Metamath Proof Explorer

Theorem cvmlift2lem7

Description: Lemma for cvmlift2 . (Contributed by Mario Carneiro, 7-May-2015)

Ref Expression
Hypotheses cvmlift2.b ⊢ 𝐵 = ∪ 𝐶
cvmlift2.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) )
cvmlift2.g ⊢ ( 𝜑 → 𝐺 ∈ ( ( II ×t II ) Cn 𝐽 ) )
cvmlift2.p ⊢ ( 𝜑 → 𝑃 ∈ 𝐵 )
cvmlift2.i ⊢ ( 𝜑 → ( 𝐹 ‘ 𝑃 ) = ( 0 𝐺 0 ) )
cvmlift2.h ⊢ 𝐻 = ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑧 𝐺 0 ) ) ∧ ( 𝑓 ‘ 0 ) = 𝑃 ) )
cvmlift2.k ⊢ 𝐾 = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 𝑦 ) )
Assertion cvmlift2lem7 ( 𝜑 → ( 𝐹 ∘ 𝐾 ) = 𝐺 )

Proof

Step Hyp Ref Expression
1 cvmlift2.b ⊢ 𝐵 = ∪ 𝐶
2 cvmlift2.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) )
3 cvmlift2.g ⊢ ( 𝜑 → 𝐺 ∈ ( ( II ×t II ) Cn 𝐽 ) )
4 cvmlift2.p ⊢ ( 𝜑 → 𝑃 ∈ 𝐵 )
5 cvmlift2.i ⊢ ( 𝜑 → ( 𝐹 ‘ 𝑃 ) = ( 0 𝐺 0 ) )
6 cvmlift2.h ⊢ 𝐻 = ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑧 𝐺 0 ) ) ∧ ( 𝑓 ‘ 0 ) = 𝑃 ) )
7 cvmlift2.k ⊢ 𝐾 = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 𝑦 ) )
8 eqid ⊢ ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) = ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) )
9 1 2 3 4 5 6 8 cvmlift2lem3 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) → ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ∈ ( II Cn 𝐶 ) ∧ ( 𝐹 ∘ ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) )
10 9 adantrr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ∈ ( II Cn 𝐶 ) ∧ ( 𝐹 ∘ ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) )
11 10 simp2d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( 𝐹 ∘ ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) )
12 11 fveq1d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( ( 𝐹 ∘ ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ) ‘ 𝑦 ) = ( ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ‘ 𝑦 ) )
13 10 simp1d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ∈ ( II Cn 𝐶 ) )
14 iiuni ⊢ ( 0 [,] 1 ) = ∪ II
15 14 1 cnf ⊢ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ∈ ( II Cn 𝐶 ) → ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) : ( 0 [,] 1 ) ⟶ 𝐵 )
16 13 15 syl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) : ( 0 [,] 1 ) ⟶ 𝐵 )
17 simprr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → 𝑦 ∈ ( 0 [,] 1 ) )
18 fvco3 ⊢ ( ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) : ( 0 [,] 1 ) ⟶ 𝐵 ∧ 𝑦 ∈ ( 0 [,] 1 ) ) → ( ( 𝐹 ∘ ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ) ‘ 𝑦 ) = ( 𝐹 ‘ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 𝑦 ) ) )
19 16 17 18 syl2anc ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( ( 𝐹 ∘ ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ) ‘ 𝑦 ) = ( 𝐹 ‘ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 𝑦 ) ) )
20 oveq2 ⊢ ( 𝑧 = 𝑦 → ( 𝑥 𝐺 𝑧 ) = ( 𝑥 𝐺 𝑦 ) )
21 eqid ⊢ ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) )
22 ovex ⊢ ( 𝑥 𝐺 𝑦 ) ∈ V
23 20 21 22 fvmpt ⊢ ( 𝑦 ∈ ( 0 [,] 1 ) → ( ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ‘ 𝑦 ) = ( 𝑥 𝐺 𝑦 ) )
24 17 23 syl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ‘ 𝑦 ) = ( 𝑥 𝐺 𝑦 ) )
25 12 19 24 3eqtr3d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( 𝐹 ‘ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 𝑦 ) ) = ( 𝑥 𝐺 𝑦 ) )
26 25 3impb ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) → ( 𝐹 ‘ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 𝑦 ) ) = ( 𝑥 𝐺 𝑦 ) )
27 26 mpoeq3dva ⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝐹 ‘ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 𝑦 ) ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑦 ) ) )
28 16 17 ffvelrnd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 𝑦 ) ∈ 𝐵 )
29 7 a1i ⊢ ( 𝜑 → 𝐾 = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 𝑦 ) ) )
30 cvmcn ⊢ ( 𝐹 ∈ ( 𝐶 CovMap 𝐽 ) → 𝐹 ∈ ( 𝐶 Cn 𝐽 ) )
31 eqid ⊢ ∪ 𝐽 = ∪ 𝐽
32 1 31 cnf ⊢ ( 𝐹 ∈ ( 𝐶 Cn 𝐽 ) → 𝐹 : 𝐵 ⟶ ∪ 𝐽 )
33 2 30 32 3syl ⊢ ( 𝜑 → 𝐹 : 𝐵 ⟶ ∪ 𝐽 )
34 33 feqmptd ⊢ ( 𝜑 → 𝐹 = ( 𝑤 ∈ 𝐵 ↦ ( 𝐹 ‘ 𝑤 ) ) )
35 fveq2 ⊢ ( 𝑤 = ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 𝑦 ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 𝑦 ) ) )
36 28 29 34 35 fmpoco ⊢ ( 𝜑 → ( 𝐹 ∘ 𝐾 ) = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝐹 ‘ ( ( ℩ 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹 ∘ 𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻 ‘ 𝑥 ) ) ) ‘ 𝑦 ) ) ) )
37 iitop ⊢ II ∈ Top
38 37 37 14 14 txunii ⊢ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) = ∪ ( II ×t II )
39 38 31 cnf ⊢ ( 𝐺 ∈ ( ( II ×t II ) Cn 𝐽 ) → 𝐺 : ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ⟶ ∪ 𝐽 )
40 ffn ⊢ ( 𝐺 : ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ⟶ ∪ 𝐽 → 𝐺 Fn ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) )
41 3 39 40 3syl ⊢ ( 𝜑 → 𝐺 Fn ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) )
42 fnov ⊢ ( 𝐺 Fn ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ↔ 𝐺 = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑦 ) ) )
43 41 42 sylib ⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑦 ) ) )
44 27 36 43 3eqtr4d ⊢ ( 𝜑 → ( 𝐹 ∘ 𝐾 ) = 𝐺 )