Metamath Proof Explorer


Theorem cvmlift2lem8

Description: Lemma for cvmlift2 . (Contributed by Mario Carneiro, 9-Mar-2015)

Ref Expression
Hypotheses cvmlift2.b 𝐵 = 𝐶
cvmlift2.f ( 𝜑𝐹 ∈ ( 𝐶 CovMap 𝐽 ) )
cvmlift2.g ( 𝜑𝐺 ∈ ( ( II ×t II ) Cn 𝐽 ) )
cvmlift2.p ( 𝜑𝑃𝐵 )
cvmlift2.i ( 𝜑 → ( 𝐹𝑃 ) = ( 0 𝐺 0 ) )
cvmlift2.h 𝐻 = ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑧 𝐺 0 ) ) ∧ ( 𝑓 ‘ 0 ) = 𝑃 ) )
cvmlift2.k 𝐾 = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻𝑥 ) ) ) ‘ 𝑦 ) )
Assertion cvmlift2lem8 ( ( 𝜑𝑋 ∈ ( 0 [,] 1 ) ) → ( 𝑋 𝐾 0 ) = ( 𝐻𝑋 ) )

Proof

Step Hyp Ref Expression
1 cvmlift2.b 𝐵 = 𝐶
2 cvmlift2.f ( 𝜑𝐹 ∈ ( 𝐶 CovMap 𝐽 ) )
3 cvmlift2.g ( 𝜑𝐺 ∈ ( ( II ×t II ) Cn 𝐽 ) )
4 cvmlift2.p ( 𝜑𝑃𝐵 )
5 cvmlift2.i ( 𝜑 → ( 𝐹𝑃 ) = ( 0 𝐺 0 ) )
6 cvmlift2.h 𝐻 = ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑧 𝐺 0 ) ) ∧ ( 𝑓 ‘ 0 ) = 𝑃 ) )
7 cvmlift2.k 𝐾 = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻𝑥 ) ) ) ‘ 𝑦 ) )
8 simpr ( ( 𝜑𝑋 ∈ ( 0 [,] 1 ) ) → 𝑋 ∈ ( 0 [,] 1 ) )
9 0elunit 0 ∈ ( 0 [,] 1 )
10 1 2 3 4 5 6 7 cvmlift2lem4 ( ( 𝑋 ∈ ( 0 [,] 1 ) ∧ 0 ∈ ( 0 [,] 1 ) ) → ( 𝑋 𝐾 0 ) = ( ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻𝑋 ) ) ) ‘ 0 ) )
11 8 9 10 sylancl ( ( 𝜑𝑋 ∈ ( 0 [,] 1 ) ) → ( 𝑋 𝐾 0 ) = ( ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻𝑋 ) ) ) ‘ 0 ) )
12 eqid ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻𝑋 ) ) ) = ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻𝑋 ) ) )
13 1 2 3 4 5 6 12 cvmlift2lem3 ( ( 𝜑𝑋 ∈ ( 0 [,] 1 ) ) → ( ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻𝑋 ) ) ) ∈ ( II Cn 𝐶 ) ∧ ( 𝐹 ∘ ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻𝑋 ) ) ) ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻𝑋 ) ) ) ‘ 0 ) = ( 𝐻𝑋 ) ) )
14 13 simp3d ( ( 𝜑𝑋 ∈ ( 0 [,] 1 ) ) → ( ( 𝑓 ∈ ( II Cn 𝐶 ) ( ( 𝐹𝑓 ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑋 𝐺 𝑧 ) ) ∧ ( 𝑓 ‘ 0 ) = ( 𝐻𝑋 ) ) ) ‘ 0 ) = ( 𝐻𝑋 ) )
15 11 14 eqtrd ( ( 𝜑𝑋 ∈ ( 0 [,] 1 ) ) → ( 𝑋 𝐾 0 ) = ( 𝐻𝑋 ) )