Step |
Hyp |
Ref |
Expression |
1 |
|
pcoval.2 |
⊢ ( 𝜑 → 𝐹 ∈ ( II Cn 𝐽 ) ) |
2 |
|
pcoval.3 |
⊢ ( 𝜑 → 𝐺 ∈ ( II Cn 𝐽 ) ) |
3 |
1 2
|
pcoval |
⊢ ( 𝜑 → ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) ) |
4 |
3
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ 1 ) = ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) ‘ 1 ) ) |
5 |
|
1elunit |
⊢ 1 ∈ ( 0 [,] 1 ) |
6 |
|
halflt1 |
⊢ ( 1 / 2 ) < 1 |
7 |
|
halfre |
⊢ ( 1 / 2 ) ∈ ℝ |
8 |
|
1re |
⊢ 1 ∈ ℝ |
9 |
7 8
|
ltnlei |
⊢ ( ( 1 / 2 ) < 1 ↔ ¬ 1 ≤ ( 1 / 2 ) ) |
10 |
6 9
|
mpbi |
⊢ ¬ 1 ≤ ( 1 / 2 ) |
11 |
|
breq1 |
⊢ ( 𝑥 = 1 → ( 𝑥 ≤ ( 1 / 2 ) ↔ 1 ≤ ( 1 / 2 ) ) ) |
12 |
10 11
|
mtbiri |
⊢ ( 𝑥 = 1 → ¬ 𝑥 ≤ ( 1 / 2 ) ) |
13 |
12
|
iffalsed |
⊢ ( 𝑥 = 1 → if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) = ( 𝐺 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) |
14 |
|
oveq2 |
⊢ ( 𝑥 = 1 → ( 2 · 𝑥 ) = ( 2 · 1 ) ) |
15 |
|
2t1e2 |
⊢ ( 2 · 1 ) = 2 |
16 |
14 15
|
eqtrdi |
⊢ ( 𝑥 = 1 → ( 2 · 𝑥 ) = 2 ) |
17 |
16
|
oveq1d |
⊢ ( 𝑥 = 1 → ( ( 2 · 𝑥 ) − 1 ) = ( 2 − 1 ) ) |
18 |
|
2m1e1 |
⊢ ( 2 − 1 ) = 1 |
19 |
17 18
|
eqtrdi |
⊢ ( 𝑥 = 1 → ( ( 2 · 𝑥 ) − 1 ) = 1 ) |
20 |
19
|
fveq2d |
⊢ ( 𝑥 = 1 → ( 𝐺 ‘ ( ( 2 · 𝑥 ) − 1 ) ) = ( 𝐺 ‘ 1 ) ) |
21 |
13 20
|
eqtrd |
⊢ ( 𝑥 = 1 → if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) = ( 𝐺 ‘ 1 ) ) |
22 |
|
eqid |
⊢ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) |
23 |
|
fvex |
⊢ ( 𝐺 ‘ 1 ) ∈ V |
24 |
21 22 23
|
fvmpt |
⊢ ( 1 ∈ ( 0 [,] 1 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) ‘ 1 ) = ( 𝐺 ‘ 1 ) ) |
25 |
5 24
|
ax-mp |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) ‘ 1 ) = ( 𝐺 ‘ 1 ) |
26 |
4 25
|
eqtrdi |
⊢ ( 𝜑 → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ 1 ) = ( 𝐺 ‘ 1 ) ) |