Theorem: continuous_sInf_dom₂
File: Mathlib/Topology/Constructions.lean

tactic: haveI := sInf tas
state_before: α✝ : Type u
β✝ : Type v
γ✝ : Type ?u.29369
δ : Type ?u.29372
ε : Type ?u.29375
ζ : Type ?u.29378
inst✝⁵ : TopologicalSpace α✝
inst✝⁴ : TopologicalSpace β✝
inst✝³ : TopologicalSpace γ✝
inst✝² : TopologicalSpace δ
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
α : Type ?u.29415
β : Type ?u.29419
γ : Type ?u.29428
f : α → β → γ
tas : Set (TopologicalSpace α)
tbs : Set (TopologicalSpace β)
ta : TopologicalSpace α
tb : TopologicalSpace β
tc : TopologicalSpace γ
ha : ta ∈ tas
hb : tb ∈ tbs
hf : Continuous fun p => f p.fst p.snd
⊢ Sort ?u.29593

tactic: haveI := sInf tbs
state_before: α✝ : Type u
β✝ : Type v
γ✝ : Type ?u.29369
δ : Type ?u.29372
ε : Type ?u.29375
ζ : Type ?u.29378
inst✝⁵ : TopologicalSpace α✝
inst✝⁴ : TopologicalSpace β✝
inst✝³ : TopologicalSpace γ✝
inst✝² : TopologicalSpace δ
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
α : Type ?u.29415
β : Type ?u.29419
γ : Type ?u.29428
f : α → β → γ
tas : Set (TopologicalSpace α)
tbs : Set (TopologicalSpace β)
ta : TopologicalSpace α
tb : TopologicalSpace β
tc : TopologicalSpace γ
ha : ta ∈ tas
hb : tb ∈ tbs
hf : Continuous fun p => f p.fst p.snd
this : TopologicalSpace α
⊢ Sort ?u.29593

tactic: exact @Continuous _ _ _ tc fun p : α × β => f p.1 p.2
state_before: α✝ : Type u
β✝ : Type v
γ✝ : Type ?u.29369
δ : Type ?u.29372
ε : Type ?u.29375
ζ : Type ?u.29378
inst✝⁵ : TopologicalSpace α✝
inst✝⁴ : TopologicalSpace β✝
inst✝³ : TopologicalSpace γ✝
inst✝² : TopologicalSpace δ
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
α : Type ?u.29415
β : Type ?u.29419
γ : Type ?u.29428
f : α → β → γ
tas : Set (TopologicalSpace α)
tbs : Set (TopologicalSpace β)
ta : TopologicalSpace α
tb : TopologicalSpace β
tc : TopologicalSpace γ
ha : ta ∈ tas
hb : tb ∈ tbs
hf : Continuous fun p => f p.fst p.snd
this✝ : TopologicalSpace α
this : TopologicalSpace β
⊢ Sort ?u.29593

tactic: have ha := continuous_sInf_dom ha continuous_id
state_before: α✝ : Type u
β✝ : Type v
γ✝ : Type ?u.29369
δ : Type ?u.29372
ε : Type ?u.29375
ζ : Type ?u.29378
inst✝⁵ : TopologicalSpace α✝
inst✝⁴ : TopologicalSpace β✝
inst✝³ : TopologicalSpace γ✝
inst✝² : TopologicalSpace δ
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β → γ
tas : Set (TopologicalSpace α)
tbs : Set (TopologicalSpace β)
ta : TopologicalSpace α
tb : TopologicalSpace β
tc : TopologicalSpace γ
ha : ta ∈ tas
hb : tb ∈ tbs
hf : Continuous fun p => f p.fst p.snd
⊢ Continuous fun p => f p.fst p.snd

tactic: have hb := continuous_sInf_dom hb continuous_id
state_before: α✝ : Type u
β✝ : Type v
γ✝ : Type ?u.29369
δ : Type ?u.29372
ε : Type ?u.29375
ζ : Type ?u.29378
inst✝⁵ : TopologicalSpace α✝
inst✝⁴ : TopologicalSpace β✝
inst✝³ : TopologicalSpace γ✝
inst✝² : TopologicalSpace δ
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β → γ
tas : Set (TopologicalSpace α)
tbs : Set (TopologicalSpace β)
ta : TopologicalSpace α
tb : TopologicalSpace β
tc : TopologicalSpace γ
ha✝ : ta ∈ tas
hb : tb ∈ tbs
hf : Continuous fun p => f p.fst p.snd
ha : Continuous id
⊢ Continuous fun p => f p.fst p.snd

tactic: have h_continuous_id := @Continuous.prod_map _ _ _ _ ta tb (sInf tas) (sInf tbs) _ _ ha hb
state_before: α✝ : Type u
β✝ : Type v
γ✝ : Type ?u.29369
δ : Type ?u.29372
ε : Type ?u.29375
ζ : Type ?u.29378
inst✝⁵ : TopologicalSpace α✝
inst✝⁴ : TopologicalSpace β✝
inst✝³ : TopologicalSpace γ✝
inst✝² : TopologicalSpace δ
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β → γ
tas : Set (TopologicalSpace α)
tbs : Set (TopologicalSpace β)
ta : TopologicalSpace α
tb : TopologicalSpace β
tc : TopologicalSpace γ
ha✝ : ta ∈ tas
hb✝ : tb ∈ tbs
hf : Continuous fun p => f p.fst p.snd
ha : Continuous id
hb : Continuous id
⊢ Continuous fun p => f p.fst p.snd

tactic: exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ hf h_continuous_id
state_before: α✝ : Type u
β✝ : Type v
γ✝ : Type ?u.29369
δ : Type ?u.29372
ε : Type ?u.29375
ζ : Type ?u.29378
inst✝⁵ : TopologicalSpace α✝
inst✝⁴ : TopologicalSpace β✝
inst✝³ : TopologicalSpace γ✝
inst✝² : TopologicalSpace δ
inst✝¹ : TopologicalSpace ε
inst✝ : TopologicalSpace ζ
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β → γ
tas : Set (TopologicalSpace α)
tbs : Set (TopologicalSpace β)
ta : TopologicalSpace α
tb : TopologicalSpace β
tc : TopologicalSpace γ
ha✝ : ta ∈ tas
hb✝ : tb ∈ tbs
hf : Continuous fun p => f p.fst p.snd
ha : Continuous id
hb : Continuous id
h_continuous_id : Continuous fun x => (id x.fst, id x.snd)
⊢ Continuous fun p => f p.fst p.snd