-- Mathlib/Topology/Constructions.lean
theorem continuous_inf_dom_left₂ {X Y Z} {f : X → Y → Z} {ta1 ta2 : TopologicalSpace X}
    {tb1 tb2 : TopologicalSpace Y} {tc1 : TopologicalSpace Z}
    (h : by haveI := ta1; haveI := tb1; exact Continuous fun p : X × Y => f p.1 p.2) : by
    haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact Continuous fun p : X × Y => f p.1 p.2

Proof state:
X✝ : Type u
Y✝ : Type v
Z✝ : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X✝
inst✝⁴ : TopologicalSpace Y✝
inst✝³ : TopologicalSpace Z✝
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
X : Type ?u.27326
Y : Type ?u.27332
Z : Type ?u.27338
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
this✝ : TopologicalSpace X
this : TopologicalSpace Y
⊢ Sort ?u.27340

Tactic:
exact Continuous fun p : X × Y => f p.1 p.2

Result:
no goals

Proof state:
X✝ : Type u
Y✝ : Type v
Z✝ : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X✝
inst✝⁴ : TopologicalSpace Y✝
inst✝³ : TopologicalSpace Z✝
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
X : Type ?u.27326
Y : Type ?u.27332
Z : Type ?u.27338
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
h : Continuous fun p => f p.1 p.2
this : TopologicalSpace X
⊢ Sort ?u.27343

Tactic:
haveI := tb1 ⊓ tb2

Result:
X✝ : Type u
Y✝ : Type v
Z✝ : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X✝
inst✝⁴ : TopologicalSpace Y✝
inst✝³ : TopologicalSpace Z✝
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
X : Type ?u.27326
Y : Type ?u.27332
Z : Type ?u.27338
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
h : Continuous fun p => f p.1 p.2
this✝ : TopologicalSpace X
this : TopologicalSpace Y
⊢ Sort ?u.27343

Proof state:
X✝ : Type u
Y✝ : Type v
Z✝ : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X✝
inst✝⁴ : TopologicalSpace Y✝
inst✝³ : TopologicalSpace Z✝
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
X : Type ?u.27326
Y : Type ?u.27332
Z : Type ?u.27338
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
this : TopologicalSpace X
⊢ Sort ?u.27340

Tactic:
haveI := tb1

Result:
X✝ : Type u
Y✝ : Type v
Z✝ : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X✝
inst✝⁴ : TopologicalSpace Y✝
inst✝³ : TopologicalSpace Z✝
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
X : Type ?u.27326
Y : Type ?u.27332
Z : Type ?u.27338
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
this✝ : TopologicalSpace X
this : TopologicalSpace Y
⊢ Sort ?u.27340

Proof state:
X✝ : Type u
Y✝ : Type v
Z✝ : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X✝
inst✝⁴ : TopologicalSpace Y✝
inst✝³ : TopologicalSpace Z✝
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
X : Type ?u.27326
Y : Type ?u.27332
Z : Type ?u.27338
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
⊢ Sort ?u.27340

Tactic:
haveI := ta1

Result:
X✝ : Type u
Y✝ : Type v
Z✝ : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X✝
inst✝⁴ : TopologicalSpace Y✝
inst✝³ : TopologicalSpace Z✝
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
X : Type ?u.27326
Y : Type ?u.27332
Z : Type ?u.27338
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
this : TopologicalSpace X
⊢ Sort ?u.27340

Proof state:
X : Type u_5
Y : Type u_6
Z : Type u_7
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
h : Continuous fun p => f p.1 p.2
ha : Continuous id
hb : Continuous id
h_continuous_id : Continuous (Prod.map id id)
⊢ Continuous fun p => f p.1 p.2

Tactic:
exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id

Result:
no goals

Proof state:
X : Type u_5
Y : Type u_6
Z : Type u_7
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
h : Continuous fun p => f p.1 p.2
⊢ Continuous fun p => f p.1 p.2

Tactic:
have ha := @continuous_inf_dom_left _ _ id ta1 ta2 ta1 (@continuous_id _ (id _))

Result:
X : Type u_5
Y : Type u_6
Z : Type u_7
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
h : Continuous fun p => f p.1 p.2
ha : Continuous id
⊢ Continuous fun p => f p.1 p.2

Proof state:
X : Type u_5
Y : Type u_6
Z : Type u_7
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
h : Continuous fun p => f p.1 p.2
ha : Continuous id
hb : Continuous id
⊢ Continuous fun p => f p.1 p.2

Tactic:
have h_continuous_id := @Continuous.prodMap _ _ _ _ ta1 tb1 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb

Result:
X : Type u_5
Y : Type u_6
Z : Type u_7
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
h : Continuous fun p => f p.1 p.2
ha : Continuous id
hb : Continuous id
h_continuous_id : Continuous (Prod.map id id)
⊢ Continuous fun p => f p.1 p.2

Proof state:
X : Type u_5
Y : Type u_6
Z : Type u_7
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
h : Continuous fun p => f p.1 p.2
ha : Continuous id
⊢ Continuous fun p => f p.1 p.2

Tactic:
have hb := @continuous_inf_dom_left _ _ id tb1 tb2 tb1 (@continuous_id _ (id _))

Result:
X : Type u_5
Y : Type u_6
Z : Type u_7
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
h : Continuous fun p => f p.1 p.2
ha : Continuous id
hb : Continuous id
⊢ Continuous fun p => f p.1 p.2

Proof state:
X✝ : Type u
Y✝ : Type v
Z✝ : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X✝
inst✝⁴ : TopologicalSpace Y✝
inst✝³ : TopologicalSpace Z✝
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
X : Type ?u.27326
Y : Type ?u.27332
Z : Type ?u.27338
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
h : Continuous fun p => f p.1 p.2
⊢ Sort ?u.27343

Tactic:
haveI := ta1 ⊓ ta2

Result:
X✝ : Type u
Y✝ : Type v
Z✝ : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X✝
inst✝⁴ : TopologicalSpace Y✝
inst✝³ : TopologicalSpace Z✝
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
X : Type ?u.27326
Y : Type ?u.27332
Z : Type ?u.27338
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
h : Continuous fun p => f p.1 p.2
this : TopologicalSpace X
⊢ Sort ?u.27343

Proof state:
X✝ : Type u
Y✝ : Type v
Z✝ : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝⁵ : TopologicalSpace X✝
inst✝⁴ : TopologicalSpace Y✝
inst✝³ : TopologicalSpace Z✝
inst✝² : TopologicalSpace W
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
X : Type ?u.27326
Y : Type ?u.27332
Z : Type ?u.27338
f : X → Y → Z
ta1 ta2 : TopologicalSpace X
tb1 tb2 : TopologicalSpace Y
tc1 : TopologicalSpace Z
h : Continuous fun p => f p.1 p.2
this✝ : TopologicalSpace X
this : TopologicalSpace Y
⊢ Sort ?u.27343

Tactic:
exact Continuous fun p : X × Y => f p.1 p.2

Result:
no goals