-- Mathlib/RingTheory/FractionalIdeal/Extended.lean
theorem extended_add : (I + J).extended L hf = (I.extended L hf) + (J.extended L hf)

Proof state:
case a.ht.intro.intro
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
y : K
hy : y ∈ ↑I ∪ ↑J
⊢ y ∈ I + J

Tactic:
exact hy.elim (fun h ↦ (mem_add I J y).mpr ⟨y, h, 0, zero_mem J, add_zero y⟩)
  (fun h ↦ (mem_add I J y).mpr ⟨0, zero_mem I, y, h, zero_add y⟩)

Result:
no goals

Proof state:
case a.ht
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
⊢ ⇑(IsLocalization.map L f hf) '' (↑I ∪ ↑J) ⊆ ↑(span B (⇑(IsLocalization.map L f hf) '' ↑(I + J)))

Tactic:
rintro _ ⟨y, hy, rfl⟩

Result:
case a.ht.intro.intro
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
y : K
hy : y ∈ ↑I ∪ ↑J
⊢ (IsLocalization.map L f hf) y ∈ ↑(span B (⇑(IsLocalization.map L f hf) '' ↑(I + J)))

Proof state:
case a.hs.intro.intro.intro.intro.intro.intro
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
i : K
hi : i ∈ I
j : K
hj : j ∈ J
hy : i + j ∈ ↑(I + J)
⊢ (IsLocalization.map L f hf) i + (IsLocalization.map L f hf) j ∈ ↑(span B (⇑(IsLocalization.map L f hf) '' (↑I ∪ ↑J)))

Tactic:
exact add_mem (Submodule.subset_span ⟨i, Set.mem_union_left _ hi, by simp⟩)
  (Submodule.subset_span ⟨j, Set.mem_union_right _ hj, by simp⟩)

Result:
no goals

Proof state:
case a.hs.intro.intro.intro.intro.intro.intro
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
i : K
hi : i ∈ I
j : K
hj : j ∈ J
hy : i + j ∈ ↑(I + J)
⊢ (IsLocalization.map L f hf) (i + j) ∈ ↑(span B (⇑(IsLocalization.map L f hf) '' (↑I ∪ ↑J)))

Tactic:
rw [RingHom.map_add]

Result:
case a.hs.intro.intro.intro.intro.intro.intro
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
i : K
hi : i ∈ I
j : K
hj : j ∈ J
hy : i + j ∈ ↑(I + J)
⊢ (IsLocalization.map L f hf) i + (IsLocalization.map L f hf) j ∈ ↑(span B (⇑(IsLocalization.map L f hf) '' (↑I ∪ ↑J)))