Step |
Hyp |
Ref |
Expression |
1 |
|
pcoval.2 |
⊢ ( 𝜑 → 𝐹 ∈ ( II Cn 𝐽 ) ) |
2 |
|
pcoval.3 |
⊢ ( 𝜑 → 𝐺 ∈ ( II Cn 𝐽 ) ) |
3 |
|
pcoval2.4 |
⊢ ( 𝜑 → ( 𝐹 ‘ 1 ) = ( 𝐺 ‘ 0 ) ) |
4 |
|
0re |
⊢ 0 ∈ ℝ |
5 |
|
1re |
⊢ 1 ∈ ℝ |
6 |
|
halfge0 |
⊢ 0 ≤ ( 1 / 2 ) |
7 |
|
1le1 |
⊢ 1 ≤ 1 |
8 |
|
iccss |
⊢ ( ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ ) ∧ ( 0 ≤ ( 1 / 2 ) ∧ 1 ≤ 1 ) ) → ( ( 1 / 2 ) [,] 1 ) ⊆ ( 0 [,] 1 ) ) |
9 |
4 5 6 7 8
|
mp4an |
⊢ ( ( 1 / 2 ) [,] 1 ) ⊆ ( 0 [,] 1 ) |
10 |
9
|
sseli |
⊢ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) → 𝑋 ∈ ( 0 [,] 1 ) ) |
11 |
1 2
|
pcovalg |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 0 [,] 1 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ 𝑋 ) = if ( 𝑋 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑋 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) ) ) |
12 |
10 11
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ 𝑋 ) = if ( 𝑋 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑋 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) ) ) |
13 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ∧ 𝑋 ≤ ( 1 / 2 ) ) ) → ( 𝐹 ‘ 1 ) = ( 𝐺 ‘ 0 ) ) |
14 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ∧ 𝑋 ≤ ( 1 / 2 ) ) ) → 𝑋 ≤ ( 1 / 2 ) ) |
15 |
|
halfre |
⊢ ( 1 / 2 ) ∈ ℝ |
16 |
15 5
|
elicc2i |
⊢ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ↔ ( 𝑋 ∈ ℝ ∧ ( 1 / 2 ) ≤ 𝑋 ∧ 𝑋 ≤ 1 ) ) |
17 |
16
|
simp2bi |
⊢ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) → ( 1 / 2 ) ≤ 𝑋 ) |
18 |
17
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ∧ 𝑋 ≤ ( 1 / 2 ) ) ) → ( 1 / 2 ) ≤ 𝑋 ) |
19 |
16
|
simp1bi |
⊢ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) → 𝑋 ∈ ℝ ) |
20 |
19
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ∧ 𝑋 ≤ ( 1 / 2 ) ) ) → 𝑋 ∈ ℝ ) |
21 |
|
letri3 |
⊢ ( ( 𝑋 ∈ ℝ ∧ ( 1 / 2 ) ∈ ℝ ) → ( 𝑋 = ( 1 / 2 ) ↔ ( 𝑋 ≤ ( 1 / 2 ) ∧ ( 1 / 2 ) ≤ 𝑋 ) ) ) |
22 |
20 15 21
|
sylancl |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ∧ 𝑋 ≤ ( 1 / 2 ) ) ) → ( 𝑋 = ( 1 / 2 ) ↔ ( 𝑋 ≤ ( 1 / 2 ) ∧ ( 1 / 2 ) ≤ 𝑋 ) ) ) |
23 |
14 18 22
|
mpbir2and |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ∧ 𝑋 ≤ ( 1 / 2 ) ) ) → 𝑋 = ( 1 / 2 ) ) |
24 |
23
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ∧ 𝑋 ≤ ( 1 / 2 ) ) ) → ( 2 · 𝑋 ) = ( 2 · ( 1 / 2 ) ) ) |
25 |
|
2cn |
⊢ 2 ∈ ℂ |
26 |
|
2ne0 |
⊢ 2 ≠ 0 |
27 |
25 26
|
recidi |
⊢ ( 2 · ( 1 / 2 ) ) = 1 |
28 |
24 27
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ∧ 𝑋 ≤ ( 1 / 2 ) ) ) → ( 2 · 𝑋 ) = 1 ) |
29 |
28
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ∧ 𝑋 ≤ ( 1 / 2 ) ) ) → ( 𝐹 ‘ ( 2 · 𝑋 ) ) = ( 𝐹 ‘ 1 ) ) |
30 |
28
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ∧ 𝑋 ≤ ( 1 / 2 ) ) ) → ( ( 2 · 𝑋 ) − 1 ) = ( 1 − 1 ) ) |
31 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
32 |
30 31
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ∧ 𝑋 ≤ ( 1 / 2 ) ) ) → ( ( 2 · 𝑋 ) − 1 ) = 0 ) |
33 |
32
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ∧ 𝑋 ≤ ( 1 / 2 ) ) ) → ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) = ( 𝐺 ‘ 0 ) ) |
34 |
13 29 33
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ∧ 𝑋 ≤ ( 1 / 2 ) ) ) → ( 𝐹 ‘ ( 2 · 𝑋 ) ) = ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) ) |
35 |
34
|
ifeq1d |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ∧ 𝑋 ≤ ( 1 / 2 ) ) ) → if ( 𝑋 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑋 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) ) = if ( 𝑋 ≤ ( 1 / 2 ) , ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) ) ) |
36 |
|
ifid |
⊢ if ( 𝑋 ≤ ( 1 / 2 ) , ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) ) = ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) |
37 |
35 36
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ∧ 𝑋 ≤ ( 1 / 2 ) ) ) → if ( 𝑋 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑋 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) ) = ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) ) |
38 |
37
|
expr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ) → ( 𝑋 ≤ ( 1 / 2 ) → if ( 𝑋 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑋 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) ) = ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) ) ) |
39 |
|
iffalse |
⊢ ( ¬ 𝑋 ≤ ( 1 / 2 ) → if ( 𝑋 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑋 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) ) = ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) ) |
40 |
38 39
|
pm2.61d1 |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ) → if ( 𝑋 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑋 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) ) = ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) ) |
41 |
12 40
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( ( 1 / 2 ) [,] 1 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ 𝑋 ) = ( 𝐺 ‘ ( ( 2 · 𝑋 ) − 1 ) ) ) |