Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: case refine_1.refine_1
A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
x : K
hx : x ∈ span A (↑↑I * ↑↑J)
y : K
hy : y ∈ ↑↑I * ↑↑J
⊢ (IsLocalization.map L f hf) y ∈ ↑(span B (⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J))

pp: case refine_1.refine_2
A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
x : K
hx : x ∈ span A (↑↑I * ↑↑J)
y z : K
x✝¹ : y ∈ span A (↑↑I * ↑↑J)
x✝ : z ∈ span A (↑↑I * ↑↑J)
hy : (IsLocalization.map L f hf) y ∈ ↑(span B (⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J))
hz : (IsLocalization.map L f hf) z ∈ ↑(span B (⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J))
⊢ (IsLocalization.map L f hf) (y + z) ∈
    ↑(span B (⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J))

pp: case refine_1.refine_3
A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
x : K
hx : x ∈ span A (↑↑I * ↑↑J)
a : A
y : K
x✝ : y ∈ span A (↑↑I * ↑↑J)
hy : (IsLocalization.map L f hf) y ∈ ↑(span B (⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J))
⊢ (IsLocalization.map L f hf) (a • y) ∈
    ↑(span B (⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J))

Tactic:
· rcases Set.mem_mul.mp hy with ⟨i, hi, j, hj, rfl⟩
      exact subset_span <| Set.mem_mul.mpr
        ⟨map_f i, ⟨i, hi, by simp⟩, map_f j, ⟨j, hj, by simp⟩, by simp⟩

Goal state after:
case refine_1.refine_2
A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
x : K
hx : x ∈ span A (↑↑I * ↑↑J)
y z : K
x✝¹ : y ∈ span A (↑↑I * ↑↑J)
x✝ : z ∈ span A (↑↑I * ↑↑J)
hy : (IsLocalization.map L f hf) y ∈ ↑(span B (⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J))
hz : (IsLocalization.map L f hf) z ∈ ↑(span B (⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J))
⊢ (IsLocalization.map L f hf) (y + z) ∈
    ↑(span B (⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J))

case refine_1.refine_3
A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I J : FractionalIdeal M K
x : K
hx : x ∈ span A (↑↑I * ↑↑J)
a : A
y : K
x✝ : y ∈ span A (↑↑I * ↑↑J)
hy : (IsLocalization.map L f hf) y ∈ ↑(span B (⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J))
⊢ (IsLocalization.map L f hf) (a • y) ∈
    ↑(span B (⇑(IsLocalization.map L f hf) '' ↑I * ⇑(IsLocalization.map L f hf) '' ↑J))