i = ideal.span {a} ⊔ P ^ (i + 1),
{ rw eq_b at c_mem,
simp only [mul_comm a],
exact ideal.mem_span_singleton_sup.mp c_mem },
refine (ideal.eq_prime_pow_of_succ_lt_of_le hP
(lt_of_le_of_ne le_sup_right _)
(sup_le (ideal.span_le.mpr (set.singleton_subset_iff.mpr a_mem))
... | lemma | ideal.exists_mul_add_mem_pow_succ | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal.eq_prime_pow_of_succ_lt_of_le",
"ideal.pow_succ_lt_pow",
"ideal.span",
"is_dedekind_domain",
"le_sup_right",
"mul_comm",
"sup_le"
] | If `a ∈ P^i \ P^(i+1)` and `c ∈ P^i`, then `a * d + e = c` for `e ∈ P^(i+1)`.
`ideal.mul_add_mem_pow_succ_unique` shows the choice of `d` is unique, up to `P`.
Inspired by [Neukirch], proposition 6.1 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.mem_prime_of_mul_mem_pow [is_dedekind_domain S]
{P : ideal S} [P_prime : P.is_prime] (hP : P ≠ ⊥) {i : ℕ}
{a b : S} (a_not_mem : a ∉ P ^ (i + 1))
(ab_mem : a * b ∈ P ^ (i + 1)) : b ∈ P | begin
simp only [← ideal.span_singleton_le_iff_mem, ← ideal.dvd_iff_le, pow_succ,
← ideal.span_singleton_mul_span_singleton] at a_not_mem ab_mem ⊢,
exact (prime_pow_succ_dvd_mul (ideal.prime_of_is_prime hP P_prime) ab_mem).resolve_left a_not_mem
end | lemma | ideal.mem_prime_of_mul_mem_pow | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal",
"ideal.dvd_iff_le",
"ideal.prime_of_is_prime",
"ideal.span_singleton_le_iff_mem",
"ideal.span_singleton_mul_span_singleton",
"is_dedekind_domain",
"pow_succ",
"prime_pow_succ_dvd_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.mul_add_mem_pow_succ_unique [is_dedekind_domain S] {i : ℕ}
(a d d' e e' : S) (a_not_mem : a ∉ P ^ (i + 1))
(e_mem : e ∈ P ^ (i + 1)) (e'_mem : e' ∈ P ^ (i + 1))
(h : (a * d + e) - (a * d' + e') ∈ P ^ (i + 1)) : d - d' ∈ P | begin
have : e' - e ∈ P ^ (i + 1) := ideal.sub_mem _ e'_mem e_mem,
have h' : a * (d - d') ∈ P ^ (i + 1),
{ convert ideal.add_mem _ h (ideal.sub_mem _ e'_mem e_mem),
ring },
exact ideal.mem_prime_of_mul_mem_pow hP a_not_mem h'
end | lemma | ideal.mul_add_mem_pow_succ_unique | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal.add_mem",
"ideal.mem_prime_of_mul_mem_pow",
"ideal.sub_mem",
"is_dedekind_domain",
"ring"
] | The choice of `d` in `ideal.exists_mul_add_mem_pow_succ` is unique, up to `P`.
Inspired by [Neukirch], proposition 6.1 | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_quot_pow_of_prime [is_dedekind_domain S] [module.finite ℤ S] [module.free ℤ S] {i : ℕ} :
card_quot (P ^ i) = card_quot P ^ i | begin
let b := module.free.choose_basis ℤ S,
classical,
induction i with i ih,
{ simp },
letI := ideal.fintype_quotient_of_free_of_ne_bot (P ^ i.succ) (pow_ne_zero _ hP),
letI := ideal.fintype_quotient_of_free_of_ne_bot (P ^ i) (pow_ne_zero _ hP),
letI := ideal.fintype_quotient_of_free_of_ne_bot P hP,
h... | lemma | card_quot_pow_of_prime | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"equiv.of_bijective",
"fintype.card_eq",
"ideal.exists_mul_add_mem_pow_succ",
"ideal.fintype_quotient_of_free_of_ne_bot",
"ideal.mul_add_mem_pow_succ_inj",
"ideal.mul_add_mem_pow_succ_unique",
"ideal.mul_mem_right",
"ideal.pow_succ_lt_pow",
"ideal.quotient.eq",
"ideal.quotient.mk_eq_mk",
"ih",
... | Multiplicity of the ideal norm, for powers of prime ideals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_quot_mul [is_dedekind_domain S] [module.free ℤ S] [module.finite ℤ S] (I J : ideal S) :
card_quot (I * J) = card_quot I * card_quot J | begin
let b := module.free.choose_basis ℤ S,
casesI is_empty_or_nonempty (module.free.choose_basis_index ℤ S),
{ haveI : subsingleton S := function.surjective.subsingleton b.repr.to_equiv.symm.surjective,
nontriviality S,
exfalso,
exact not_nontrivial_iff_subsingleton.mpr ‹subsingleton S› ‹nontrivial ... | theorem | card_quot_mul | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"card_quot_mul_of_coprime",
"card_quot_pow_of_prime",
"function.surjective.subsingleton",
"ideal",
"ideal.is_prime",
"ideal.is_prime_of_prime",
"ideal.mul_top",
"infinite",
"infinite.of_surjective",
"is_dedekind_domain",
"is_empty_or_nonempty",
"le_sup_left",
"le_sup_right",
"module.finite... | Multiplicativity of the ideal norm in number rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.abs_norm [infinite S] [is_dedekind_domain S]
[module.free ℤ S] [module.finite ℤ S] :
ideal S →*₀ ℕ | { to_fun := submodule.card_quot,
map_mul' := λ I J, by rw card_quot_mul,
map_one' := by rw [ideal.one_eq_top, card_quot_top],
map_zero' := by rw [ideal.zero_eq_bot, card_quot_bot] } | def | ideal.abs_norm | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"card_quot_mul",
"ideal",
"ideal.one_eq_top",
"ideal.zero_eq_bot",
"infinite",
"is_dedekind_domain",
"module.finite",
"module.free",
"submodule.card_quot"
] | The absolute norm of the ideal `I : ideal R` is the cardinality of the quotient `R ⧸ I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_norm_apply (I : ideal S) : abs_norm I = card_quot I | rfl | lemma | ideal.abs_norm_apply | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"abs_norm",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_norm_bot : abs_norm (⊥ : ideal S) = 0 | by rw [← ideal.zero_eq_bot, _root_.map_zero] | lemma | ideal.abs_norm_bot | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"abs_norm",
"ideal",
"ideal.zero_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_norm_top : abs_norm (⊤ : ideal S) = 1 | by rw [← ideal.one_eq_top, _root_.map_one] | lemma | ideal.abs_norm_top | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"abs_norm",
"ideal",
"ideal.one_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_norm_eq_one_iff {I : ideal S} : abs_norm I = 1 ↔ I = ⊤ | by rw [abs_norm_apply, card_quot_eq_one_iff] | lemma | ideal.abs_norm_eq_one_iff | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"abs_norm",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_norm_ne_zero_iff (I : ideal S) : ideal.abs_norm I ≠ 0 ↔ finite (S ⧸ I) | ⟨λ h,nat.finite_of_card_ne_zero h,
λ h, (@add_subgroup.finite_index_of_finite_quotient _ _ _ h).finite_index⟩ | lemma | ideal.abs_norm_ne_zero_iff | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"finite",
"ideal",
"ideal.abs_norm",
"nat.finite_of_card_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_abs_det_equiv (I : ideal S) {E : Type*} [add_equiv_class E S I] (e : E) :
int.nat_abs (linear_map.det
((submodule.subtype I).restrict_scalars ℤ ∘ₗ add_monoid_hom.to_int_linear_map (e : S →+ I))) =
ideal.abs_norm I | begin
-- `S ⧸ I` might be infinite if `I = ⊥`, but then `e` can't be an equiv.
by_cases hI : I = ⊥,
{ unfreezingI { subst hI },
have : (1 : S) ≠ 0 := one_ne_zero,
have : (1 : S) = 0 := equiv_like.injective e (subsingleton.elim _ _),
contradiction },
let ι := module.free.choose_basis_index ℤ S,
le... | theorem | ideal.nat_abs_det_equiv | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"abs_norm",
"add_equiv.to_int_linear_equiv",
"add_equiv_class",
"add_monoid_hom.to_int_linear_map",
"basis.repr_self",
"classical.dec_eq",
"equiv.refl",
"equiv.refl_apply",
"equiv_like.injective",
"finset.univ",
"finsupp.single_eq_of_ne",
"finsupp.single_eq_same",
"finsupp.smul_single",
"f... | Let `e : S ≃ I` be an additive isomorphism (therefore a `ℤ`-linear equiv).
Then an alternative way to compute the norm of `I` is given by taking the determinant of `e`.
See `nat_abs_det_basis_change` for a more familiar formulation of this result. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_abs_det_basis_change {ι : Type*} [fintype ι] [decidable_eq ι]
(b : basis ι ℤ S) (I : ideal S) (bI : basis ι ℤ I) :
(b.det (coe ∘ bI)).nat_abs = ideal.abs_norm I | begin
let e := b.equiv bI (equiv.refl _),
calc (b.det ((submodule.subtype I).restrict_scalars ℤ ∘ bI)).nat_abs
= (linear_map.det ((submodule.subtype I).restrict_scalars ℤ ∘ₗ (e : S →ₗ[ℤ] I))).nat_abs
: by rw basis.det_comp_basis
... = _ : nat_abs_det_equiv I e
end | theorem | ideal.nat_abs_det_basis_change | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"basis",
"basis.det_comp_basis",
"equiv.refl",
"fintype",
"ideal",
"ideal.abs_norm",
"linear_map.det",
"restrict_scalars",
"submodule.subtype"
] | Let `b` be a basis for `S` over `ℤ` and `bI` a basis for `I` over `ℤ` of the same dimension.
Then an alternative way to compute the norm of `I` is given by taking the determinant of `bI`
over `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_norm_span_singleton (r : S) :
abs_norm (span ({r} : set S)) = (algebra.norm ℤ r).nat_abs | begin
rw algebra.norm_apply,
by_cases hr : r = 0,
{ simp only [hr, ideal.span_zero, algebra.coe_lmul_eq_mul, eq_self_iff_true, ideal.abs_norm_bot,
linear_map.det_zero'', set.singleton_zero, _root_.map_zero, int.nat_abs_zero] },
letI := ideal.fintype_quotient_of_free_of_ne_bot (span {r}) (mt span_singleton... | lemma | ideal.abs_norm_span_singleton | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"abs_norm",
"algebra.coe_lmul_eq_mul",
"algebra.norm",
"algebra.norm_apply",
"equiv.refl",
"ideal.abs_norm_bot",
"ideal.fintype_quotient_of_free_of_ne_bot",
"ideal.span_zero",
"linear_map.det_zero''",
"module.free.choose_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_norm_dvd_abs_norm_of_le {I J : ideal S} (h : J ≤ I) : I.abs_norm ∣ J.abs_norm | map_dvd abs_norm (dvd_iff_le.mpr h) | lemma | ideal.abs_norm_dvd_abs_norm_of_le | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"abs_norm",
"ideal",
"map_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_norm_dvd_norm_of_mem {I : ideal S} {x : S} (h : x ∈ I) : ↑I.abs_norm ∣ algebra.norm ℤ x | begin
rw [← int.dvd_nat_abs, ← abs_norm_span_singleton x, int.coe_nat_dvd],
exact abs_norm_dvd_abs_norm_of_le ((span_singleton_le_iff_mem _).mpr h)
end | lemma | ideal.abs_norm_dvd_norm_of_mem | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"algebra.norm",
"ideal",
"int.coe_nat_dvd",
"int.dvd_nat_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_norm_span_insert (r : S) (s : set S) :
abs_norm (span (insert r s)) ∣ gcd (abs_norm (span s)) (algebra.norm ℤ r).nat_abs | (dvd_gcd_iff _ _ _).mpr
⟨abs_norm_dvd_abs_norm_of_le (span_mono (set.subset_insert _ _)),
trans
(abs_norm_dvd_abs_norm_of_le (span_mono (set.singleton_subset_iff.mpr (set.mem_insert _ _))))
(by rw abs_norm_span_singleton)⟩ | lemma | ideal.abs_norm_span_insert | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"abs_norm",
"algebra.norm",
"dvd_gcd_iff",
"set.mem_insert",
"set.subset_insert"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_of_irreducible_abs_norm {I : ideal S} (hI : irreducible I.abs_norm) :
irreducible I | irreducible_iff.mpr
⟨λ h, hI.not_unit (by simpa only [ideal.is_unit_iff, nat.is_unit_iff, abs_norm_eq_one_iff]
using h),
by rintro a b rfl; simpa only [ideal.is_unit_iff, nat.is_unit_iff, abs_norm_eq_one_iff]
using hI.is_unit_or_is_unit (_root_.map_mul abs_norm a b)⟩ | lemma | ideal.irreducible_of_irreducible_abs_norm | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"abs_norm",
"ideal",
"ideal.is_unit_iff",
"irreducible",
"nat.is_unit_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime_of_irreducible_abs_norm {I : ideal S} (hI : irreducible I.abs_norm) :
I.is_prime | is_prime_of_prime (unique_factorization_monoid.irreducible_iff_prime.mp
(irreducible_of_irreducible_abs_norm hI)) | lemma | ideal.is_prime_of_irreducible_abs_norm | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal",
"irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_of_irreducible_abs_norm_span {a : S} (ha : a ≠ 0)
(hI : irreducible (ideal.span ({a} : set S)).abs_norm) :
prime a | (ideal.span_singleton_prime ha).mp (is_prime_of_irreducible_abs_norm hI) | lemma | ideal.prime_of_irreducible_abs_norm_span | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"abs_norm",
"ideal.span",
"ideal.span_singleton_prime",
"irreducible",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_norm_mem (I : ideal S) : ↑I.abs_norm ∈ I | by rw [abs_norm_apply, card_quot, ← ideal.quotient.eq_zero_iff_mem, map_nat_cast,
quotient.index_eq_zero] | lemma | ideal.abs_norm_mem | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal",
"ideal.quotient.eq_zero_iff_mem",
"map_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_abs_norm_le (I : ideal S) :
ideal.span { (ideal.abs_norm I : S) } ≤ I | by simp only [ideal.span_le, set.singleton_subset_iff, set_like.mem_coe, ideal.abs_norm_mem I] | lemma | ideal.span_singleton_abs_norm_le | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal",
"ideal.abs_norm",
"ideal.abs_norm_mem",
"ideal.span",
"ideal.span_le",
"set.singleton_subset_iff",
"set_like.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_set_of_abs_norm_eq [char_zero S] {n : ℕ} (hn : 0 < n) :
{ I : ideal S | ideal.abs_norm I = n }.finite | begin
let f := λ I : ideal S, ideal.map (ideal.quotient.mk (@ideal.span S _ {n})) I,
refine @set.finite.of_finite_image _ _ _ f _ _,
{ suffices : finite (S ⧸ @ideal.span S _ {n}),
{ let g := (coe : ideal (S ⧸ @ideal.span S _ {n}) → set (S ⧸ @ideal.span S _ {n})),
refine @set.finite.of_finite_image _ _ _... | lemma | ideal.finite_set_of_abs_norm_eq | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"algebra.norm_eq_zero_iff",
"char_zero",
"finite",
"ideal",
"ideal.abs_norm",
"ideal.comap",
"ideal.map",
"ideal.quotient.mk",
"ideal.span",
"int.nat_abs_eq_zero",
"nat.cast_eq_zero",
"set.finite'",
"set.finite.of_finite_image",
"set.finite.subset",
"set.finite_univ",
"set.subset_univ"... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_norm (I : ideal S) : ideal R | ideal.span (algebra.norm R '' (I : set S)) | def | ideal.span_norm | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"algebra.norm",
"ideal",
"ideal.span"
] | `ideal.span_norm R (I : ideal S)` is the ideal generated by mapping `algebra.norm R` over `I`.
See also `ideal.rel_norm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_norm_bot
[nontrivial S] [module.free R S] [module.finite R S] :
span_norm R (⊥ : ideal S) = ⊥ | span_eq_bot.mpr (λ x hx, by simpa using hx) | lemma | ideal.span_norm_bot | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal",
"module.finite",
"module.free",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_norm_eq_bot_iff [is_domain R] [is_domain S]
[module.free R S] [module.finite R S] {I : ideal S} :
span_norm R I = ⊥ ↔ I = ⊥ | begin
simp only [span_norm, ideal.span_eq_bot, set.mem_image, set_like.mem_coe, forall_exists_index,
and_imp, forall_apply_eq_imp_iff₂,
algebra.norm_eq_zero_iff_of_basis (module.free.choose_basis R S), @eq_bot_iff _ _ _ I,
set_like.le_def],
refl
end | lemma | ideal.span_norm_eq_bot_iff | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"algebra.norm_eq_zero_iff_of_basis",
"and_imp",
"eq_bot_iff",
"forall_apply_eq_imp_iff₂",
"forall_exists_index",
"ideal",
"ideal.span_eq_bot",
"is_domain",
"module.finite",
"module.free",
"module.free.choose_basis",
"set.mem_image",
"set_like.le_def",
"set_like.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_mem_span_norm {I : ideal S} (x : S) (hx : x ∈ I) : algebra.norm R x ∈ I.span_norm R | subset_span (set.mem_image_of_mem _ hx) | lemma | ideal.norm_mem_span_norm | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"algebra.norm",
"ideal",
"set.mem_image_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_norm_singleton {r : S} :
span_norm R (span ({r} : set S)) = span {algebra.norm R r} | le_antisymm
(span_le.mpr (λ x hx, mem_span_singleton.mpr begin
obtain ⟨x, hx', rfl⟩ := (set.mem_image _ _ _).mp hx,
exact map_dvd _ (mem_span_singleton.mp hx')
end))
((span_singleton_le_iff_mem _).mpr (norm_mem_span_norm _ _ (mem_span_singleton_self _))) | lemma | ideal.span_norm_singleton | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"algebra.norm",
"map_dvd",
"set.mem_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_norm_top : span_norm R (⊤ : ideal S) = ⊤ | by simp [← ideal.span_singleton_one] | lemma | ideal.span_norm_top | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal",
"ideal.span_singleton_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_span_norm (I : ideal S) {T : Type*} [comm_ring T] (f : R →+* T) :
map f (span_norm R I) = span ((f ∘ algebra.norm R) '' (I : set S)) | by rw [span_norm, map_span, set.image_image] | lemma | ideal.map_span_norm | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"algebra.norm",
"comm_ring",
"ideal",
"set.image_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_norm_mono {I J : ideal S} (h : I ≤ J) : span_norm R I ≤ span_norm R J | ideal.span_mono (set.monotone_image h) | lemma | ideal.span_norm_mono | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal",
"ideal.span_mono",
"set.monotone_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_norm_localization (I : ideal S) [module.finite R S] [module.free R S]
(M : submonoid R) {Rₘ : Type*} (Sₘ : Type*)
[comm_ring Rₘ] [algebra R Rₘ] [comm_ring Sₘ] [algebra S Sₘ]
[algebra Rₘ Sₘ] [algebra R Sₘ] [is_scalar_tower R Rₘ Sₘ] [is_scalar_tower R S Sₘ]
[is_localization M Rₘ] [is_localization (algebra.al... | begin
casesI h : subsingleton_or_nontrivial R,
{ haveI := is_localization.unique R Rₘ M,
simp },
let b := module.free.choose_basis R S,
rw map_span_norm,
refine span_eq_span (set.image_subset_iff.mpr _) (set.image_subset_iff.mpr _),
{ rintros a' ha',
simp only [set.mem_preimage, submodule_span_eq, ←... | lemma | ideal.span_norm_localization | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"algebra",
"algebra.algebra_map_submonoid",
"algebra.norm",
"algebra.norm_algebra_map_of_basis",
"algebra.norm_localization",
"algebra_map",
"comm_ring",
"fintype.card",
"ideal",
"is_localization",
"is_localization.mem_map_algebra_map_iff",
"is_localization.unique",
"is_scalar_tower",
"is_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_norm_mul_span_norm_le (I J : ideal S) :
span_norm R I * span_norm R J ≤ span_norm R (I * J) | begin
rw [span_norm, span_norm, span_norm, ideal.span_mul_span', ← set.image_mul],
refine ideal.span_mono (set.monotone_image _),
rintros _ ⟨x, y, hxI, hyJ, rfl⟩,
exact ideal.mul_mem_mul hxI hyJ
end | lemma | ideal.span_norm_mul_span_norm_le | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal",
"ideal.mul_mem_mul",
"ideal.span_mono",
"ideal.span_mul_span'",
"set.image_mul",
"set.monotone_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_norm_mul_of_bot_or_top [is_domain R] [is_domain S]
[module.free R S] [module.finite R S]
(eq_bot_or_top : ∀ I : ideal R, I = ⊥ ∨ I = ⊤)
(I J : ideal S) :
span_norm R (I * J) = span_norm R I * span_norm R J | begin
refine le_antisymm _ (span_norm_mul_span_norm_le _ _ _),
cases eq_bot_or_top (span_norm R I) with hI hI,
{ rw [hI, span_norm_eq_bot_iff.mp hI, bot_mul, span_norm_bot],
exact bot_le },
rw [hI, ideal.top_mul],
cases eq_bot_or_top (span_norm R J) with hJ hJ,
{ rw [hJ, span_norm_eq_bot_iff.mp hJ, mul_... | lemma | ideal.span_norm_mul_of_bot_or_top | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"bot_le",
"ideal",
"ideal.top_mul",
"is_domain",
"le_top",
"module.finite",
"module.free"
] | This condition `eq_bot_or_top` is equivalent to being a field.
However, `span_norm_mul_of_field` is harder to apply since we'd need to upgrade a `comm_ring R`
instance to a `field R` instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_norm_mul_of_field {K : Type*} [field K] [algebra K S] [is_domain S]
[module.finite K S] (I J : ideal S) :
span_norm K (I * J) = span_norm K I * span_norm K J | span_norm_mul_of_bot_or_top K eq_bot_or_top I J | lemma | ideal.span_norm_mul_of_field | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"algebra",
"field",
"ideal",
"is_domain",
"module.finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_norm_mul (I J : ideal S) : span_norm R (I * J) = span_norm R I * span_norm R J | begin
nontriviality R,
casesI subsingleton_or_nontrivial S,
{ have : ∀ I : ideal S, I = ⊤ := λ I, subsingleton.elim I ⊤,
simp [this I, this J, this (I * J)] },
refine eq_of_localization_maximal _,
unfreezingI { intros P hP },
by_cases hP0 : P = ⊥,
{ unfreezingI { subst hP0 },
rw span_norm_mul_of_b... | lemma | ideal.span_norm_mul | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"algebra",
"algebra.algebra_map_submonoid",
"bot_le",
"classical.dec_eq",
"ideal",
"ideal.map_mul",
"is_dedekind_domain",
"is_dedekind_domain.is_principal_ideal_ring_localization_over_prime",
"is_domain",
"is_localization.is_dedekind_domain",
"is_localization.is_domain_localization",
"is_local... | Multiplicativity of `ideal.span_norm`. simp-normal form is `map_mul (ideal.rel_norm R)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_norm : ideal S →*₀ ideal R | { to_fun := span_norm R,
map_zero' := span_norm_bot R,
map_one' := by rw [one_eq_top, span_norm_top R, one_eq_top],
map_mul' := span_norm_mul R } | def | ideal.rel_norm | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal"
] | The relative norm `ideal.rel_norm R (I : ideal S)`, where `R` and `S` are Dedekind domains,
and `S` is an extension of `R` that is finite and free as a module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_norm_apply (I : ideal S) :
rel_norm R I = span (algebra.norm R '' (I : set S) : set R) | rfl | lemma | ideal.rel_norm_apply | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"algebra.norm",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_norm_eq (I : ideal S) : span_norm R I = rel_norm R I | rfl | lemma | ideal.span_norm_eq | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_norm_bot : rel_norm R (⊥ : ideal S) = ⊥ | by simpa only [zero_eq_bot] using map_zero (rel_norm R : ideal S →*₀ _) | lemma | ideal.rel_norm_bot | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_norm_top : rel_norm R (⊤ : ideal S) = ⊤ | by simpa only [one_eq_top] using map_one (rel_norm R : ideal S →*₀ _) | lemma | ideal.rel_norm_top | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal",
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_norm_eq_bot_iff {I : ideal S} : rel_norm R I = ⊥ ↔ I = ⊥ | span_norm_eq_bot_iff | lemma | ideal.rel_norm_eq_bot_iff | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_mem_rel_norm (I : ideal S) {x : S} (hx : x ∈ I) : algebra.norm R x ∈ rel_norm R I | norm_mem_span_norm R x hx | lemma | ideal.norm_mem_rel_norm | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"algebra.norm",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_norm_singleton (r : S) :
rel_norm R (span ({r} : set S)) = span {algebra.norm R r} | span_norm_singleton R | lemma | ideal.rel_norm_singleton | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"algebra.norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_rel_norm (I : ideal S) {T : Type*} [comm_ring T] (f : R →+* T) :
map f (rel_norm R I) = span ((f ∘ algebra.norm R) '' (I : set S)) | map_span_norm R I f | lemma | ideal.map_rel_norm | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"algebra.norm",
"comm_ring",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_norm_mono {I J : ideal S} (h : I ≤ J) : rel_norm R I ≤ rel_norm R J | span_norm_mono R h | lemma | ideal.rel_norm_mono | ring_theory.ideal | src/ring_theory/ideal/norm.lean | [
"algebra.char_p.quotient",
"data.finsupp.fintype",
"data.int.absolute_value",
"data.int.associated",
"linear_algebra.free_module.determinant",
"linear_algebra.free_module.ideal_quotient",
"ring_theory.dedekind_domain.pid",
"ring_theory.local_properties",
"ring_theory.localization.norm"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_smul' : has_smul (ideal R) (submodule R M) | ⟨submodule.map₂ (linear_map.lsmul R M)⟩ | instance | submodule.has_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"
] | [
"has_smul",
"ideal",
"linear_map.lsmul",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.ideal.smul_eq_mul (I J : ideal R) : I • J = I * J | rfl | lemma | ideal.smul_eq_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"
] | This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to
apply. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
annihilator (N : submodule R M) : ideal R | (linear_map.lsmul R N).ker | def | submodule.annihilator | 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.lsmul",
"submodule"
] | `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0:M) | ⟨λ hr n hn, congr_arg subtype.val (linear_map.ext_iff.1 (linear_map.mem_ker.1 hr) ⟨n, hn⟩),
λ h, linear_map.mem_ker.2 $ linear_map.ext $ λ n, subtype.eq $ h n.1 n.2⟩ | theorem | submodule.mem_annihilator | 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.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (linear_map.id : M →ₗ[R] M)) ⊥ | mem_annihilator.trans ⟨λ H n hn, (mem_bot R).2 $ H n hn, λ H n hn, (mem_bot R).1 $ H hn⟩ | theorem | submodule.mem_annihilator' | 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 | |
mem_annihilator_span (s : set M) (r : R) :
r ∈ (submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 | begin
rw submodule.mem_annihilator,
split,
{ intros h n, exact h _ (submodule.subset_span n.prop) },
{ intros h n hn,
apply submodule.span_induction hn,
{ intros x hx, exact h ⟨x, hx⟩ },
{ exact smul_zero _ },
{ intros x y hx hy, rw [smul_add, hx, hy, zero_add] },
{ intros a x hx, rw [smul_c... | lemma | submodule.mem_annihilator_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"
] | [
"smul_add",
"smul_zero",
"submodule.mem_annihilator",
"submodule.span",
"submodule.span_induction",
"submodule.subset_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_annihilator_span_singleton (g : M) (r : R) :
r ∈ (submodule.span R ({g} : set M)).annihilator ↔ r • g = 0 | by simp [mem_annihilator_span] | lemma | submodule.mem_annihilator_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"
] | [
"submodule.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
annihilator_bot : (⊥ : submodule R M).annihilator = ⊤ | (ideal.eq_top_iff_one _).2 $ mem_annihilator'.2 bot_le | theorem | submodule.annihilator_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"
] | [
"bot_le",
"ideal.eq_top_iff_one",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ | ⟨λ H, eq_bot_iff.2 $ λ (n:M) hn, (mem_bot R).2 $
one_smul R n ▸ mem_annihilator.1 ((ideal.eq_top_iff_one _).1 H) n hn,
λ H, H.symm ▸ annihilator_bot⟩ | theorem | submodule.annihilator_eq_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"
] | [
"ideal.eq_top_iff_one",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator | λ r hrp, mem_annihilator.2 $ λ n hn, mem_annihilator.1 hrp n $ h hn | theorem | submodule.annihilator_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 | |
annihilator_supr (ι : Sort w) (f : ι → submodule R M) :
(annihilator ⨆ i, f i) = ⨅ i, annihilator (f i) | le_antisymm (le_infi $ λ i, annihilator_mono $ le_supr _ _)
(λ r H, mem_annihilator'.2 $ supr_le $ λ i,
have _ := (mem_infi _).1 H i, mem_annihilator'.1 this) | theorem | submodule.annihilator_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"
] | [
"le_infi",
"le_supr",
"submodule",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N | apply_mem_map₂ _ hr hn | theorem | submodule.smul_mem_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_le {P : submodule R M} : I • N ≤ P ↔ ∀ (r ∈ I) (n ∈ N), r • n ∈ P | map₂_le | theorem | submodule.smul_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N)
(Hb : ∀ (r ∈ I) (n ∈ N), p (r • n))
(H1 : ∀ x y, p x → p y → p (x + y)) : p x | begin
have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem,
refine submodule.supr_induction _ H _ H0 H1,
rintros ⟨i, hi⟩ m ⟨j, hj, (rfl : i • _ = m) ⟩,
exact Hb _ hi _ hj,
end | theorem | submodule.smul_induction_on | 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.supr_induction",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_induction_on' {x : M} (hx : x ∈ I • N)
{p : Π x, x ∈ I • N → Prop}
(Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N),
p (r • n) (smul_mem_smul hr hn))
(H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (submodule.add_mem _ ‹_› ‹_›)) :
p x hx | begin
refine exists.elim _ (λ (h : x ∈ I • N) (H : p x h), H),
exact smul_induction_on hx
(λ a ha x hx, ⟨_, Hb _ ha _ hx⟩)
(λ x y ⟨_, hx⟩ ⟨_, hy⟩, ⟨_, H1 _ _ _ _ hx hy⟩),
end | theorem | submodule.smul_induction_on' | 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.add_mem"
] | Dependent version of `submodule.smul_induction_on`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_smul_span_singleton {I : ideal R} {m : M} {x : M} :
x ∈ I • span R ({m} : set M) ↔ ∃ y ∈ I, y • m = x | ⟨λ hx, smul_induction_on hx
(λ r hri n hnm,
let ⟨s, hs⟩ := mem_span_singleton.1 hnm in ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩)
(λ m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩,
⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩),
λ ⟨y, hyi, hy⟩, hy ▸ smul_mem_smul hyi (subset_span $ set.mem_sin... | theorem | submodule.mem_smul_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"
] | [
"add_smul",
"ideal",
"set.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_le_right : I • N ≤ N | smul_le.2 $ λ r hr n, N.smul_mem r | theorem | submodule.smul_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 | |
smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P | map₂_le_map₂ hij hnp | theorem | submodule.smul_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 | |
smul_mono_left (h : I ≤ J) : I • N ≤ J • N | map₂_le_map₂_left h | theorem | submodule.smul_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mono_right (h : N ≤ P) : I • N ≤ I • P | map₂_le_map₂_right h | theorem | submodule.smul_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_smul_top (I : ideal R) (f : R →ₗ[R] M) :
submodule.map f I ≤ I • (⊤ : submodule R M) | begin
rintros _ ⟨y, hy, rfl⟩,
rw [← mul_one y, ← smul_eq_mul, f.map_smul],
exact smul_mem_smul hy mem_top
end | lemma | submodule.map_le_smul_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_one",
"smul_eq_mul",
"submodule",
"submodule.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
annihilator_smul (N : submodule R M) : annihilator N • N = ⊥ | eq_bot_iff.2 (smul_le.2 (λ r, mem_annihilator.1)) | theorem | submodule.annihilator_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"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
annihilator_mul (I : ideal R) : annihilator I * I = ⊥ | annihilator_smul I | theorem | submodule.annihilator_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 | |
mul_annihilator (I : ideal R) : I * annihilator I = ⊥ | by rw [mul_comm, annihilator_mul] | theorem | submodule.mul_annihilator | 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 | |
smul_bot : I • (⊥ : submodule R M) = ⊥ | map₂_bot_right _ _ | theorem | submodule.smul_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_smul : (⊥ : ideal R) • N = ⊥ | map₂_bot_left _ _ | theorem | submodule.bot_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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