Theorem: StoneCech.projective
File: Mathlib/Topology/ExtremallyDisconnected.lean

Proof steps:
state_before: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
⊢ CompactT2.Projective (StoneCech X)
state_after: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
⊢ ∃ h, Continuous h ∧ g ∘ h = f
tactic: intro Y Z _tsY _tsZ _csY _t2Y _csZ _csZ f g hf hg g_sur
state_before: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
⊢ ∃ h, Continuous h ∧ g ∘ h = f
state_after: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
s : Z → Y := fun z => Classical.choose ⋯
⊢ ∃ h, Continuous h ∧ g ∘ h = f
tactic: let s : Z → Y := fun z => Classical.choose <| g_sur z
state_before: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
s : Z → Y := fun z => Classical.choose ⋯
⊢ ∃ h, Continuous h ∧ g ∘ h = f
state_after: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
s : Z → Y := fun z => Classical.choose ⋯
hs : g ∘ s = id
⊢ ∃ h, Continuous h ∧ g ∘ h = f
tactic: have hs : g ∘ s = id := funext fun z => Classical.choose_spec (g_sur z)
state_before: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
s : Z → Y := fun z => Classical.choose ⋯
hs : g ∘ s = id
⊢ ∃ h, Continuous h ∧ g ∘ h = f
state_after: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
s : Z → Y := fun z => Classical.choose ⋯
hs : g ∘ s = id
t : X → Y := s ∘ f ∘ stoneCechUnit
⊢ ∃ h, Continuous h ∧ g ∘ h = f
tactic: let t := s ∘ f ∘ stoneCechUnit
state_before: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
s : Z → Y := fun z => Classical.choose ⋯
hs : g ∘ s = id
t : X → Y := s ∘ f ∘ stoneCechUnit
⊢ ∃ h, Continuous h ∧ g ∘ h = f
state_after: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
s : Z → Y := fun z => Classical.choose ⋯
hs : g ∘ s = id
t : X → Y := s ∘ f ∘ stoneCechUnit
ht : Continuous t
⊢ ∃ h, Continuous h ∧ g ∘ h = f
tactic: have ht : Continuous t := continuous_of_discreteTopology
state_before: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
s : Z → Y := fun z => Classical.choose ⋯
hs : g ∘ s = id
t : X → Y := s ∘ f ∘ stoneCechUnit
ht : Continuous t
⊢ ∃ h, Continuous h ∧ g ∘ h = f
state_after: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
s : Z → Y := fun z => Classical.choose ⋯
hs : g ∘ s = id
t : X → Y := s ∘ f ∘ stoneCechUnit
ht : Continuous t
h : StoneCech X → Y := stoneCechExtend ht
⊢ ∃ h, Continuous h ∧ g ∘ h = f
tactic: let h : StoneCech X → Y := stoneCechExtend ht
state_before: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
s : Z → Y := fun z => Classical.choose ⋯
hs : g ∘ s = id
t : X → Y := s ∘ f ∘ stoneCechUnit
ht : Continuous t
h : StoneCech X → Y := stoneCechExtend ht
⊢ ∃ h, Continuous h ∧ g ∘ h = f
state_after: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
s : Z → Y := fun z => Classical.choose ⋯
hs : g ∘ s = id
t : X → Y := s ∘ f ∘ stoneCechUnit
ht : Continuous t
h : StoneCech X → Y := stoneCechExtend ht
hh : Continuous h
⊢ ∃ h, Continuous h ∧ g ∘ h = f
tactic: have hh : Continuous h := continuous_stoneCechExtend ht
state_before: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
s : Z → Y := fun z => Classical.choose ⋯
hs : g ∘ s = id
t : X → Y := s ∘ f ∘ stoneCechUnit
ht : Continuous t
h : StoneCech X → Y := stoneCechExtend ht
hh : Continuous h
⊢ ∃ h, Continuous h ∧ g ∘ h = f
state_after: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
s : Z → Y := fun z => Classical.choose ⋯
hs : g ∘ s = id
t : X → Y := s ∘ f ∘ stoneCechUnit
ht : Continuous t
h : StoneCech X → Y := stoneCechExtend ht
hh : Continuous h
⊢ (g ∘ h) ∘ stoneCechUnit = f ∘ stoneCechUnit
tactic: refine ⟨h, hh, denseRange_stoneCechUnit.equalizer (hg.comp hh) hf ?_⟩
state_before: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
Y Z : Type u
_tsY : TopologicalSpace Y
_tsZ : TopologicalSpace Z
_csY : CompactSpace Y
_t2Y : T2Space Y
_csZ✝ : CompactSpace Z
_csZ : T2Space Z
f : StoneCech X → Z
g : Y → Z
hf : Continuous f
hg : Continuous g
g_sur : Surjective g
s : Z → Y := fun z => Classical.choose ⋯
hs : g ∘ s = id
t : X → Y := s ∘ f ∘ stoneCechUnit
ht : Continuous t
h : StoneCech X → Y := stoneCechExtend ht
hh : Continuous h
⊢ (g ∘ h) ∘ stoneCechUnit = f ∘ stoneCechUnit
state_after: no goals
tactic: rw [comp.assoc, stoneCechExtend_extends ht, ← comp.assoc, hs, id_comp]