-- Mathlib/RingTheory/Valuation/LocalSubring.lean
lemma bijective_rangeRestrict_comp_of_valuationRing [IsDomain R] [ValuationRing R]
    [IsLocalRing S] [Algebra R K] [IsFractionRing R K]
    (f : R →+* S) (g : S →+* K) (h : g.comp f = algebraMap R K) [IsLocalHom f] :
    Function.Bijective (g.rangeRestrict.comp f)

Proof state:
case mk.intro
R : Type u_1
S : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Field K
inst✝⁵ : IsDomain R
inst✝⁴ : ValuationRing R
inst✝³ : IsLocalRing S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
f : R →+* S
g : S →+* K
h : g.comp f = algebraMap R K
inst✝ : IsLocalHom f
V : ValuationSubring K := { toSubring := (algebraMap R K).range, mem_or_inv_mem' := ⋯ }
this : LocalSubring.range g ≤ V.toLocalSubring
x : S
⊢ ∃ a, (g.rangeRestrict.comp f) a = ⟨g x, ⋯⟩

Tactic:
obtain ⟨y, hy⟩ := this.1 ⟨x, rfl⟩

Result:
case mk.intro.intro
R : Type u_1
S : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Field K
inst✝⁵ : IsDomain R
inst✝⁴ : ValuationRing R
inst✝³ : IsLocalRing S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
f : R →+* S
g : S →+* K
h : g.comp f = algebraMap R K
inst✝ : IsLocalHom f
V : ValuationSubring K := { toSubring := (algebraMap R K).range, mem_or_inv_mem' := ⋯ }
this : LocalSubring.range g ≤ V.toLocalSubring
x : S
y : R
hy : (algebraMap R K) y = g x
⊢ ∃ a, (g.rangeRestrict.comp f) a = ⟨g x, ⋯⟩

Proof state:
case h.e'_3
R : Type u_1
S : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Field K
inst✝⁵ : IsDomain R
inst✝⁴ : ValuationRing R
inst✝³ : IsLocalRing S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
f : R →+* S
g : S →+* K
h : g.comp f = algebraMap R K
inst✝ : IsLocalHom f
⊢ Subtype.val ∘ ⇑(g.rangeRestrict.comp f) = ⇑(algebraMap R K)

Tactic:
rw [← h]

Result:
case h.e'_3
R : Type u_1
S : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Field K
inst✝⁵ : IsDomain R
inst✝⁴ : ValuationRing R
inst✝³ : IsLocalRing S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
f : R →+* S
g : S →+* K
h : g.comp f = algebraMap R K
inst✝ : IsLocalHom f
⊢ Subtype.val ∘ ⇑(g.rangeRestrict.comp f) = ⇑(g.comp f)

Proof state:
R : Type u_1
S : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Field K
inst✝⁵ : IsDomain R
inst✝⁴ : ValuationRing R
inst✝³ : IsLocalRing S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
f : R →+* S
g : S →+* K
h : g.comp f = algebraMap R K
inst✝ : IsLocalHom f
V : ValuationSubring K := { toSubring := (algebraMap R K).range, mem_or_inv_mem' := ⋯ }
this : LocalSubring.range g ≤ V.toLocalSubring
x : S
y : R
hy : (algebraMap R K) y = g x
⊢ ↑((g.rangeRestrict.comp f) y) = ↑⟨g x, ⋯⟩

Tactic:
simpa [← h] using hy

Result:
no goals

Proof state:
case refine_2.a.mk.intro.h.a
R : Type u_1
S : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Field K
inst✝⁵ : IsDomain R
inst✝⁴ : ValuationRing R
inst✝³ : IsLocalRing S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
f : R →+* S
g : S →+* K
h : g.comp f = algebraMap R K
inst✝ : IsLocalHom f
V : ValuationSubring K := { toSubring := (algebraMap R K).range, mem_or_inv_mem' := ⋯ }
H : (algebraMap R K).range ≤ g.range
a : R
ha : IsUnit ⟨(algebraMap R K) a, ⋯⟩
⊢ IsUnit (g.rangeRestrict (f a))

Tactic:
convert ha

Result:
case h.e'_3.h.e'_3
R : Type u_1
S : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Field K
inst✝⁵ : IsDomain R
inst✝⁴ : ValuationRing R
inst✝³ : IsLocalRing S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
f : R →+* S
g : S →+* K
h : g.comp f = algebraMap R K
inst✝ : IsLocalHom f
V : ValuationSubring K := { toSubring := (algebraMap R K).range, mem_or_inv_mem' := ⋯ }
H : (algebraMap R K).range ≤ g.range
a : R
ha : IsUnit ⟨(algebraMap R K) a, ⋯⟩
⊢ ↑(g.rangeRestrict (f a)) = (algebraMap R K) a

Proof state:
case refine_2.a
R : Type u_1
S : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Field K
inst✝⁵ : IsDomain R
inst✝⁴ : ValuationRing R
inst✝³ : IsLocalRing S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
f : R →+* S
g : S →+* K
h : g.comp f = algebraMap R K
inst✝ : IsLocalHom f
V : ValuationSubring K := { toSubring := (algebraMap R K).range, mem_or_inv_mem' := ⋯ }
H : (algebraMap R K).range ≤ g.range
⊢ V.toLocalSubring ≤ LocalSubring.range g

Tactic:
refine ⟨H, ⟨?_⟩⟩

Result:
case refine_2.a
R : Type u_1
S : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Field K
inst✝⁵ : IsDomain R
inst✝⁴ : ValuationRing R
inst✝³ : IsLocalRing S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
f : R →+* S
g : S →+* K
h : g.comp f = algebraMap R K
inst✝ : IsLocalHom f
V : ValuationSubring K := { toSubring := (algebraMap R K).range, mem_or_inv_mem' := ⋯ }
H : (algebraMap R K).range ≤ g.range
⊢ ∀ (a : ↥V.toLocalSubring.toSubring), IsUnit ((Subring.inclusion H) a) → IsUnit a

Proof state:
case refine_2.a.mk.intro.h
R : Type u_1
S : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Field K
inst✝⁵ : IsDomain R
inst✝⁴ : ValuationRing R
inst✝³ : IsLocalRing S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
f : R →+* S
g : S →+* K
h : g.comp f = algebraMap R K
inst✝ : IsLocalHom f
V : ValuationSubring K := { toSubring := (algebraMap R K).range, mem_or_inv_mem' := ⋯ }
H : (algebraMap R K).range ≤ g.range
a : R
ha : IsUnit ⟨(algebraMap R K) a, ⋯⟩
⊢ IsUnit (f a)

Tactic:
apply (IsLocalHom.of_surjective g.rangeRestrict g.rangeRestrict_surjective).1

Result:
case refine_2.a.mk.intro.h.a
R : Type u_1
S : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Field K
inst✝⁵ : IsDomain R
inst✝⁴ : ValuationRing R
inst✝³ : IsLocalRing S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
f : R →+* S
g : S →+* K
h : g.comp f = algebraMap R K
inst✝ : IsLocalHom f
V : ValuationSubring K := { toSubring := (algebraMap R K).range, mem_or_inv_mem' := ⋯ }
H : (algebraMap R K).range ≤ g.range
a : R
ha : IsUnit ⟨(algebraMap R K) a, ⋯⟩
⊢ IsUnit (g.rangeRestrict (f a))

Proof state:
case refine_1
R : Type u_1
S : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Field K
inst✝⁵ : IsDomain R
inst✝⁴ : ValuationRing R
inst✝³ : IsLocalRing S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
f : R →+* S
g : S →+* K
h : g.comp f = algebraMap R K
inst✝ : IsLocalHom f
⊢ Function.Injective ⇑(g.rangeRestrict.comp f)

Tactic:
exact .of_comp (f := Subtype.val) (by convert (IsFractionRing.injective R K); rw [← h]; rfl)

Result:
no goals

Proof state:
case h.e'_3.h.e'_3
R : Type u_1
S : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Field K
inst✝⁵ : IsDomain R
inst✝⁴ : ValuationRing R
inst✝³ : IsLocalRing S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
f : R →+* S
g : S →+* K
h : g.comp f = algebraMap R K
inst✝ : IsLocalHom f
V : ValuationSubring K := { toSubring := (algebraMap R K).range, mem_or_inv_mem' := ⋯ }
H : (algebraMap R K).range ≤ g.range
a : R
ha : IsUnit ⟨(algebraMap R K) a, ⋯⟩
⊢ ↑(g.rangeRestrict (f a)) = (algebraMap R K) a

Tactic:
simp [← h]

Result:
no goals