statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
coe_zero : ((0 : power_series R) : laurent_series R) = 0 | (of_power_series ℤ R).map_zero | lemma | power_series.coe_zero | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : power_series R) : laurent_series R) = 1 | (of_power_series ℤ R).map_one | lemma | power_series.coe_one | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"map_one",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add : ((f + g : power_series R) : laurent_series R) = f + g | (of_power_series ℤ R).map_add _ _ | lemma | power_series.coe_add | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub : ((f' - g' : power_series R') : laurent_series R') = f' - g' | (of_power_series ℤ R').map_sub _ _ | lemma | power_series.coe_sub | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg : ((-f' : power_series R') : laurent_series R') = -f' | (of_power_series ℤ R').map_neg _ | lemma | power_series.coe_neg | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul : ((f * g : power_series R) : laurent_series R) = f * g | (of_power_series ℤ R).map_mul _ _ | lemma | power_series.coe_mul | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"map_mul",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_coe (i : ℤ) :
((f : power_series R) : laurent_series R).coeff i =
if i < 0 then 0 else power_series.coeff R i.nat_abs f | begin
cases i,
{ rw [int.nat_abs_of_nat_core, int.of_nat_eq_coe, coeff_coe_power_series,
if_neg (int.coe_nat_nonneg _).not_lt] },
{ rw [coe_power_series, of_power_series_apply, emb_domain_notin_image_support,
if_pos (int.neg_succ_lt_zero _)],
simp only [not_exists, rel_embedding.coe_fn_mk, set... | lemma | power_series.coeff_coe | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"function.embedding.coe_fn_mk",
"int.coe_nat_nonneg",
"laurent_series",
"not_and",
"not_exists",
"power_series",
"power_series.coeff",
"rel_embedding.coe_fn_mk",
"set.mem_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_C (r : R) : ((C R r : power_series R) : laurent_series R) =
hahn_series.C r | of_power_series_C _ | lemma | power_series.coe_C | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"hahn_series.C",
"laurent_series",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_X : ((X : power_series R) : laurent_series R) = single 1 1 | of_power_series_X | lemma | power_series.coe_X | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul {S : Type*} [semiring S] [module R S]
(r : R) (x : power_series S) : ((r • x : power_series S) : laurent_series S) = r • x | by { ext, simp [coeff_coe, coeff_smul, smul_ite] } | lemma | power_series.coe_smul | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"module",
"power_series",
"semiring",
"smul_ite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bit0 :
((bit0 f : power_series R) : laurent_series R) = bit0 f | (of_power_series ℤ R).map_bit0 _ | lemma | power_series.coe_bit0 | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"map_bit0",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bit1 :
((bit1 f : power_series R) : laurent_series R) = bit1 f | (of_power_series ℤ R).map_bit1 _ | lemma | power_series.coe_bit1 | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"map_bit1",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow (n : ℕ) :
((f ^ n : power_series R) : laurent_series R) = f ^ n | (of_power_series ℤ R).map_pow _ _ | lemma | power_series.coe_pow | ring_theory | src/ring_theory/laurent_series.lean | [
"ring_theory.hahn_series",
"ring_theory.localization.fraction_ring"
] | [
"laurent_series",
"map_pow",
"power_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_preserves : Prop | ∀ {R : Type u} [hR : comm_ring R] (M : by exactI submonoid R) (S : Type u) [hS : comm_ring S]
[by exactI algebra R S] [by exactI is_localization M S], @P R hR → @P S hS | def | localization_preserves | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra",
"comm_ring",
"is_localization",
"submonoid"
] | A property `P` of comm rings is said to be preserved by localization
if `P` holds for `M⁻¹R` whenever `P` holds for `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_localization_maximal : Prop | ∀ (R : Type u) [comm_ring R],
by exactI (∀ (J : ideal R) (hJ : J.is_maximal), by exactI P (localization.at_prime J)) → P R | def | of_localization_maximal | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"comm_ring",
"ideal",
"localization.at_prime"
] | A property `P` of comm rings satisfies `of_localization_maximal` if
if `P` holds for `R` whenever `P` holds for `Rₘ` for all maximal ideal `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.localization_preserves | ∀ ⦃R S : Type u⦄ [comm_ring R] [comm_ring S] (f : by exactI R →+* S) (M : by exactI submonoid R)
(R' S' : Type u) [comm_ring R'] [comm_ring S'] [by exactI algebra R R']
[by exactI algebra S S'] [by exactI is_localization M R']
[by exactI is_localization (M.map f) S'],
by exactI (P f → P (is_localization... | def | ring_hom.localization_preserves | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra",
"comm_ring",
"is_localization",
"is_localization.map",
"submonoid",
"submonoid.le_comap_map"
] | A property `P` of ring homs is said to be preserved by localization
if `P` holds for `M⁻¹R →+* M⁻¹S` whenever `P` holds for `R →+* S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.of_localization_finite_span | ∀ ⦃R S : Type u⦄ [comm_ring R] [comm_ring S] (f : by exactI R →+* S)
(s : finset R) (hs : by exactI ideal.span (s : set R) = ⊤)
(H : by exactI (∀ (r : s), P (localization.away_map f r))), by exactI P f | def | ring_hom.of_localization_finite_span | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"comm_ring",
"finset",
"ideal.span",
"localization.away_map"
] | A property `P` of ring homs satisfies `ring_hom.of_localization_finite_span`
if `P` holds for `R →+* S` whenever there exists a finite set `{ r }` that spans `R` such that
`P` holds for `Rᵣ →+* Sᵣ`.
Note that this is equivalent to `ring_hom.of_localization_span` via
`ring_hom.of_localization_span_iff_finite`, but this... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.of_localization_span | ∀ ⦃R S : Type u⦄ [comm_ring R] [comm_ring S] (f : by exactI R →+* S)
(s : set R) (hs : by exactI ideal.span s = ⊤)
(H : by exactI (∀ (r : s), P (localization.away_map f r))), by exactI P f | def | ring_hom.of_localization_span | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"comm_ring",
"ideal.span",
"localization.away_map"
] | A property `P` of ring homs satisfies `ring_hom.of_localization_finite_span`
if `P` holds for `R →+* S` whenever there exists a set `{ r }` that spans `R` such that
`P` holds for `Rᵣ →+* Sᵣ`.
Note that this is equivalent to `ring_hom.of_localization_finite_span` via
`ring_hom.of_localization_span_iff_finite`, but this... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.holds_for_localization_away : Prop | ∀ ⦃R : Type u⦄ (S : Type u) [comm_ring R] [comm_ring S] [by exactI algebra R S] (r : R)
[by exactI is_localization.away r S], by exactI P (algebra_map R S) | def | ring_hom.holds_for_localization_away | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra",
"algebra_map",
"comm_ring",
"is_localization.away"
] | A property `P` of ring homs satisfies `ring_hom.holds_for_localization_away`
if `P` holds for each localization map `R →+* Rᵣ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.of_localization_finite_span_target : Prop | ∀ ⦃R S : Type u⦄ [comm_ring R] [comm_ring S] (f : by exactI R →+* S)
(s : finset S) (hs : by exactI ideal.span (s : set S) = ⊤)
(H : by exactI (∀ (r : s), P ((algebra_map S (localization.away (r : S))).comp f))),
by exactI P f | def | ring_hom.of_localization_finite_span_target | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra_map",
"comm_ring",
"finset",
"ideal.span",
"localization.away"
] | A property `P` of ring homs satisfies `ring_hom.of_localization_finite_span_target`
if `P` holds for `R →+* S` whenever there exists a finite set `{ r }` that spans `S` such that
`P` holds for `R →+* Sᵣ`.
Note that this is equivalent to `ring_hom.of_localization_span_target` via
`ring_hom.of_localization_span_target_i... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.of_localization_span_target : Prop | ∀ ⦃R S : Type u⦄ [comm_ring R] [comm_ring S] (f : by exactI R →+* S)
(s : set S) (hs : by exactI ideal.span s = ⊤)
(H : by exactI (∀ (r : s), P ((algebra_map S (localization.away (r : S))).comp f))),
by exactI P f | def | ring_hom.of_localization_span_target | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra_map",
"comm_ring",
"ideal.span",
"localization.away"
] | A property `P` of ring homs satisfies `ring_hom.of_localization_span_target`
if `P` holds for `R →+* S` whenever there exists a set `{ r }` that spans `S` such that
`P` holds for `R →+* Sᵣ`.
Note that this is equivalent to `ring_hom.of_localization_finite_span_target` via
`ring_hom.of_localization_span_target_iff_fini... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.of_localization_prime : Prop | ∀ ⦃R S : Type u⦄ [comm_ring R] [comm_ring S] (f : by exactI R →+* S),
by exactI (∀ (J : ideal S) (hJ : J.is_prime),
by exactI P (localization.local_ring_hom _ J f rfl)) → P f | def | ring_hom.of_localization_prime | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"comm_ring",
"ideal",
"localization.local_ring_hom"
] | A property `P` of ring homs satisfies `of_localization_prime` if
if `P` holds for `R` whenever `P` holds for `Rₘ` for all prime ideals `p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.property_is_local : Prop | (localization_preserves : ring_hom.localization_preserves @P)
(of_localization_span_target : ring_hom.of_localization_span_target @P)
(stable_under_composition : ring_hom.stable_under_composition @P)
(holds_for_localization_away : ring_hom.holds_for_localization_away @P) | structure | ring_hom.property_is_local | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"localization_preserves",
"ring_hom.holds_for_localization_away",
"ring_hom.localization_preserves",
"ring_hom.of_localization_span_target",
"ring_hom.stable_under_composition"
] | A property of ring homs is local if it is preserved by localizations and compositions, and for
each `{ r }` that spans `S`, we have `P (R →+* S) ↔ ∀ r, P (R →+* Sᵣ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.of_localization_span_iff_finite :
ring_hom.of_localization_span @P ↔ ring_hom.of_localization_finite_span @P | begin
delta ring_hom.of_localization_span ring_hom.of_localization_finite_span,
apply forall₅_congr, -- TODO: Using `refine` here breaks `resetI`.
introsI,
split,
{ intros h s, exact h s },
{ intros h s hs hs',
obtain ⟨s', h₁, h₂⟩ := (ideal.span_eq_top_iff_finite s).mp hs,
exact h s' h₂ (λ x, hs' ⟨_... | lemma | ring_hom.of_localization_span_iff_finite | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"forall₅_congr",
"ideal.span_eq_top_iff_finite",
"ring_hom.of_localization_finite_span",
"ring_hom.of_localization_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.of_localization_span_target_iff_finite :
ring_hom.of_localization_span_target @P ↔ ring_hom.of_localization_finite_span_target @P | begin
delta ring_hom.of_localization_span_target ring_hom.of_localization_finite_span_target,
apply forall₅_congr, -- TODO: Using `refine` here breaks `resetI`.
introsI,
split,
{ intros h s, exact h s },
{ intros h s hs hs',
obtain ⟨s', h₁, h₂⟩ := (ideal.span_eq_top_iff_finite s).mp hs,
exact h s' h... | lemma | ring_hom.of_localization_span_target_iff_finite | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"forall₅_congr",
"ideal.span_eq_top_iff_finite",
"ring_hom.of_localization_finite_span_target",
"ring_hom.of_localization_span_target"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.ring_hom.property_is_local.respects_iso (hP : ring_hom.property_is_local @P) :
ring_hom.respects_iso @P | begin
apply hP.stable_under_composition.respects_iso,
introv,
resetI,
letI := e.to_ring_hom.to_algebra,
apply_with hP.holds_for_localization_away { instances := ff },
apply is_localization.away_of_is_unit_of_bijective _ is_unit_one,
exact e.bijective
end | lemma | ring_hom.property_is_local.respects_iso | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"is_localization.away_of_is_unit_of_bijective",
"is_unit_one",
"ring_hom.property_is_local",
"ring_hom.respects_iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.localization_preserves.away
(H : ring_hom.localization_preserves @P) (r : R) [is_localization.away r R']
[is_localization.away (f r) S'] (hf : P f) :
P (by exactI is_localization.away.map R' S' f r) | begin
resetI,
haveI : is_localization ((submonoid.powers r).map f) S',
{ rw submonoid.map_powers, assumption },
exact H f (submonoid.powers r) R' S' hf,
end | lemma | ring_hom.localization_preserves.away | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"is_localization",
"is_localization.away",
"is_localization.away.map",
"ring_hom.localization_preserves",
"submonoid.map_powers",
"submonoid.powers"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.property_is_local.of_localization_span (hP : ring_hom.property_is_local @P) :
ring_hom.of_localization_span @P | begin
introv R hs hs',
resetI,
apply_fun (ideal.map f) at hs,
rw [ideal.map_span, ideal.map_top] at hs,
apply hP.of_localization_span_target _ _ hs,
rintro ⟨_, r, hr, rfl⟩,
have := hs' ⟨r, hr⟩,
convert hP.stable_under_composition _ _ (hP.holds_for_localization_away (localization.away r) r)
(hs' ⟨r, ... | lemma | ring_hom.property_is_local.of_localization_span | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"ideal.map",
"ideal.map_span",
"ideal.map_top",
"is_localization.map_comp",
"localization.away",
"ring_hom.of_localization_span",
"ring_hom.property_is_local"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.le_of_localization_maximal {I J : ideal R}
(h : ∀ (P : ideal R) (hP : P.is_maximal),
ideal.map (algebra_map R (by exactI localization.at_prime P)) I ≤
ideal.map (algebra_map R (by exactI localization.at_prime P)) J) :
I ≤ J | begin
intros x hx,
suffices : J.colon (ideal.span {x}) = ⊤,
{ simpa using submodule.mem_colon.mp
(show (1 : R) ∈ J.colon (ideal.span {x}), from this.symm ▸ submodule.mem_top)
x (ideal.mem_span_singleton_self x) },
refine not.imp_symm (J.colon (ideal.span {x})).exists_le_maximal _,
push_neg,
intr... | lemma | ideal.le_of_localization_maximal | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra_map",
"ideal",
"ideal.map",
"ideal.mem_map_of_mem",
"ideal.mem_span_singleton_self",
"ideal.span",
"is_localization.map_eq_zero_iff",
"is_localization.mem_map_algebra_map_iff",
"localization.at_prime",
"mul_assoc",
"mul_comm",
"mul_left_comm",
"not.imp_symm",
"submodule.mem_top",
... | Let `I J : ideal R`. If the localization of `I` at each maximal ideal `P` is included in
the localization of `J` at `P`, then `I ≤ J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.eq_of_localization_maximal {I J : ideal R}
(h : ∀ (P : ideal R) (hP : P.is_maximal),
ideal.map (algebra_map R (by exactI localization.at_prime P)) I =
ideal.map (algebra_map R (by exactI localization.at_prime P)) J) :
I = J | le_antisymm
(ideal.le_of_localization_maximal (λ P hP, (h P hP).le))
(ideal.le_of_localization_maximal (λ P hP, (h P hP).ge)) | theorem | ideal.eq_of_localization_maximal | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra_map",
"ideal",
"ideal.le_of_localization_maximal",
"ideal.map",
"localization.at_prime"
] | Let `I J : ideal R`. If the localization of `I` at each maximal ideal `P` is equal to
the localization of `J` at `P`, then `I = J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_eq_bot_of_localization' (I : ideal R)
(h : ∀ (J : ideal R) (hJ : J.is_maximal),
ideal.map (algebra_map R (by exactI (localization.at_prime J))) I = ⊥) : I = ⊥ | ideal.eq_of_localization_maximal (λ P hP, (by simpa using h P hP)) | lemma | ideal_eq_bot_of_localization' | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra_map",
"ideal",
"ideal.eq_of_localization_maximal",
"ideal.map",
"localization.at_prime"
] | An ideal is trivial if its localization at every maximal ideal is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal_eq_bot_of_localization (I : ideal R)
(h : ∀ (J : ideal R) (hJ : J.is_maximal),
by exactI is_localization.coe_submodule (localization.at_prime J) I = ⊥) : I = ⊥ | ideal_eq_bot_of_localization' _ (λ P hP, (ideal.map_eq_bot_iff_le_ker _).mpr (λ x hx,
by { rw [ring_hom.mem_ker, ← submodule.mem_bot R, ← h P hP, is_localization.mem_coe_submodule],
exact ⟨x, hx, rfl⟩ })) | lemma | ideal_eq_bot_of_localization | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"ideal",
"ideal.map_eq_bot_iff_le_ker",
"ideal_eq_bot_of_localization'",
"is_localization.coe_submodule",
"is_localization.mem_coe_submodule",
"localization.at_prime",
"ring_hom.mem_ker",
"submodule.mem_bot"
] | An ideal is trivial if its localization at every maximal ideal is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_of_localization (r : R)
(h : ∀ (J : ideal R) (hJ : J.is_maximal),
by exactI algebra_map R (localization.at_prime J) r = 0) : r = 0 | begin
rw ← ideal.span_singleton_eq_bot,
apply ideal_eq_bot_of_localization,
intros J hJ,
delta is_localization.coe_submodule,
erw [submodule.map_span, submodule.span_eq_bot],
rintro _ ⟨_, h', rfl⟩,
cases set.mem_singleton_iff.mpr h',
exact h J hJ,
end | lemma | eq_zero_of_localization | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra_map",
"ideal",
"ideal.span_singleton_eq_bot",
"ideal_eq_bot_of_localization",
"is_localization.coe_submodule",
"localization.at_prime",
"submodule.map_span",
"submodule.span_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_is_reduced : localization_preserves (λ R hR, by exactI is_reduced R) | begin
introv R _ _,
resetI,
constructor,
rintro x ⟨(_|n), e⟩,
{ simpa using congr_arg (*x) e },
obtain ⟨⟨y, m⟩, hx⟩ := is_localization.surj M x,
dsimp only at hx,
let hx' := congr_arg (^ n.succ) hx,
simp only [mul_pow, e, zero_mul, ← ring_hom.map_pow] at hx',
rw [← (algebra_map R S).map_zero] at hx'... | lemma | localization_is_reduced | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra_map",
"is_localization.map_eq_zero_iff",
"is_nilpotent.eq_zero",
"is_reduced",
"localization_preserves",
"mul_assoc",
"mul_comm",
"mul_left_comm",
"mul_left_inj",
"mul_pow",
"mul_zero",
"pow_succ",
"ring_hom.map_pow",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_reduced_of_localization_maximal :
of_localization_maximal (λ R hR, by exactI is_reduced R) | begin
introv R h,
constructor,
intros x hx,
apply eq_zero_of_localization,
intros J hJ,
specialize h J hJ,
resetI,
exact (hx.map $ algebra_map R $ localization.at_prime J).eq_zero,
end | lemma | is_reduced_of_localization_maximal | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra_map",
"eq_zero_of_localization",
"is_reduced",
"localization.at_prime",
"of_localization_maximal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_preserves_surjective :
ring_hom.localization_preserves (λ R S _ _ f, function.surjective f) | begin
introv R H x,
resetI,
obtain ⟨x, ⟨_, s, hs, rfl⟩, rfl⟩ := is_localization.mk'_surjective (M.map f) x,
obtain ⟨y, rfl⟩ := H x,
use is_localization.mk' R' y ⟨s, hs⟩,
rw is_localization.map_mk',
refl,
end | lemma | localization_preserves_surjective | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"is_localization.map_mk'",
"is_localization.mk'",
"is_localization.mk'_surjective",
"ring_hom.localization_preserves"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_of_localization_span :
ring_hom.of_localization_span (λ R S _ _ f, function.surjective f) | begin
introv R e H,
rw [← set.range_iff_surjective, set.eq_univ_iff_forall],
resetI,
letI := f.to_algebra,
intro x,
apply submodule.mem_of_span_eq_top_of_smul_pow_mem (algebra.of_id R S).to_linear_map.range s e,
intro r,
obtain ⟨a, e'⟩ := H r (algebra_map _ _ x),
obtain ⟨b, ⟨_, n, rfl⟩, rfl⟩ := is_loc... | lemma | surjective_of_localization_span | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra.of_id",
"algebra_map",
"is_localization.eq_mk'_iff_mul_eq",
"is_localization.map_mk'",
"is_localization.mk'_surjective",
"map_mul",
"map_pow",
"mul_comm",
"pow_add",
"ring_hom.of_localization_span",
"set.eq_univ_iff_forall",
"set.range_iff_surjective",
"submodule.mem_of_span_eq_top_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_finite : ring_hom.localization_preserves @ring_hom.finite | begin
introv R hf,
-- Setting up the `algebra` and `is_scalar_tower` instances needed
resetI,
letI := f.to_algebra,
letI := ((algebra_map S S').comp f).to_algebra,
let f' : R' →+* S' := is_localization.map S' f (submonoid.le_comap_map M),
letI := f'.to_algebra,
haveI : is_scalar_tower R R' S' :=
is_... | lemma | localization_finite | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"alg_hom.mk'",
"algebra.smul_def",
"algebra_map",
"eq_top_iff",
"finset.coe_image",
"is_localization.map",
"is_localization.map_comp",
"is_localization.map_mk'",
"is_localization.mk'",
"is_localization.mk'_eq_mul_mk'_one",
"is_localization.mk'_surjective",
"is_scalar_tower",
"is_scalar_tower... | If `S` is a finite `R`-algebra, then `S' = M⁻¹S` is a finite `R' = M⁻¹R`-algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
localization_away_map_finite (r : R) [is_localization.away r R']
[is_localization.away (f r) S'] (hf : f.finite) :
(is_localization.away.map R' S' f r).finite | localization_finite.away r hf | lemma | localization_away_map_finite | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"finite",
"is_localization.away",
"is_localization.away.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization.smul_mem_finset_integer_multiple_span [algebra R S]
[algebra R S'] [is_scalar_tower R S S']
[is_localization (M.map (algebra_map R S)) S'] (x : S)
(s : finset S') (hx : algebra_map S S' x ∈ submodule.span R (s : set S')) :
∃ m : M, m • x ∈ submodule.span R
(is_localization.finset_integer... | begin
let g : S →ₐ[R] S' := alg_hom.mk' (algebra_map S S')
(λ c x, by simp [algebra.algebra_map_eq_smul_one]),
-- We first obtain the `y' ∈ M` such that `s' = y' • s` is falls in the image of `S` in `S'`.
let y := is_localization.common_denom_of_finset (M.map (algebra_map R S)) s,
have hx₁ : (y : S) • ↑s =... | lemma | is_localization.smul_mem_finset_integer_multiple_span | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"alg_hom.mk'",
"algebra",
"algebra.algebra_map_eq_smul_one",
"algebra.smul_def",
"algebra_map",
"finset",
"is_localization",
"is_localization.common_denom_of_finset",
"is_localization.finset_integer_multiple",
"is_localization.finset_integer_multiple_image",
"is_scalar_tower",
"one_smul",
"s... | Let `S` be an `R`-algebra, `M` an submonoid of `R`, and `S' = M⁻¹S`.
If the image of some `x : S` falls in the span of some finite `s ⊆ S'` over `R`,
then there exists some `m : M` such that `m • x` falls in the
span of `finset_integer_multiple _ s` over `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multiple_mem_span_of_mem_localization_span [algebra R' S] [algebra R S]
[is_scalar_tower R R' S] [is_localization M R']
(s : set S) (x : S) (hx : x ∈ submodule.span R' s) :
∃ t : M, t • x ∈ submodule.span R s | begin
classical,
obtain ⟨s', hss', hs'⟩ := submodule.mem_span_finite_of_mem_span hx,
rsuffices ⟨t, ht⟩ : ∃ t : M, t • x ∈ submodule.span R (s' : set S),
{ exact ⟨t, submodule.span_mono hss' ht⟩ },
clear hx hss' s,
revert x,
apply s'.induction_on,
{ intros x hx, use 1, simpa using hx },
rintros a s ha ... | lemma | multiple_mem_span_of_mem_localization_span | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra",
"algebra.smul_def",
"algebra_map",
"finset.coe_image",
"finset.coe_insert",
"finset.image_insert",
"is_localization",
"is_scalar_tower",
"is_scalar_tower.algebra_map_apply",
"mul_assoc",
"mul_comm",
"ring_hom.map_mul",
"smul_add",
"smul_smul",
"submodule.mem_span_finite_of_mem... | If `S` is an `R' = M⁻¹R` algebra, and `x ∈ span R' s`,
then `t • x ∈ span R s` for some `t : M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multiple_mem_adjoin_of_mem_localization_adjoin [algebra R' S] [algebra R S]
[is_scalar_tower R R' S] [is_localization M R']
(s : set S) (x : S) (hx : x ∈ algebra.adjoin R' s) :
∃ t : M, t • x ∈ algebra.adjoin R s | begin
change ∃ (t : M), t • x ∈ (algebra.adjoin R s).to_submodule,
change x ∈ (algebra.adjoin R' s).to_submodule at hx,
simp_rw [algebra.adjoin_eq_span] at hx ⊢,
exact multiple_mem_span_of_mem_localization_span M R' _ _ hx
end | lemma | multiple_mem_adjoin_of_mem_localization_adjoin | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra",
"algebra.adjoin",
"algebra.adjoin_eq_span",
"is_localization",
"is_scalar_tower",
"multiple_mem_span_of_mem_localization_span"
] | If `S` is an `R' = M⁻¹R` algebra, and `x ∈ adjoin R' s`,
then `t • x ∈ adjoin R s` for some `t : M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_of_localization_span : ring_hom.of_localization_span @ring_hom.finite | begin
rw ring_hom.of_localization_span_iff_finite,
introv R hs H,
-- We first setup the instances
resetI,
letI := f.to_algebra,
letI := λ (r : s), (localization.away_map f r).to_algebra,
haveI : ∀ r : s, is_localization ((submonoid.powers (r : R)).map (algebra_map R S))
(localization.away (f r)),
{ ... | lemma | finite_of_localization_span | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra.smul_def",
"algebra_map",
"eq_top_iff",
"is_localization",
"is_localization.finset_integer_multiple",
"is_localization.map_comp",
"is_localization.smul_mem_finset_integer_multiple_span",
"is_scalar_tower",
"is_scalar_tower.algebra_map_apply",
"is_scalar_tower.of_algebra_map_eq'",
"le_su... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_finite_type : ring_hom.localization_preserves @ring_hom.finite_type | begin
introv R hf,
-- mirrors the proof of `localization_map_finite`
resetI,
letI := f.to_algebra,
letI := ((algebra_map S S').comp f).to_algebra,
let f' : R' →+* S' := is_localization.map S' f (submonoid.le_comap_map M),
letI := f'.to_algebra,
haveI : is_scalar_tower R R' S' :=
is_scalar_tower.of_a... | lemma | localization_finite_type | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"alg_hom.mk'",
"algebra.adjoin",
"algebra.adjoin_le_iff",
"algebra.smul_def",
"algebra.subset_adjoin",
"algebra_map",
"eq_top_iff",
"finset.coe_image",
"is_localization.map",
"is_localization.map_comp",
"is_localization.map_mk'",
"is_localization.mk'",
"is_localization.mk'_eq_mul_mk'_one",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_away_map_finite_type (r : R) [is_localization.away r R']
[is_localization.away (f r) S'] (hf : f.finite_type) :
(is_localization.away.map R' S' f r).finite_type | localization_finite_type.away r hf | lemma | localization_away_map_finite_type | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"is_localization.away",
"is_localization.away.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization.exists_smul_mem_of_mem_adjoin [algebra R S]
[algebra R S'] [is_scalar_tower R S S'] (M : submonoid S)
[is_localization M S'] (x : S) (s : finset S') (A : subalgebra R S)
(hA₁ : (is_localization.finset_integer_multiple M s : set S) ⊆ A)
(hA₂ : M ≤ A.to_submonoid)
(hx : algebra_map S S' x ∈ alge... | begin
let g : S →ₐ[R] S' := is_scalar_tower.to_alg_hom R S S',
let y := is_localization.common_denom_of_finset M s,
have hx₁ : (y : S) • ↑s = g '' _ := (is_localization.finset_integer_multiple_image _ s).symm,
obtain ⟨n, hn⟩ := algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin (y : S) (s : set S')
(A.map g)... | lemma | is_localization.exists_smul_mem_of_mem_adjoin | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra",
"algebra.adjoin",
"algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin",
"algebra.smul_def",
"algebra_map",
"finset",
"is_localization",
"is_localization.common_denom_of_finset",
"is_localization.finset_integer_multiple",
"is_localization.finset_integer_multiple_image",
"is_scalar_tower... | Let `S` be an `R`-algebra, `M` a submonoid of `S`, `S' = M⁻¹S`.
Suppose the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
and `A` is an `R`-subalgebra of `S` containing both `M` and the numerators of `s`.
Then, there exists some `m : M` such that `m • x` falls in `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization.lift_mem_adjoin_finset_integer_multiple [algebra R S]
[algebra R S'] [is_scalar_tower R S S']
[is_localization (M.map (algebra_map R S)) S'] (x : S)
(s : finset S') (hx : algebra_map S S' x ∈ algebra.adjoin R (s : set S')) :
∃ m : M, m • x ∈ algebra.adjoin R
(is_localization.finset_integ... | begin
obtain ⟨⟨_, a, ha, rfl⟩, e⟩ := is_localization.exists_smul_mem_of_mem_adjoin
(M.map (algebra_map R S)) x s (algebra.adjoin R _) algebra.subset_adjoin _ hx,
{ exact ⟨⟨a, ha⟩, by simpa [submonoid.smul_def] using e⟩ },
{ rintros _ ⟨a, ha, rfl⟩, exact subalgebra.algebra_map_mem _ a }
end | lemma | is_localization.lift_mem_adjoin_finset_integer_multiple | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra",
"algebra.adjoin",
"algebra.subset_adjoin",
"algebra_map",
"finset",
"is_localization",
"is_localization.exists_smul_mem_of_mem_adjoin",
"is_localization.finset_integer_multiple",
"is_scalar_tower",
"subalgebra.algebra_map_mem",
"submonoid.smul_def"
] | Let `S` be an `R`-algebra, `M` an submonoid of `R`, and `S' = M⁻¹S`.
If the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
then there exists some `m : M` such that `m • x` falls in the
adjoin of `finset_integer_multiple _ s` over `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_type_of_localization_span : ring_hom.of_localization_span @ring_hom.finite_type | begin
rw ring_hom.of_localization_span_iff_finite,
introv R hs H,
-- mirrors the proof of `finite_of_localization_span`
resetI,
letI := f.to_algebra,
letI := λ (r : s), (localization.away_map f r).to_algebra,
haveI : ∀ r : s, is_localization ((submonoid.powers (r : R)).map (algebra_map R S))
(localiza... | lemma | finite_type_of_localization_span | ring_theory | src/ring_theory/local_properties.lean | [
"ring_theory.finite_type",
"ring_theory.localization.at_prime",
"ring_theory.localization.away.basic",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.nilpotent",
"ring_theory.ring_hom_properties"
] | [
"algebra.adjoin",
"algebra.adjoin_attach_bUnion",
"algebra.smul_def",
"algebra_map",
"eq_top_iff",
"is_localization",
"is_localization.finset_integer_multiple",
"is_localization.lift_mem_adjoin_finset_integer_multiple",
"is_localization.map_comp",
"is_scalar_tower",
"is_scalar_tower.algebra_map_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_bilinear : A →ₗ[R] matrix n n R →ₗ[R] matrix n n A | (algebra.lsmul R (matrix n n A)).to_linear_map.compl₂ (algebra.linear_map R A).map_matrix | def | matrix_equiv_tensor.to_fun_bilinear | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"algebra.linear_map",
"algebra.lsmul",
"matrix"
] | (Implementation detail).
The function underlying `(A ⊗[R] matrix n n R) →ₐ[R] matrix n n A`,
as an `R`-bilinear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_bilinear_apply (a : A) (m : matrix n n R) :
to_fun_bilinear R A n a m = a • m.map (algebra_map R A) | rfl | lemma | matrix_equiv_tensor.to_fun_bilinear_apply | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"algebra_map",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_linear : A ⊗[R] matrix n n R →ₗ[R] matrix n n A | tensor_product.lift (to_fun_bilinear R A n) | def | matrix_equiv_tensor.to_fun_linear | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"matrix",
"tensor_product.lift"
] | (Implementation detail).
The function underlying `(A ⊗[R] matrix n n R) →ₐ[R] matrix n n A`,
as an `R`-linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_alg_hom : (A ⊗[R] matrix n n R) →ₐ[R] matrix n n A | alg_hom_of_linear_map_tensor_product
(to_fun_linear R A n)
begin
intros,
simp_rw [to_fun_linear, lift.tmul, to_fun_bilinear_apply, mul_eq_mul, matrix.map_mul],
ext,
dsimp,
simp_rw [matrix.mul_apply, pi.smul_apply, matrix.map_apply, smul_eq_mul, finset.mul_sum,
_root_.mul_assoc, algebra.left_comm],
end
beg... | def | matrix_equiv_tensor.to_fun_alg_hom | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"algebra.algebra_map_eq_smul_one",
"algebra.left_comm",
"algebra_map",
"algebra_map_smul",
"finset.mul_sum",
"map_one",
"matrix",
"matrix.map_apply",
"matrix.map_mul",
"matrix.map_one",
"matrix.mul_apply",
"pi.smul_apply",
"smul_eq_mul"
] | The function `(A ⊗[R] matrix n n R) →ₐ[R] matrix n n A`, as an algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_alg_hom_apply (a : A) (m : matrix n n R) :
to_fun_alg_hom R A n (a ⊗ₜ m) = a • m.map (algebra_map R A) | by simp [to_fun_alg_hom, alg_hom_of_linear_map_tensor_product, to_fun_linear] | lemma | matrix_equiv_tensor.to_fun_alg_hom_apply | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"algebra_map",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_fun (M : matrix n n A) : A ⊗[R] matrix n n R | ∑ (p : n × n), M p.1 p.2 ⊗ₜ (std_basis_matrix p.1 p.2 1) | def | matrix_equiv_tensor.inv_fun | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"inv_fun",
"matrix"
] | (Implementation detail.)
The bare function `matrix n n A → A ⊗[R] matrix n n R`.
(We don't need to show that it's an algebra map, thankfully --- just that it's an inverse.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_fun_zero : inv_fun R A n 0 = 0 | by simp [inv_fun] | lemma | matrix_equiv_tensor.inv_fun_zero | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_fun_add (M N : matrix n n A) :
inv_fun R A n (M + N) = inv_fun R A n M + inv_fun R A n N | by simp [inv_fun, add_tmul, finset.sum_add_distrib] | lemma | matrix_equiv_tensor.inv_fun_add | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"inv_fun",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_fun_smul (a : A) (M : matrix n n A) :
inv_fun R A n (a • M) = (a ⊗ₜ 1) * inv_fun R A n M | by simp [inv_fun,finset.mul_sum] | lemma | matrix_equiv_tensor.inv_fun_smul | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"finset.mul_sum",
"inv_fun",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_fun_algebra_map (M : matrix n n R) :
inv_fun R A n (M.map (algebra_map R A)) = 1 ⊗ₜ M | begin
dsimp [inv_fun],
simp only [algebra.algebra_map_eq_smul_one, smul_tmul, ←tmul_sum, mul_boole],
congr,
conv_rhs {rw matrix_eq_sum_std_basis M},
convert finset.sum_product, simp,
end | lemma | matrix_equiv_tensor.inv_fun_algebra_map | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"algebra.algebra_map_eq_smul_one",
"algebra_map",
"inv_fun",
"matrix",
"mul_boole"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_inv (M : matrix n n A) : (to_fun_alg_hom R A n) (inv_fun R A n M) = M | begin
simp only [inv_fun, alg_hom.map_sum, std_basis_matrix, apply_ite ⇑(algebra_map R A), smul_eq_mul,
mul_boole, to_fun_alg_hom_apply, ring_hom.map_zero, ring_hom.map_one, matrix.map_apply,
pi.smul_def],
convert finset.sum_product, apply matrix_eq_sum_std_basis,
end | lemma | matrix_equiv_tensor.right_inv | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"alg_hom.map_sum",
"algebra_map",
"apply_ite",
"inv_fun",
"matrix",
"matrix.map_apply",
"mul_boole",
"pi.smul_def",
"ring_hom.map_one",
"ring_hom.map_zero",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inv (M : A ⊗[R] matrix n n R) : inv_fun R A n (to_fun_alg_hom R A n M) = M | begin
induction M using tensor_product.induction_on with a m x y hx hy,
{ simp, },
{ simp, },
{ simp [alg_hom.map_sum, hx, hy], },
end | lemma | matrix_equiv_tensor.left_inv | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"alg_hom.map_sum",
"inv_fun",
"matrix",
"tensor_product.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv : (A ⊗[R] matrix n n R) ≃ matrix n n A | { to_fun := to_fun_alg_hom R A n,
inv_fun := inv_fun R A n,
left_inv := left_inv R A n,
right_inv := right_inv R A n, } | def | matrix_equiv_tensor.equiv | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"equiv",
"inv_fun",
"matrix"
] | (Implementation detail)
The equivalence, ignoring the algebra structure, `(A ⊗[R] matrix n n R) ≃ matrix n n A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
matrix_equiv_tensor : matrix n n A ≃ₐ[R] (A ⊗[R] matrix n n R) | alg_equiv.symm { ..(matrix_equiv_tensor.to_fun_alg_hom R A n), ..(matrix_equiv_tensor.equiv R A n) } | def | matrix_equiv_tensor | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"alg_equiv.symm",
"matrix",
"matrix_equiv_tensor.equiv",
"matrix_equiv_tensor.to_fun_alg_hom"
] | The `R`-algebra isomorphism `matrix n n A ≃ₐ[R] (A ⊗[R] matrix n n R)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
matrix_equiv_tensor_apply (M : matrix n n A) :
matrix_equiv_tensor R A n M =
∑ (p : n × n), M p.1 p.2 ⊗ₜ (std_basis_matrix p.1 p.2 1) | rfl | lemma | matrix_equiv_tensor_apply | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"matrix",
"matrix_equiv_tensor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
matrix_equiv_tensor_apply_std_basis (i j : n) (x : A):
matrix_equiv_tensor R A n (std_basis_matrix i j x) =
x ⊗ₜ (std_basis_matrix i j 1) | begin
have t : ∀ (p : n × n), (i = p.1 ∧ j = p.2) ↔ (p = (i, j)) := by tidy,
simp [ite_tmul, t, std_basis_matrix],
end | lemma | matrix_equiv_tensor_apply_std_basis | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"matrix_equiv_tensor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
matrix_equiv_tensor_apply_symm (a : A) (M : matrix n n R) :
(matrix_equiv_tensor R A n).symm (a ⊗ₜ M) =
M.map (λ x, a * algebra_map R A x) | begin
simp [matrix_equiv_tensor, to_fun_alg_hom, alg_hom_of_linear_map_tensor_product, to_fun_linear],
refl,
end | lemma | matrix_equiv_tensor_apply_symm | ring_theory | src/ring_theory/matrix_algebra.lean | [
"data.matrix.basis",
"ring_theory.tensor_product"
] | [
"algebra_map",
"matrix",
"matrix_equiv_tensor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity [monoid α] [decidable_rel ((∣) : α → α → Prop)] (a b : α) : part_enat | part_enat.find $ λ n, ¬a ^ (n + 1) ∣ b | def | multiplicity | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"monoid",
"part_enat",
"part_enat.find"
] | `multiplicity a b` returns the largest natural number `n` such that
`a ^ n ∣ b`, as an `part_enat` or natural with infinity. If `∀ n, a ^ n ∣ b`,
then it returns `⊤` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite (a b : α) : Prop | ∃ n : ℕ, ¬a ^ (n + 1) ∣ b | def | multiplicity.finite | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite"
] | `multiplicity.finite a b` indicates that the multiplicity of `a` in `b` is finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_iff_dom [decidable_rel ((∣) : α → α → Prop)] {a b : α} :
finite a b ↔ (multiplicity a b).dom | iff.rfl | lemma | multiplicity.finite_iff_dom | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_def {a b : α} : finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b | iff.rfl | lemma | multiplicity.finite_def | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_dvd_one_of_finite_one_right {a : α} : finite a 1 → ¬a ∣ 1 | λ ⟨n, hn⟩ ⟨d, hd⟩, hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩ | lemma | multiplicity.not_dvd_one_of_finite_one_right | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"pow_mul_pow_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.coe_nat_multiplicity (a b : ℕ) :
multiplicity (a : ℤ) (b : ℤ) = multiplicity a b | begin
apply part.ext',
{ repeat { rw [← finite_iff_dom, finite_def] },
norm_cast },
{ intros h1 h2,
apply _root_.le_antisymm; { apply nat.find_mono, norm_cast, simp } }
end | theorem | multiplicity.int.coe_nat_multiplicity | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity",
"nat.find_mono",
"part.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_finite_iff_forall {a b : α} : (¬ finite a b) ↔ ∀ n : ℕ, a ^ n ∣ b | ⟨λ h n, nat.cases_on n (by { rw pow_zero, exact one_dvd _ }) (by simpa [finite, not_not] using h),
by simp [finite, multiplicity, not_not]; tauto⟩ | lemma | multiplicity.not_finite_iff_forall | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"multiplicity",
"not_not",
"one_dvd",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_unit_of_finite {a b : α} (h : finite a b) : ¬is_unit a | let ⟨n, hn⟩ := h in hn ∘ is_unit.dvd ∘ is_unit.pow (n + 1) | lemma | multiplicity.not_unit_of_finite | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"is_unit",
"is_unit.dvd",
"is_unit.pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_of_finite_mul_right {a b c : α} : finite a (b * c) → finite a b | λ ⟨n, hn⟩, ⟨n, λ h, hn (h.trans (dvd_mul_right _ _))⟩ | lemma | multiplicity.finite_of_finite_mul_right | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"dvd_mul_right",
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : part_enat) ≤ multiplicity a b → a ^ k ∣ b | by { rw ← part_enat.some_eq_coe, exact
nat.cases_on k (λ _, by { rw pow_zero, exact one_dvd _ })
(λ k ⟨h₁, h₂⟩, by_contradiction (λ hk, (nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk))) } | lemma | multiplicity.pow_dvd_of_le_multiplicity | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"by_contradiction",
"multiplicity",
"one_dvd",
"part_enat",
"part_enat.some_eq_coe",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_multiplicity_dvd {a b : α} (h : finite a b) : a ^ get (multiplicity a b) h ∣ b | pow_dvd_of_le_multiplicity (by rw part_enat.coe_get) | lemma | multiplicity.pow_multiplicity_dvd | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"multiplicity",
"part_enat.coe_get"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b | λ h, by rw [part_enat.lt_coe_iff] at hm; exact nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h) | lemma | multiplicity.is_greatest | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"is_greatest",
"multiplicity",
"part_enat.lt_coe_iff",
"pow_dvd_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_greatest' {a b : α} {m : ℕ} (h : finite a b) (hm : get (multiplicity a b) h < m) :
¬a ^ m ∣ b | is_greatest (by rwa [← part_enat.coe_lt_coe, part_enat.coe_get] at hm) | lemma | multiplicity.is_greatest' | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"is_greatest",
"multiplicity",
"part_enat.coe_get",
"part_enat.coe_lt_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_of_dvd {a b : α} (hfin : finite a b) (hdiv : a ∣ b) : 0 < (multiplicity a b).get hfin | begin
refine zero_lt_iff.2 (λ h, _),
simpa [hdiv] using (is_greatest' hfin (lt_one_iff.mpr h)),
end | lemma | multiplicity.pos_of_dvd | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
(k : part_enat) = multiplicity a b | le_antisymm (le_of_not_gt (λ hk', is_greatest hk' hk)) $
have finite a b, from ⟨k, hsucc⟩,
by { rw [part_enat.le_coe_iff], exact ⟨this, nat.find_min' _ hsucc⟩ } | lemma | multiplicity.unique | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"is_greatest",
"multiplicity",
"part_enat",
"part_enat.le_coe_iff",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬ a ^ (k + 1) ∣ b) :
k = get (multiplicity a b) ⟨k, hsucc⟩ | by rw [← part_enat.coe_inj, part_enat.coe_get, unique hk hsucc] | lemma | multiplicity.unique' | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity",
"part_enat.coe_get",
"part_enat.coe_inj",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_multiplicity_of_pow_dvd {a b : α}
{k : ℕ} (hk : a ^ k ∣ b) : (k : part_enat) ≤ multiplicity a b | le_of_not_gt $ λ hk', is_greatest hk' hk | lemma | multiplicity.le_multiplicity_of_pow_dvd | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"is_greatest",
"multiplicity",
"part_enat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_dvd_iff_le_multiplicity {a b : α}
{k : ℕ} : a ^ k ∣ b ↔ (k : part_enat) ≤ multiplicity a b | ⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩ | lemma | multiplicity.pow_dvd_iff_le_multiplicity | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity",
"part_enat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_lt_iff_neg_dvd {a b : α} {k : ℕ} :
multiplicity a b < (k : part_enat) ↔ ¬ a ^ k ∣ b | by { rw [pow_dvd_iff_le_multiplicity, not_le] } | lemma | multiplicity.multiplicity_lt_iff_neg_dvd | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity",
"part_enat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : part_enat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b | begin
rw [← part_enat.some_eq_coe],
exact ⟨λ h, let ⟨h₁, h₂⟩ := eq_some_iff.1 h in
h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest
(by { rw [part_enat.lt_coe_iff], exact ⟨h₁, lt_succ_self _⟩ })⟩,
λ h, eq_some_iff.2 ⟨⟨n, h.2⟩, eq.symm $ unique' h.1 h.2⟩⟩
end | lemma | multiplicity.eq_coe_iff | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"is_greatest",
"multiplicity",
"part_enat",
"part_enat.lt_coe_iff",
"part_enat.some_eq_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_top_iff {a b : α} :
multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b | (part_enat.find_eq_top_iff _).trans $
by { simp only [not_not],
exact ⟨λ h n, nat.cases_on n (by { rw pow_zero, exact one_dvd _}) (λ n, h _), λ h n, h _⟩ } | lemma | multiplicity.eq_top_iff | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"eq_top_iff",
"multiplicity",
"not_not",
"one_dvd",
"part_enat.find_eq_top_iff",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_left {a : α} (b : α) (ha : is_unit a) : multiplicity a b = ⊤ | eq_top_iff.2 (λ _, is_unit.dvd (ha.pow _)) | lemma | multiplicity.is_unit_left | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"is_unit",
"is_unit.dvd",
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_left (b : α) : multiplicity 1 b = ⊤ | is_unit_left b is_unit_one | lemma | multiplicity.one_left | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"is_unit_one",
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
get_one_right {a : α} (ha : finite a 1) : get (multiplicity a 1) ha = 0 | begin
rw [part_enat.get_eq_iff_eq_coe, eq_coe_iff, pow_zero],
simp [not_dvd_one_of_finite_one_right ha],
end | lemma | multiplicity.get_one_right | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"multiplicity",
"part_enat.get_eq_iff_eq_coe",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ | is_unit_left a u.is_unit | lemma | multiplicity.unit_left | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬ a ∣ b | by { rw [← nat.cast_zero, eq_coe_iff], simp } | lemma | multiplicity.multiplicity_eq_zero | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity",
"nat.cast_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b | multiplicity_eq_zero.not_left | lemma | multiplicity.multiplicity_ne_zero | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬ finite a b | part.eq_none_iff' | lemma | multiplicity.eq_top_iff_not_finite | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"multiplicity",
"part.eq_none_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ finite a b | by rw [ne.def, eq_top_iff_not_finite, not_not] | lemma | multiplicity.ne_top_iff_finite | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"multiplicity",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ finite a b | by rw [lt_top_iff_ne_top, ne_top_iff_finite] | lemma | multiplicity.lt_top_iff_finite | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"lt_top_iff_ne_top",
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : finite a b) :
∃ (c : α), b = a ^ ((multiplicity a b).get hfin) * c ∧ ¬ a ∣ c | begin
obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin,
refine ⟨c, hc, _⟩,
rintro ⟨k, hk⟩,
rw [hk, ← mul_assoc, ← pow_succ'] at hc,
have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩,
exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁,
end | lemma | multiplicity.exists_eq_pow_mul_and_not_dvd | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"mul_assoc",
"multiplicity",
"multiplicity.pow_multiplicity_dvd",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_le_multiplicity_iff {a b c d : α} : multiplicity a b ≤ multiplicity c d ↔
(∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d) | ⟨λ h n hab, (pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h)),
λ h, if hab : finite a b
then by rw [← part_enat.coe_get (finite_iff_dom.1 hab)];
exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _))
else
have ∀ n : ℕ, c ^ n ∣ d, from λ n, h n (not_finite_iff_forall... | lemma | multiplicity.multiplicity_le_multiplicity_iff | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"finite",
"multiplicity",
"part_enat.coe_get"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_eq_multiplicity_iff {a b c d : α} : multiplicity a b = multiplicity c d ↔
(∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d) | ⟨λ h n, ⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩,
λ h, le_antisymm (multiplicity_le_multiplicity_iff.mpr (λ n, (h n).mp))
(multiplicity_le_multiplicity_iff.mpr (λ n, (h n).mpr))⟩ | lemma | multiplicity.multiplicity_eq_multiplicity_iff | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) :
multiplicity a b ≤ multiplicity a c | multiplicity_le_multiplicity_iff.2 $ λ n hb, hb.trans h | lemma | multiplicity.multiplicity_le_multiplicity_of_dvd_right | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_associated_right {a b c : α} (h : associated b c) :
multiplicity a b = multiplicity a c | le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd)
(multiplicity_le_multiplicity_of_dvd_right h.symm.dvd) | lemma | multiplicity.eq_of_associated_right | ring_theory | src/ring_theory/multiplicity.lean | [
"algebra.associated",
"algebra.big_operators.basic",
"ring_theory.valuation.basic"
] | [
"associated",
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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