-- Mathlib/RingTheory/RingHom/Locally.lean
/-- If `P` is preserved by localization away, then so is `Locally P`. -/
lemma locally_localizationAwayPreserves (hPl : LocalizationAwayPreserves P) :
    LocalizationAwayPreserves (Locally P)

Proof state:
case intro.intro
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
s : Set S
hsone : Ideal.span s = ⊤
hs : ∀ t ∈ s, (fun {R S} [CommRing R] [CommRing S] => P) ((algebraMap S (Localization.Away t)).comp f)
⊢ Locally (fun {R S} [CommRing R] [CommRing S] => P) (IsLocalization.Away.map R' S' f r)

Tactic:
rw [locally_iff_exists hPl.respectsIso]

Result:
case intro.intro
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
s : Set S
hsone : Ideal.span s = ⊤
hs : ∀ t ∈ s, (fun {R S} [CommRing R] [CommRing S] => P) ((algebraMap S (Localization.Away t)).comp f)
⊢ ∃ ι s,
    ∃ (_ : Ideal.span (Set.range s) = ⊤),
      ∃ Sₜ x x_1,
        ∃ (_ : ∀ (i : ι), IsLocalization.Away (s i) (Sₜ i)),
          ∀ (i : ι), P ((algebraMap S' (Sₜ i)).comp (IsLocalization.Away.map R' S' f r))

Proof state:
case intro.intro.refine_1
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
s : Set S
hsone : Ideal.span s = ⊤
hs : ∀ t ∈ s, (fun {R S} [CommRing R] [CommRing S] => P) ((algebraMap S (Localization.Away t)).comp f)
rₐ : (a : ↑s) → Localization.Away ↑a := fun a => (algebraMap S (Localization.Away ↑a)) (f r)
Sₐ : ↑s → Type u := fun a => Localization.Away (rₐ a)
this✝² : ∀ (a : ↑s), IsLocalization.Away (((algebraMap S (Localization.Away ↑a)).comp f) r) (Sₐ a)
this✝¹ :
  ∀ (a : ↑s),
    IsLocalization (Algebra.algebraMapSubmonoid (Localization.Away ↑a) (Submonoid.map f (Submonoid.powers r))) (Sₐ a)
H : ∀ (a : ↑s), Submonoid.powers (f r) ≤ Submonoid.comap (algebraMap S (Localization.Away ↑a)) (Submonoid.powers (rₐ a))
this✝ : (a : ↑s) → Algebra S' (Sₐ a) :=
  fun a => (IsLocalization.map (Sₐ a) (algebraMap S (Localization.Away ↑a)) ⋯).toAlgebra
this : ∀ (a : ↑s), IsScalarTower S S' (Sₐ a)
⊢ Ideal.span (Set.range fun a => (algebraMap S S') ↑a) = ⊤

Tactic:
rw [← Set.image_eq_range, ← Ideal.map_span, hsone, Ideal.map_top]

Result:
no goals

Proof state:
case intro.intro.refine_3
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
s : Set S
hsone : Ideal.span s = ⊤
hs : ∀ t ∈ s, (fun {R S} [CommRing R] [CommRing S] => P) ((algebraMap S (Localization.Away t)).comp f)
rₐ : (a : ↑s) → Localization.Away ↑a := fun a => (algebraMap S (Localization.Away ↑a)) (f r)
Sₐ : ↑s → Type u := fun a => Localization.Away (rₐ a)
this✝² : ∀ (a : ↑s), IsLocalization.Away (((algebraMap S (Localization.Away ↑a)).comp f) r) (Sₐ a)
this✝¹ :
  ∀ (a : ↑s),
    IsLocalization (Algebra.algebraMapSubmonoid (Localization.Away ↑a) (Submonoid.map f (Submonoid.powers r))) (Sₐ a)
H : ∀ (a : ↑s), Submonoid.powers (f r) ≤ Submonoid.comap (algebraMap S (Localization.Away ↑a)) (Submonoid.powers (rₐ a))
this✝ : (a : ↑s) → Algebra S' (Sₐ a) :=
  fun a => (IsLocalization.map (Sₐ a) (algebraMap S (Localization.Away ↑a)) ⋯).toAlgebra
this : ∀ (a : ↑s), IsScalarTower S S' (Sₐ a)
a : ↑s
⊢ P ((algebraMap S' (Sₐ a)).comp (IsLocalization.Away.map R' S' f r))

Tactic:
rw [algebraMap_toAlgebra, IsLocalization.Away.map, IsLocalization.map_comp_map]

Result:
case intro.intro.refine_3
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
s : Set S
hsone : Ideal.span s = ⊤
hs : ∀ t ∈ s, (fun {R S} [CommRing R] [CommRing S] => P) ((algebraMap S (Localization.Away t)).comp f)
rₐ : (a : ↑s) → Localization.Away ↑a := fun a => (algebraMap S (Localization.Away ↑a)) (f r)
Sₐ : ↑s → Type u := fun a => Localization.Away (rₐ a)
this✝² : ∀ (a : ↑s), IsLocalization.Away (((algebraMap S (Localization.Away ↑a)).comp f) r) (Sₐ a)
this✝¹ :
  ∀ (a : ↑s),
    IsLocalization (Algebra.algebraMapSubmonoid (Localization.Away ↑a) (Submonoid.map f (Submonoid.powers r))) (Sₐ a)
H : ∀ (a : ↑s), Submonoid.powers (f r) ≤ Submonoid.comap (algebraMap S (Localization.Away ↑a)) (Submonoid.powers (rₐ a))
this✝ : (a : ↑s) → Algebra S' (Sₐ a) :=
  fun a => (IsLocalization.map (Sₐ a) (algebraMap S (Localization.Away ↑a)) ⋯).toAlgebra
this : ∀ (a : ↑s), IsScalarTower S S' (Sₐ a)
a : ↑s
⊢ P (IsLocalization.map (Sₐ a) ((algebraMap S (Localization.Away ↑a)).comp f) ⋯)

Proof state:
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
s : Set S
hsone : Ideal.span s = ⊤
hs : ∀ t ∈ s, (fun {R S} [CommRing R] [CommRing S] => P) ((algebraMap S (Localization.Away t)).comp f)
rₐ : (a : ↑s) → Localization.Away ↑a := fun a => (algebraMap S (Localization.Away ↑a)) (f r)
Sₐ : ↑s → Type u := fun a => Localization.Away (rₐ a)
this : ∀ (a : ↑s), IsLocalization.Away (((algebraMap S (Localization.Away ↑a)).comp f) r) (Sₐ a)
a : ↑s
⊢ IsLocalization (Algebra.algebraMapSubmonoid (Localization.Away ↑a) (Submonoid.map f (Submonoid.powers r))) (Sₐ a)

Tactic:
convert inferInstanceAs (IsLocalization.Away (rₐ a) (Sₐ a))

Result:
case h.e.h.e'_3.h
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
s : Set S
hsone : Ideal.span s = ⊤
hs : ∀ t ∈ s, (fun {R S} [CommRing R] [CommRing S] => P) ((algebraMap S (Localization.Away t)).comp f)
rₐ : (a : ↑s) → Localization.Away ↑a := fun a => (algebraMap S (Localization.Away ↑a)) (f r)
Sₐ : ↑s → Type u := fun a => Localization.Away (rₐ a)
this : ∀ (a : ↑s), IsLocalization.Away (((algebraMap S (Localization.Away ↑a)).comp f) r) (Sₐ a)
a : ↑s
⊢ Algebra.algebraMapSubmonoid (Localization.Away ↑a) (Submonoid.map f (Submonoid.powers r)) = Submonoid.powers (rₐ a)