Step |
Hyp |
Ref |
Expression |
1 |
|
sconnpht2.1 |
⊢ ( 𝜑 → 𝐽 ∈ SConn ) |
2 |
|
sconnpht2.2 |
⊢ ( 𝜑 → 𝐹 ∈ ( II Cn 𝐽 ) ) |
3 |
|
sconnpht2.3 |
⊢ ( 𝜑 → 𝐺 ∈ ( II Cn 𝐽 ) ) |
4 |
|
sconnpht2.4 |
⊢ ( 𝜑 → ( 𝐹 ‘ 0 ) = ( 𝐺 ‘ 0 ) ) |
5 |
|
sconnpht2.5 |
⊢ ( 𝜑 → ( 𝐹 ‘ 1 ) = ( 𝐺 ‘ 1 ) ) |
6 |
|
eqid |
⊢ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) |
7 |
6
|
pcorevcl |
⊢ ( 𝐺 ∈ ( II Cn 𝐽 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ∈ ( II Cn 𝐽 ) ∧ ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ‘ 0 ) = ( 𝐺 ‘ 1 ) ∧ ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ‘ 1 ) = ( 𝐺 ‘ 0 ) ) ) |
8 |
3 7
|
syl |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ∈ ( II Cn 𝐽 ) ∧ ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ‘ 0 ) = ( 𝐺 ‘ 1 ) ∧ ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ‘ 1 ) = ( 𝐺 ‘ 0 ) ) ) |
9 |
8
|
simp1d |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ∈ ( II Cn 𝐽 ) ) |
10 |
8
|
simp2d |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ‘ 0 ) = ( 𝐺 ‘ 1 ) ) |
11 |
5 10
|
eqtr4d |
⊢ ( 𝜑 → ( 𝐹 ‘ 1 ) = ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ‘ 0 ) ) |
12 |
2 9 11
|
pcocn |
⊢ ( 𝜑 → ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ∈ ( II Cn 𝐽 ) ) |
13 |
2 9
|
pco0 |
⊢ ( 𝜑 → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ‘ 0 ) = ( 𝐹 ‘ 0 ) ) |
14 |
2 9
|
pco1 |
⊢ ( 𝜑 → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ‘ 1 ) = ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ‘ 1 ) ) |
15 |
8
|
simp3d |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ‘ 1 ) = ( 𝐺 ‘ 0 ) ) |
16 |
4 15
|
eqtr4d |
⊢ ( 𝜑 → ( 𝐹 ‘ 0 ) = ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ‘ 1 ) ) |
17 |
14 16
|
eqtr4d |
⊢ ( 𝜑 → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ‘ 1 ) = ( 𝐹 ‘ 0 ) ) |
18 |
13 17
|
eqtr4d |
⊢ ( 𝜑 → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ‘ 0 ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ‘ 1 ) ) |
19 |
|
sconnpht |
⊢ ( ( 𝐽 ∈ SConn ∧ ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ∈ ( II Cn 𝐽 ) ∧ ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ‘ 0 ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ‘ 1 ) ) → ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ( ≃ph ‘ 𝐽 ) ( ( 0 [,] 1 ) × { ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ‘ 0 ) } ) ) |
20 |
1 12 18 19
|
syl3anc |
⊢ ( 𝜑 → ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ( ≃ph ‘ 𝐽 ) ( ( 0 [,] 1 ) × { ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ‘ 0 ) } ) ) |
21 |
13
|
sneqd |
⊢ ( 𝜑 → { ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ‘ 0 ) } = { ( 𝐹 ‘ 0 ) } ) |
22 |
21
|
xpeq2d |
⊢ ( 𝜑 → ( ( 0 [,] 1 ) × { ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ‘ 0 ) } ) = ( ( 0 [,] 1 ) × { ( 𝐹 ‘ 0 ) } ) ) |
23 |
20 22
|
breqtrd |
⊢ ( 𝜑 → ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ( ≃ph ‘ 𝐽 ) ( ( 0 [,] 1 ) × { ( 𝐹 ‘ 0 ) } ) ) |
24 |
|
eqid |
⊢ ( ( 0 [,] 1 ) × { ( 𝐹 ‘ 0 ) } ) = ( ( 0 [,] 1 ) × { ( 𝐹 ‘ 0 ) } ) |
25 |
6 24 2 3 4 5
|
pcophtb |
⊢ ( 𝜑 → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( 1 − 𝑥 ) ) ) ) ( ≃ph ‘ 𝐽 ) ( ( 0 [,] 1 ) × { ( 𝐹 ‘ 0 ) } ) ↔ 𝐹 ( ≃ph ‘ 𝐽 ) 𝐺 ) ) |
26 |
23 25
|
mpbid |
⊢ ( 𝜑 → 𝐹 ( ≃ph ‘ 𝐽 ) 𝐺 ) |