statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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integral_closure : integral_closure O R = ⊥ | bot_unique $ λ r hr, let ⟨x, hx⟩ := hv.3 (hv.mem_of_integral hr) in algebra.mem_bot.2 ⟨x, hx⟩ | lemma | valuation.integers.integral_closure | ring_theory.valuation | src/ring_theory/valuation/integral.lean | [
"ring_theory.integrally_closed",
"ring_theory.valuation.integers"
] | [
"bot_unique",
"integral_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrally_closed : is_integrally_closed O | (is_integrally_closed.integral_closure_eq_bot_iff K).mp (valuation.integers.integral_closure hv) | lemma | valuation.integers.integrally_closed | ring_theory.valuation | src/ring_theory/valuation/integral.lean | [
"ring_theory.integrally_closed",
"ring_theory.valuation.integers"
] | [
"is_integrally_closed",
"is_integrally_closed.integral_closure_eq_bot_iff",
"valuation.integers.integral_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
on_quot_val {J : ideal R} (hJ : J ≤ supp v) :
R ⧸ J → Γ₀ | λ q, quotient.lift_on' q v $ λ a b h,
calc v a = v (b + -(-a + b)) : by simp
... = v b :
v.map_add_supp b $ (ideal.neg_mem_iff _).2 $ hJ $ quotient_add_group.left_rel_apply.mp h | def | valuation.on_quot_val | ring_theory.valuation | src/ring_theory/valuation/quotient.lean | [
"ring_theory.valuation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ideal.neg_mem_iff",
"quotient.lift_on'"
] | If `hJ : J ⊆ supp v` then `on_quot_val hJ` is the induced function on R/J as a function.
Note: it's just the function; the valuation is `on_quot hJ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
on_quot {J : ideal R} (hJ : J ≤ supp v) :
valuation (R ⧸ J) Γ₀ | { to_fun := v.on_quot_val hJ,
map_zero' := v.map_zero,
map_one' := v.map_one,
map_mul' := λ xbar ybar, quotient.ind₂' v.map_mul xbar ybar,
map_add_le_max' := λ xbar ybar, quotient.ind₂' v.map_add xbar ybar } | def | valuation.on_quot | ring_theory.valuation | src/ring_theory/valuation/quotient.lean | [
"ring_theory.valuation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"quotient.ind₂'",
"valuation"
] | The extension of valuation v on R to valuation on R/J if J ⊆ supp v | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
on_quot_comap_eq {J : ideal R} (hJ : J ≤ supp v) :
(v.on_quot hJ).comap (ideal.quotient.mk J) = v | ext $ λ r, rfl | lemma | valuation.on_quot_comap_eq | ring_theory.valuation | src/ring_theory/valuation/quotient.lean | [
"ring_theory.valuation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_le_supp_comap (J : ideal R) (v : valuation (R ⧸ J) Γ₀) :
J ≤ (v.comap (ideal.quotient.mk J)).supp | by { rw [comap_supp, ← ideal.map_le_iff_le_comap], simp } | lemma | valuation.self_le_supp_comap | ring_theory.valuation | src/ring_theory/valuation/quotient.lean | [
"ring_theory.valuation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ideal.map_le_iff_le_comap",
"ideal.quotient.mk",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_on_quot_eq (J : ideal R) (v : valuation (R ⧸ J) Γ₀) :
(v.comap (ideal.quotient.mk J)).on_quot (v.self_le_supp_comap J) = v | ext $ by { rintro ⟨x⟩, refl } | lemma | valuation.comap_on_quot_eq | ring_theory.valuation | src/ring_theory/valuation/quotient.lean | [
"ring_theory.valuation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ideal.quotient.mk",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supp_quot {J : ideal R} (hJ : J ≤ supp v) :
supp (v.on_quot hJ) = (supp v).map (ideal.quotient.mk J) | begin
apply le_antisymm,
{ rintro ⟨x⟩ hx,
apply ideal.subset_span,
exact ⟨x, hx, rfl⟩ },
{ rw ideal.map_le_iff_le_comap,
intros x hx, exact hx }
end | lemma | valuation.supp_quot | ring_theory.valuation | src/ring_theory/valuation/quotient.lean | [
"ring_theory.valuation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ideal.map_le_iff_le_comap",
"ideal.quotient.mk",
"ideal.subset_span"
] | The quotient valuation on R/J has support supp(v)/J if J ⊆ supp v. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
supp_quot_supp : supp (v.on_quot le_rfl) = 0 | by { rw supp_quot, exact ideal.map_quotient_self _ } | lemma | valuation.supp_quot_supp | ring_theory.valuation | src/ring_theory/valuation/quotient.lean | [
"ring_theory.valuation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal.map_quotient_self",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
on_quot_val {J : ideal R} (hJ : J ≤ supp v) : (R ⧸ J) → Γ₀ | v.on_quot_val hJ | def | add_valuation.on_quot_val | ring_theory.valuation | src/ring_theory/valuation/quotient.lean | [
"ring_theory.valuation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal"
] | If `hJ : J ⊆ supp v` then `on_quot_val hJ` is the induced function on R/J as a function.
Note: it's just the function; the valuation is `on_quot hJ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
on_quot {J : ideal R} (hJ : J ≤ supp v) :
add_valuation (R ⧸ J) Γ₀ | v.on_quot hJ | def | add_valuation.on_quot | ring_theory.valuation | src/ring_theory/valuation/quotient.lean | [
"ring_theory.valuation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"add_valuation",
"ideal"
] | The extension of valuation v on R to valuation on R/J if J ⊆ supp v | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
on_quot_comap_eq {J : ideal R} (hJ : J ≤ supp v) :
(v.on_quot hJ).comap (ideal.quotient.mk J) = v | v.on_quot_comap_eq hJ | lemma | add_valuation.on_quot_comap_eq | ring_theory.valuation | src/ring_theory/valuation/quotient.lean | [
"ring_theory.valuation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_supp {S : Type*} [comm_ring S] (f : S →+* R) :
supp (v.comap f) = ideal.comap f v.supp | v.comap_supp f | lemma | add_valuation.comap_supp | ring_theory.valuation | src/ring_theory/valuation/quotient.lean | [
"ring_theory.valuation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"comm_ring",
"ideal.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_le_supp_comap (J : ideal R) (v : add_valuation (R ⧸ J) Γ₀) :
J ≤ (v.comap (ideal.quotient.mk J)).supp | v.self_le_supp_comap J | lemma | add_valuation.self_le_supp_comap | ring_theory.valuation | src/ring_theory/valuation/quotient.lean | [
"ring_theory.valuation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"add_valuation",
"ideal",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_on_quot_eq (J : ideal R) (v : add_valuation (R ⧸ J) Γ₀) :
(v.comap (ideal.quotient.mk J)).on_quot (v.self_le_supp_comap J) = v | v.comap_on_quot_eq J | lemma | add_valuation.comap_on_quot_eq | ring_theory.valuation | src/ring_theory/valuation/quotient.lean | [
"ring_theory.valuation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"add_valuation",
"ideal",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supp_quot {J : ideal R} (hJ : J ≤ supp v) :
supp (v.on_quot hJ) = (supp v).map (ideal.quotient.mk J) | v.supp_quot hJ | lemma | add_valuation.supp_quot | ring_theory.valuation | src/ring_theory/valuation/quotient.lean | [
"ring_theory.valuation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ideal.quotient.mk"
] | The quotient valuation on R/J has support supp(v)/J if J ⊆ supp v. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
supp_quot_supp : supp (v.on_quot le_rfl) = 0 | v.supp_quot_supp | lemma | add_valuation.supp_quot_supp | ring_theory.valuation | src/ring_theory/valuation/quotient.lean | [
"ring_theory.valuation.basic",
"ring_theory.ideal.quotient_operations"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decomposition_subgroup (A : valuation_subring L) :
subgroup (L ≃ₐ[K] L) | mul_action.stabilizer (L ≃ₐ[K] L) A | def | valuation_subring.decomposition_subgroup | ring_theory.valuation | src/ring_theory/valuation/ramification_group.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.valuation.valuation_subring"
] | [
"mul_action.stabilizer",
"subgroup",
"valuation_subring"
] | The decomposition subgroup defined as the stabilizer of the action
on the type of all valuation subrings of the field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_mul_action (A : valuation_subring L) :
sub_mul_action (A.decomposition_subgroup K) L | { carrier := A,
smul_mem' := λ g l h, set.mem_of_mem_of_subset (set.smul_mem_smul_set h) g.prop.le } | def | valuation_subring.sub_mul_action | ring_theory.valuation | src/ring_theory/valuation/ramification_group.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.valuation.valuation_subring"
] | [
"set.mem_of_mem_of_subset",
"set.smul_mem_smul_set",
"sub_mul_action",
"valuation_subring"
] | The valuation subring `A` (considered as a subset of `L`)
is stable under the action of the decomposition group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
decomposition_subgroup_mul_semiring_action (A : valuation_subring L) :
mul_semiring_action (A.decomposition_subgroup K) A | { smul_add := λ g k l, subtype.ext $ smul_add g k l,
smul_zero := λ g, subtype.ext $ smul_zero g,
smul_one := λ g, subtype.ext $ smul_one g,
smul_mul := λ g k l, subtype.ext $ smul_mul' g k l,
..(sub_mul_action.mul_action (A.sub_mul_action K)) } | instance | valuation_subring.decomposition_subgroup_mul_semiring_action | ring_theory.valuation | src/ring_theory/valuation/ramification_group.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.valuation.valuation_subring"
] | [
"mul_semiring_action",
"smul_add",
"smul_mul'",
"smul_zero",
"subtype.ext",
"valuation_subring"
] | The multiplicative action of the decomposition subgroup on `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inertia_subgroup (A : valuation_subring L) :
subgroup (A.decomposition_subgroup K) | monoid_hom.ker $
mul_semiring_action.to_ring_aut (A.decomposition_subgroup K) (local_ring.residue_field A) | def | valuation_subring.inertia_subgroup | ring_theory.valuation | src/ring_theory/valuation/ramification_group.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.valuation.valuation_subring"
] | [
"local_ring.residue_field",
"monoid_hom.ker",
"mul_semiring_action.to_ring_aut",
"subgroup",
"valuation_subring"
] | The inertia subgroup defined as the kernel of the group homomorphism from
the decomposition subgroup to the group of automorphisms of the residue field of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation_ring (A : Type u) [comm_ring A] [is_domain A] : Prop | (cond [] : ∀ a b : A, ∃ c : A, a * c = b ∨ b * c = a) | class | valuation_ring | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"comm_ring",
"is_domain"
] | An integral domain is called a `valuation ring` provided that for any pair
of elements `a b : A`, either `a` divides `b` or vice versa. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
value_group : Type v | quotient (mul_action.orbit_rel Aˣ K) | def | valuation_ring.value_group | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"mul_action.orbit_rel"
] | The value group of the valuation ring `A`. Note: this is actually a group with zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_total (a b : value_group A K) : a ≤ b ∨ b ≤ a | begin
rcases a with ⟨a⟩, rcases b with ⟨b⟩,
obtain ⟨xa,ya,hya,rfl⟩ : ∃ (a b : A), _ := is_fraction_ring.div_surjective a,
obtain ⟨xb,yb,hyb,rfl⟩ : ∃ (a b : A), _ := is_fraction_ring.div_surjective b,
have : (algebra_map A K) ya ≠ 0 :=
is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors hya,
have : (a... | lemma | valuation_ring.le_total | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"algebra.smul_def",
"algebra_map",
"is_fraction_ring.div_surjective",
"is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors",
"ring",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation : valuation K (value_group A K) | { to_fun := quotient.mk',
map_zero' := rfl,
map_one' := rfl,
map_mul' := λ _ _, rfl,
map_add_le_max' := begin
intros a b,
obtain ⟨xa,ya,hya,rfl⟩ : ∃ (a b : A), _ := is_fraction_ring.div_surjective a,
obtain ⟨xb,yb,hyb,rfl⟩ : ∃ (a b : A), _ := is_fraction_ring.div_surjective b,
have : (algebra_ma... | def | valuation_ring.valuation | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"algebra.smul_def",
"algebra_map",
"is_fraction_ring.div_surjective",
"is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors",
"map_one",
"quotient.mk'",
"ring",
"ring_hom.map_add",
"ring_hom.map_mul",
"valuation"
] | Any valuation ring induces a valuation on its fraction field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_integer_iff (x : K) : x ∈ (valuation A K).integer ↔ ∃ a : A, algebra_map A K a = x | begin
split,
{ rintros ⟨c,rfl⟩,
use c,
rw [algebra.smul_def, mul_one] },
{ rintro ⟨c,rfl⟩,
use c,
rw [algebra.smul_def, mul_one] }
end | lemma | valuation_ring.mem_integer_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"algebra.smul_def",
"algebra_map",
"mul_one",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_integer : A ≃+* (valuation A K).integer | ring_equiv.of_bijective (show A →ₙ+* (valuation A K).integer, from
{ to_fun := λ a, ⟨algebra_map A K a, (mem_integer_iff _ _ _).mpr ⟨a,rfl⟩⟩,
map_mul' := λ _ _, by { ext1, exact (algebra_map A K).map_mul _ _ },
map_zero' := by { ext1, exact (algebra_map A K).map_zero },
map_add' := λ _ _, by { ext1, exact (algebr... | def | valuation_ring.equiv_integer | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"algebra_map",
"is_fraction_ring.injective",
"map_mul",
"ring_equiv.of_bijective",
"valuation"
] | The valuation ring `A` is isomorphic to the ring of integers of its associated valuation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_equiv_integer_apply (a : A) : (equiv_integer A K a : K) = algebra_map A K a | rfl | lemma | valuation_ring.coe_equiv_integer_apply | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_algebra_map_eq : (valuation A K).integer = (algebra_map A K).range | by { ext, exact mem_integer_iff _ _ _ } | lemma | valuation_ring.range_algebra_map_eq | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"algebra_map",
"valuation"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_dvd_total :
valuation_ring R ↔ is_total R (∣) | begin
classical,
refine ⟨λ H, ⟨λ a b, _⟩, λ H, ⟨λ a b, _⟩⟩; resetI,
{ obtain ⟨c,rfl|rfl⟩ := @@valuation_ring.cond _ _ H a b; simp },
{ obtain (⟨c, rfl⟩|⟨c, rfl⟩) := @is_total.total _ _ H a b; use c; simp }
end | lemma | valuation_ring.iff_dvd_total | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"valuation_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_ideal_total :
valuation_ring R ↔ is_total (ideal R) (≤) | begin
classical,
refine ⟨λ _, by exactI ⟨le_total⟩, λ H, iff_dvd_total.mpr ⟨λ a b, _⟩⟩,
have := @is_total.total _ _ H (ideal.span {a}) (ideal.span {b}),
simp_rw ideal.span_singleton_le_span_singleton at this,
exact this.symm
end | lemma | valuation_ring.iff_ideal_total | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"ideal",
"ideal.span",
"ideal.span_singleton_le_span_singleton",
"valuation_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_total [h : valuation_ring R] (x y : R) : x ∣ y ∨ y ∣ x | @@is_total.total _ (iff_dvd_total.mp h) x y | lemma | valuation_ring.dvd_total | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"valuation_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_irreducible [valuation_ring R] ⦃p q : R⦄
(hp : irreducible p) (hq : irreducible q) : associated p q | begin
have := dvd_total p q,
rw [irreducible.dvd_comm hp hq, or_self] at this,
exact associated_of_dvd_dvd (irreducible.dvd_symm hq hp this) this,
end | lemma | valuation_ring.unique_irreducible | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"associated",
"associated_of_dvd_dvd",
"irreducible",
"irreducible.dvd_comm",
"irreducible.dvd_symm",
"valuation_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_is_integer_or_is_integer :
valuation_ring R ↔ ∀ x : K, is_localization.is_integer R x ∨ is_localization.is_integer R x⁻¹ | begin
split,
{ introsI H x,
obtain ⟨x : R, y, hy, rfl⟩ := is_fraction_ring.div_surjective x,
any_goals { apply_instance },
have := (map_ne_zero_iff _ (is_fraction_ring.injective R K)).mpr (non_zero_divisors.ne_zero hy),
obtain ⟨s, rfl|rfl⟩ := valuation_ring.cond x y,
{ exact or.inr ⟨s, eq_inv_of... | lemma | valuation_ring.iff_is_integer_or_is_integer | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"algebra_map",
"div_eq_one_iff_eq",
"eq_div_iff",
"eq_inv_of_mul_eq_one_left",
"inv_div",
"is_fraction_ring.div_surjective",
"is_fraction_ring.injective",
"is_localization.is_integer",
"map_mul",
"mul_comm",
"mul_div",
"mul_zero",
"non_zero_divisors.ne_zero",
"valuation_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integer_or_is_integer [h : valuation_ring R] (x : K) :
is_localization.is_integer R x ∨ is_localization.is_integer R x⁻¹ | (iff_is_integer_or_is_integer R K).mp h x | lemma | valuation_ring.is_integer_or_is_integer | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"is_localization.is_integer",
"valuation_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_local_bezout_domain :
valuation_ring R ↔ local_ring R ∧ is_bezout R | begin
classical,
refine ⟨λ H, by exactI ⟨infer_instance, infer_instance⟩, _⟩,
rintro ⟨h₁, h₂⟩,
resetI,
refine iff_dvd_total.mpr ⟨λ a b, _⟩,
obtain ⟨g, e : _ = ideal.span _⟩ := is_bezout.span_pair_is_principal a b,
obtain ⟨a, rfl⟩ := ideal.mem_span_singleton'.mp
(show a ∈ ideal.span {g}, by { rw [← e],... | lemma | valuation_ring.iff_local_bezout_domain | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"eq_or_ne",
"ideal.span",
"ideal.subset_span",
"is_bezout",
"is_bezout.span_pair_is_principal",
"is_unit_of_mul_is_unit_right",
"local_ring",
"mul_dvd_mul_right",
"mul_left_injective₀",
"valuation_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tfae (R : Type u) [comm_ring R] [is_domain R] :
tfae [valuation_ring R,
∀ x : fraction_ring R, is_localization.is_integer R x ∨ is_localization.is_integer R x⁻¹,
is_total R (∣),
is_total (ideal R) (≤),
local_ring R ∧ is_bezout R] | begin
tfae_have : 1 ↔ 2, { exact iff_is_integer_or_is_integer R _ },
tfae_have : 1 ↔ 3, { exact iff_dvd_total },
tfae_have : 1 ↔ 4, { exact iff_ideal_total },
tfae_have : 1 ↔ 5, { exact iff_local_bezout_domain },
tfae_finish
end | lemma | valuation_ring.tfae | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"comm_ring",
"fraction_ring",
"ideal",
"is_bezout",
"is_domain",
"is_localization.is_integer",
"local_ring",
"valuation_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.function.surjective.valuation_ring {R S : Type*} [comm_ring R] [is_domain R]
[valuation_ring R] [comm_ring S] [is_domain S] (f : R →+* S) (hf : function.surjective f) :
valuation_ring S | ⟨λ a b, begin
obtain ⟨⟨a, rfl⟩, ⟨b, rfl⟩⟩ := ⟨hf a, hf b⟩,
obtain ⟨c, rfl|rfl⟩ := valuation_ring.cond a b,
exacts [⟨f c, or.inl $ (map_mul _ _ _).symm⟩, ⟨f c, or.inr $ (map_mul _ _ _).symm⟩],
end⟩ | lemma | function.surjective.valuation_ring | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"comm_ring",
"is_domain",
"map_mul",
"valuation_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_integers : valuation_ring 𝒪 | begin
constructor,
intros a b,
cases le_total (v (algebra_map 𝒪 K a)) (v (algebra_map 𝒪 K b)),
{ obtain ⟨c,hc⟩ := valuation.integers.dvd_of_le hh h,
use c, exact or.inr hc.symm },
{ obtain ⟨c,hc⟩ := valuation.integers.dvd_of_le hh h,
use c, exact or.inl hc.symm }
end | lemma | valuation_ring.of_integers | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"algebra_map",
"valuation.integers.dvd_of_le",
"valuation_ring"
] | If `𝒪` satisfies `v.integers 𝒪` where `v` is a valuation on a field, then `𝒪`
is a valuation ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_field : valuation_ring K | begin
constructor,
intros a b,
by_cases b = 0,
{ use 0, left, simp [h] },
{ use a * b⁻¹, right, field_simp, rw mul_comm }
end | instance | valuation_ring.of_field | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"mul_comm",
"valuation_ring"
] | A field is a valuation ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_discrete_valuation_ring : valuation_ring A | begin
constructor,
intros a b,
by_cases ha : a = 0, { use 0, right, simp [ha] },
by_cases hb : b = 0, { use 0, left, simp [hb] },
obtain ⟨ϖ,hϖ⟩ := discrete_valuation_ring.exists_irreducible A,
obtain ⟨m,u,rfl⟩ := discrete_valuation_ring.eq_unit_mul_pow_irreducible ha hϖ,
obtain ⟨n,v,rfl⟩ := discrete_valua... | instance | valuation_ring.of_discrete_valuation_ring | ring_theory.valuation | src/ring_theory/valuation/valuation_ring.lean | [
"ring_theory.valuation.integers",
"ring_theory.ideal.local_ring",
"ring_theory.localization.fraction_ring",
"ring_theory.localization.integer",
"ring_theory.discrete_valuation_ring.basic",
"ring_theory.bezout",
"tactic.field_simp"
] | [
"discrete_valuation_ring.eq_unit_mul_pow_irreducible",
"discrete_valuation_ring.exists_irreducible",
"mul_assoc",
"mul_comm",
"mul_one",
"pow_add",
"units.coe_mul",
"units.mul_inv",
"valuation_ring"
] | A DVR is a valuation ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation_subring extends subring K | (mem_or_inv_mem' : ∀ x : K, x ∈ carrier ∨ x⁻¹ ∈ carrier) | structure | valuation_subring | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"subring"
] | A valuation subring of a field `K` is a subring `A` such that for every `x : K`,
either `x ∈ A` or `x⁻¹ ∈ A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_carrier (x : K) : x ∈ A.carrier ↔ x ∈ A | iff.refl _ | lemma | valuation_subring.mem_carrier | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_subring (x : K) : x ∈ A.to_subring ↔ x ∈ A | iff.refl _ | lemma | valuation_subring.mem_to_subring | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (A B : valuation_subring K)
(h : ∀ x, x ∈ A ↔ x ∈ B) : A = B | set_like.ext h | lemma | valuation_subring.ext | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"set_like.ext",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mem : (0 : K) ∈ A | A.to_subring.zero_mem | lemma | valuation_subring.zero_mem | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mem : (1 : K) ∈ A | A.to_subring.one_mem | lemma | valuation_subring.one_mem | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mem (x y : K) : x ∈ A → y ∈ A → x + y ∈ A | A.to_subring.add_mem | lemma | valuation_subring.add_mem | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem (x y : K) : x ∈ A → y ∈ A → x * y ∈ A | A.to_subring.mul_mem | lemma | valuation_subring.mul_mem | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_mem (x : K) : x ∈ A → (-x) ∈ A | A.to_subring.neg_mem | lemma | valuation_subring.neg_mem | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_or_inv_mem (x : K) : x ∈ A ∨ x⁻¹ ∈ A | A.mem_or_inv_mem' _ | lemma | valuation_subring.mem_or_inv_mem | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_subring_injective : function.injective (to_subring : valuation_subring K → subring K) | λ x y h, by { cases x, cases y, congr' } | lemma | valuation_subring.to_subring_injective | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"subring",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_top (x : K) : x ∈ (⊤ : valuation_subring K) | trivial | lemma | valuation_subring.mem_top | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_top : A ≤ ⊤ | λ a ha, mem_top _ | lemma | valuation_subring.le_top | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_apply (a : A) : algebra_map A K a = a | rfl | lemma | valuation_subring.algebra_map_apply | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"algebra_map",
"algebra_map_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
value_group | valuation_ring.value_group A K | def | valuation_subring.value_group | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_ring.value_group"
] | The value group of the valuation associated to `A`. Note: it is actually a group with zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
valuation : valuation K A.value_group | valuation_ring.valuation A K | def | valuation_subring.valuation | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation",
"valuation_ring.valuation"
] | Any valuation subring of `K` induces a natural valuation on `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inhabited_value_group : inhabited A.value_group | ⟨A.valuation 0⟩ | instance | valuation_subring.inhabited_value_group | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation_le_one (a : A) : A.valuation a ≤ 1 | (valuation_ring.mem_integer_iff A K _).2 ⟨a, rfl⟩ | lemma | valuation_subring.valuation_le_one | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_ring.mem_integer_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_of_valuation_le_one (x : K) (h : A.valuation x ≤ 1) : x ∈ A | let ⟨a, ha⟩ := (valuation_ring.mem_integer_iff A K x).1 h in ha ▸ a.2 | lemma | valuation_subring.mem_of_valuation_le_one | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_ring.mem_integer_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation_le_one_iff (x : K) : A.valuation x ≤ 1 ↔ x ∈ A | ⟨mem_of_valuation_le_one _ _, λ ha, A.valuation_le_one ⟨x, ha⟩⟩ | lemma | valuation_subring.valuation_le_one_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation_eq_iff (x y : K) : A.valuation x = A.valuation y ↔
∃ a : Aˣ, (a : K) * y = x | quotient.eq' | lemma | valuation_subring.valuation_eq_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"quotient.eq'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation_le_iff (x y : K) : A.valuation x ≤ A.valuation y ↔
∃ a : A, (a : K) * y = x | iff.rfl | lemma | valuation_subring.valuation_le_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation_surjective : function.surjective A.valuation | surjective_quot_mk _ | lemma | valuation_subring.valuation_surjective | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"surjective_quot_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation_unit (a : Aˣ) : A.valuation a = 1 | by { rw [← A.valuation.map_one, valuation_eq_iff], use a, simp } | lemma | valuation_subring.valuation_unit | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation_eq_one_iff (a : A) : is_unit a ↔ A.valuation a = 1 | ⟨ λ h, A.valuation_unit h.unit,
λ h, begin
have ha : (a : K) ≠ 0,
{ intro c, rw [c, A.valuation.map_zero] at h, exact zero_ne_one h },
have ha' : (a : K)⁻¹ ∈ A,
{ rw [← valuation_le_one_iff, map_inv₀, h, inv_one] },
apply is_unit_of_mul_eq_one a ⟨a⁻¹, ha'⟩, ext, field_simp,
end ⟩ | lemma | valuation_subring.valuation_eq_one_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"inv_one",
"is_unit",
"is_unit_of_mul_eq_one",
"map_inv₀",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation_lt_one_or_eq_one (a : A) : A.valuation a < 1 ∨ A.valuation a = 1 | lt_or_eq_of_le (A.valuation_le_one a) | lemma | valuation_subring.valuation_lt_one_or_eq_one | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation_lt_one_iff (a : A) : a ∈ local_ring.maximal_ideal A ↔ A.valuation a < 1 | begin
rw local_ring.mem_maximal_ideal,
dsimp [nonunits], rw valuation_eq_one_iff,
exact (A.valuation_le_one a).lt_iff_ne.symm,
end | lemma | valuation_subring.valuation_lt_one_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"local_ring.maximal_ideal",
"local_ring.mem_maximal_ideal",
"nonunits"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_subring (R : subring K) (hR : ∀ x : K, x ∈ R ∨ x⁻¹ ∈ R) : valuation_subring K | { mem_or_inv_mem' := hR, ..R } | def | valuation_subring.of_subring | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"subring",
"valuation_subring"
] | A subring `R` of `K` such that for all `x : K` either `x ∈ R` or `x⁻¹ ∈ R` is
a valuation subring of `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_of_subring (R : subring K) (hR : ∀ x : K, x ∈ R ∨ x⁻¹ ∈ R) (x : K) :
x ∈ of_subring R hR ↔ x ∈ R | iff.refl _ | lemma | valuation_subring.mem_of_subring | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_le (R : valuation_subring K) (S : subring K) (h : R.to_subring ≤ S) :
valuation_subring K | { mem_or_inv_mem' := λ x, (R.mem_or_inv_mem x).imp (@h x) (@h _), ..S} | def | valuation_subring.of_le | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"subring",
"valuation_subring"
] | An overring of a valuation ring is a valuation ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inclusion (R S : valuation_subring K) (h : R ≤ S) : R →+* S | subring.inclusion h | def | valuation_subring.inclusion | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"subring.inclusion",
"valuation_subring"
] | The ring homomorphism induced by the partial order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype (R : valuation_subring K) : R →+* K | subring.subtype R.to_subring | def | valuation_subring.subtype | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"subring.subtype",
"valuation_subring"
] | The canonical ring homomorphism from a valuation ring to its field of fractions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_of_le (R S : valuation_subring K) (h : R ≤ S) :
R.value_group →*₀ S.value_group | { to_fun := quotient.map' id $ λ x y ⟨u, hu⟩, ⟨units.map (R.inclusion S h).to_monoid_hom u, hu⟩,
map_zero' := rfl,
map_one' := rfl,
map_mul' := by { rintro ⟨⟩ ⟨⟩, refl } } | def | valuation_subring.map_of_le | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"quotient.map'",
"valuation_subring"
] | The canonical map on value groups induced by a coarsening of valuation rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone_map_of_le (R S : valuation_subring K) (h : R ≤ S) :
monotone (R.map_of_le S h) | by { rintros ⟨⟩ ⟨⟩ ⟨a, ha⟩, exact ⟨R.inclusion S h a, ha⟩ } | lemma | valuation_subring.monotone_map_of_le | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"monotone",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_of_le_comp_valuation (R S : valuation_subring K) (h : R ≤ S) :
R.map_of_le S h ∘ R.valuation = S.valuation | by { ext, refl } | lemma | valuation_subring.map_of_le_comp_valuation | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_of_le_valuation_apply (R S : valuation_subring K) (h : R ≤ S) (x : K) :
R.map_of_le S h (R.valuation x) = S.valuation x | rfl | lemma | valuation_subring.map_of_le_valuation_apply | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_of_le (R S : valuation_subring K) (h : R ≤ S) : ideal R | (local_ring.maximal_ideal S).comap (R.inclusion S h) | def | valuation_subring.ideal_of_le | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"local_ring.maximal_ideal",
"valuation_subring"
] | The ideal corresponding to a coarsening of a valuation ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime_ideal_of_le (R S : valuation_subring K) (h : R ≤ S) :
(ideal_of_le R S h).is_prime | (local_ring.maximal_ideal S).comap_is_prime _ | instance | valuation_subring.prime_ideal_of_le | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"local_ring.maximal_ideal",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_prime (A : valuation_subring K) (P : ideal A) [P.is_prime] :
valuation_subring K | of_le A (localization.subalgebra.of_field K _ P.prime_compl_le_non_zero_divisors).to_subring $
λ a ha, subalgebra.algebra_map_mem _ (⟨a, ha⟩ : A) | def | valuation_subring.of_prime | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"localization.subalgebra.of_field",
"subalgebra.algebra_map_mem",
"valuation_subring"
] | The coarsening of a valuation ring associated to a prime ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_prime_algebra (A : valuation_subring K) (P : ideal A) [P.is_prime] :
algebra A (A.of_prime P) | subalgebra.algebra _ | instance | valuation_subring.of_prime_algebra | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"algebra",
"ideal",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_prime_scalar_tower (A : valuation_subring K) (P : ideal A) [P.is_prime] :
is_scalar_tower A (A.of_prime P) K | is_scalar_tower.subalgebra' A K K _ | instance | valuation_subring.of_prime_scalar_tower | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"is_scalar_tower",
"is_scalar_tower.subalgebra'",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_prime_localization (A : valuation_subring K) (P : ideal A) [P.is_prime] :
is_localization.at_prime (A.of_prime P) P | by apply localization.subalgebra.is_localization_of_field K P.prime_compl
P.prime_compl_le_non_zero_divisors | instance | valuation_subring.of_prime_localization | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"is_localization.at_prime",
"localization.subalgebra.is_localization_of_field",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_prime (A : valuation_subring K) (P : ideal A) [P.is_prime] :
A ≤ of_prime A P | λ a ha, subalgebra.algebra_map_mem _ (⟨a, ha⟩ : A) | lemma | valuation_subring.le_of_prime | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"subalgebra.algebra_map_mem",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_prime_valuation_eq_one_iff_mem_prime_compl
(A : valuation_subring K)
(P : ideal A) [P.is_prime] (x : A) :
(of_prime A P).valuation x = 1 ↔ x ∈ P.prime_compl | begin
rw [← is_localization.at_prime.is_unit_to_map_iff (A.of_prime P) P x, valuation_eq_one_iff], refl,
end | lemma | valuation_subring.of_prime_valuation_eq_one_iff_mem_prime_compl | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"is_localization.at_prime.is_unit_to_map_iff",
"valuation",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_of_le_of_prime (A : valuation_subring K) (P : ideal A) [P.is_prime] :
ideal_of_le A (of_prime A P) (le_of_prime A P) = P | by { ext, apply is_localization.at_prime.to_map_mem_maximal_iff } | lemma | valuation_subring.ideal_of_le_of_prime | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal",
"is_localization.at_prime.to_map_mem_maximal_iff",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_prime_ideal_of_le (R S : valuation_subring K) (h : R ≤ S) :
of_prime R (ideal_of_le R S h) = S | begin
ext x, split,
{ rintro ⟨a, r, hr, rfl⟩, apply mul_mem, { exact h a.2 },
{ rw [← valuation_le_one_iff, map_inv₀, ← inv_one, inv_le_inv₀],
{ exact not_lt.1 ((not_iff_not.2 $ valuation_lt_one_iff S _).1 hr) },
{ intro hh, erw [valuation.zero_iff, subring.coe_eq_zero_iff] at hh,
apply hr, ... | lemma | valuation_subring.of_prime_ideal_of_le | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal.zero_mem",
"inv_le_inv₀",
"inv_one",
"is_unit_of_mul_eq_one",
"map_inv₀",
"mem_nonunits_iff",
"nonunits",
"not_not",
"one_ne_zero",
"subring.coe_eq_zero_iff",
"valuation.zero_iff",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_prime_le_of_le (P Q : ideal A) [P.is_prime] [Q.is_prime]
(h : P ≤ Q) : of_prime A Q ≤ of_prime A P | λ x ⟨a, s, hs, he⟩, ⟨a, s, λ c, hs (h c), he⟩ | lemma | valuation_subring.of_prime_le_of_le | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal_of_le_le_of_le (R S : valuation_subring K)
(hR : A ≤ R) (hS : A ≤ S) (h : R ≤ S) :
ideal_of_le A S hS ≤ ideal_of_le A R hR | λ x hx, (valuation_lt_one_iff R _).2 begin
by_contra c, push_neg at c, replace c := monotone_map_of_le R S h c,
rw [(map_of_le _ _ _).map_one, map_of_le_valuation_apply] at c,
apply not_le_of_lt ((valuation_lt_one_iff S _).1 hx) c,
end | lemma | valuation_subring.ideal_of_le_le_of_le | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"by_contra",
"map_one",
"not_le_of_lt",
"valuation_subring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_spectrum_equiv :
prime_spectrum A ≃ { S | A ≤ S } | { to_fun := λ P, ⟨of_prime A P.as_ideal, le_of_prime _ _⟩,
inv_fun := λ S, ⟨ideal_of_le _ S S.2, infer_instance⟩,
left_inv := λ P, by { ext1, simp },
right_inv := λ S, by { ext1, simp } } | def | valuation_subring.prime_spectrum_equiv | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"inv_fun",
"prime_spectrum"
] | The equivalence between coarsenings of a valuation ring and its prime ideals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime_spectrum_order_equiv : (prime_spectrum A)ᵒᵈ ≃o {S | A ≤ S} | { map_rel_iff' := λ P Q,
⟨ λ h, begin
have := ideal_of_le_le_of_le A _ _ _ _ h,
iterate 2 { erw ideal_of_le_of_prime at this },
exact this,
end,
λ h, by { apply of_prime_le_of_le, exact h } ⟩,
..(prime_spectrum_equiv A) } | def | valuation_subring.prime_spectrum_order_equiv | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"prime_spectrum"
] | An ordered variant of `prime_spectrum_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_order_overring : linear_order { S | A ≤ S } | { le_total :=
let i : is_total (prime_spectrum A) (≤) := ⟨λ ⟨x, _⟩ ⟨y, _⟩, has_le.le.is_total.total x y⟩ in
by exactI (prime_spectrum_order_equiv A).symm.to_rel_embedding.is_total.total,
decidable_le := infer_instance,
..(infer_instance : partial_order _) } | instance | valuation_subring.linear_order_overring | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"prime_spectrum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation_subring : valuation_subring K | { mem_or_inv_mem' := begin
intros x,
cases le_or_lt (v x) 1,
{ left, exact h },
{ right, change v x⁻¹ ≤ 1,
rw [map_inv₀ v, ← inv_one, inv_le_inv₀],
{ exact le_of_lt h },
{ intro c, simpa [c] using h },
{ exact one_ne_zero } }
end,
.. v.integer } | def | valuation.valuation_subring | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"inv_le_inv₀",
"inv_one",
"map_inv₀",
"one_ne_zero",
"valuation_subring"
] | The valuation subring associated to a valuation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_valuation_subring_iff (x : K) : x ∈ v.valuation_subring ↔ v x ≤ 1 | iff.refl _ | lemma | valuation.mem_valuation_subring_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equiv_iff_valuation_subring : v₁.is_equiv v₂ ↔
v₁.valuation_subring = v₂.valuation_subring | begin
split,
{ intros h, ext x, specialize h x 1, simpa using h },
{ intros h, apply is_equiv_of_val_le_one,
intros x,
have : x ∈ v₁.valuation_subring ↔ x ∈ v₂.valuation_subring, by rw h,
simpa using this }
end | lemma | valuation.is_equiv_iff_valuation_subring | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equiv_valuation_valuation_subring :
v.is_equiv v.valuation_subring.valuation | begin
rw [is_equiv_iff_val_le_one],
intro x,
rw valuation_subring.valuation_le_one_iff,
refl,
end | lemma | valuation.is_equiv_valuation_valuation_subring | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"valuation_subring.valuation_le_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
valuation_subring_valuation : A.valuation.valuation_subring = A | by { ext, rw ← A.valuation_le_one_iff, refl } | lemma | valuation_subring.valuation_subring_valuation | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_group : subgroup Kˣ | (A.valuation.to_monoid_with_zero_hom.to_monoid_hom.comp (units.coe_hom K)).ker | def | valuation_subring.unit_group | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"subgroup",
"units.coe_hom"
] | The unit group of a valuation subring, as a subgroup of `Kˣ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_unit_group_iff (x : Kˣ) : x ∈ A.unit_group ↔ A.valuation x = 1 | iff.rfl | lemma | valuation_subring.mem_unit_group_iff | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_group_mul_equiv : A.unit_group ≃* Aˣ | { to_fun := λ x,
{ val := ⟨x, mem_of_valuation_le_one A _ x.prop.le⟩,
inv := ⟨↑(x⁻¹), mem_of_valuation_le_one _ _ (x⁻¹).prop.le⟩,
val_inv := subtype.ext (units.mul_inv x),
inv_val := subtype.ext (units.inv_mul x) },
inv_fun := λ x, ⟨units.map A.subtype.to_monoid_hom x, A.valuation_unit x⟩,
left_inv :=... | def | valuation_subring.unit_group_mul_equiv | ring_theory.valuation | src/ring_theory/valuation/valuation_subring.lean | [
"ring_theory.valuation.valuation_ring",
"ring_theory.localization.as_subring",
"ring_theory.subring.pointwise",
"algebraic_geometry.prime_spectrum.basic"
] | [
"inv_fun",
"subtype.ext",
"units.inv_mul",
"units.mul_inv"
] | For a valuation subring `A`, `A.unit_group` agrees with the units of `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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