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 |
1 2
|
pcoval |
⊢ ( 𝜑 → ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) ) |
5 |
|
iitopon |
⊢ II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) |
6 |
5
|
a1i |
⊢ ( 𝜑 → II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) ) |
7 |
6
|
cnmptid |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ 𝑥 ) ∈ ( II Cn II ) ) |
8 |
|
0elunit |
⊢ 0 ∈ ( 0 [,] 1 ) |
9 |
8
|
a1i |
⊢ ( 𝜑 → 0 ∈ ( 0 [,] 1 ) ) |
10 |
6 6 9
|
cnmptc |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ 0 ) ∈ ( II Cn II ) ) |
11 |
|
eqid |
⊢ ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) ) |
12 |
|
eqid |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) |
13 |
|
eqid |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) |
14 |
|
dfii2 |
⊢ II = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] 1 ) ) |
15 |
|
0re |
⊢ 0 ∈ ℝ |
16 |
15
|
a1i |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
17 |
|
1re |
⊢ 1 ∈ ℝ |
18 |
17
|
a1i |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
19 |
|
halfre |
⊢ ( 1 / 2 ) ∈ ℝ |
20 |
|
halfge0 |
⊢ 0 ≤ ( 1 / 2 ) |
21 |
|
halflt1 |
⊢ ( 1 / 2 ) < 1 |
22 |
19 17 21
|
ltleii |
⊢ ( 1 / 2 ) ≤ 1 |
23 |
|
elicc01 |
⊢ ( ( 1 / 2 ) ∈ ( 0 [,] 1 ) ↔ ( ( 1 / 2 ) ∈ ℝ ∧ 0 ≤ ( 1 / 2 ) ∧ ( 1 / 2 ) ≤ 1 ) ) |
24 |
19 20 22 23
|
mpbir3an |
⊢ ( 1 / 2 ) ∈ ( 0 [,] 1 ) |
25 |
24
|
a1i |
⊢ ( 𝜑 → ( 1 / 2 ) ∈ ( 0 [,] 1 ) ) |
26 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 𝐹 ‘ 1 ) = ( 𝐺 ‘ 0 ) ) |
27 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → 𝑦 = ( 1 / 2 ) ) |
28 |
27
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 2 · 𝑦 ) = ( 2 · ( 1 / 2 ) ) ) |
29 |
|
2cn |
⊢ 2 ∈ ℂ |
30 |
|
2ne0 |
⊢ 2 ≠ 0 |
31 |
29 30
|
recidi |
⊢ ( 2 · ( 1 / 2 ) ) = 1 |
32 |
28 31
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 2 · 𝑦 ) = 1 ) |
33 |
32
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 𝐹 ‘ ( 2 · 𝑦 ) ) = ( 𝐹 ‘ 1 ) ) |
34 |
32
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( ( 2 · 𝑦 ) − 1 ) = ( 1 − 1 ) ) |
35 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
36 |
34 35
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( ( 2 · 𝑦 ) − 1 ) = 0 ) |
37 |
36
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 𝐺 ‘ ( ( 2 · 𝑦 ) − 1 ) ) = ( 𝐺 ‘ 0 ) ) |
38 |
26 33 37
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 𝐹 ‘ ( 2 · 𝑦 ) ) = ( 𝐺 ‘ ( ( 2 · 𝑦 ) − 1 ) ) ) |
39 |
|
retopon |
⊢ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) |
40 |
|
iccssre |
⊢ ( ( 0 ∈ ℝ ∧ ( 1 / 2 ) ∈ ℝ ) → ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ ) |
41 |
15 19 40
|
mp2an |
⊢ ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ |
42 |
|
resttopon |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) ) |
43 |
39 41 42
|
mp2an |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) |
44 |
43
|
a1i |
⊢ ( 𝜑 → ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) ) |
45 |
44 6
|
cnmpt1st |
⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ 𝑦 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ×t II ) Cn ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ) ) |
46 |
12
|
iihalf1cn |
⊢ ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) ↦ ( 2 · 𝑥 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) Cn II ) |
47 |
46
|
a1i |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) ↦ ( 2 · 𝑥 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) Cn II ) ) |
48 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 2 · 𝑥 ) = ( 2 · 𝑦 ) ) |
49 |
44 6 45 44 47 48
|
cnmpt21 |
⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 2 · 𝑦 ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ×t II ) Cn II ) ) |
50 |
44 6 49 1
|
cnmpt21f |
⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐹 ‘ ( 2 · 𝑦 ) ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ×t II ) Cn 𝐽 ) ) |
51 |
|
iccssre |
⊢ ( ( ( 1 / 2 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ ) |
52 |
19 17 51
|
mp2an |
⊢ ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ |
53 |
|
resttopon |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) ) |
54 |
39 52 53
|
mp2an |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) |
55 |
54
|
a1i |
⊢ ( 𝜑 → ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) ) |
56 |
55 6
|
cnmpt1st |
⊢ ( 𝜑 → ( 𝑦 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ 𝑦 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ×t II ) Cn ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ) ) |
57 |
13
|
iihalf2cn |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) ↦ ( ( 2 · 𝑥 ) − 1 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) Cn II ) |
58 |
57
|
a1i |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) ↦ ( ( 2 · 𝑥 ) − 1 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) Cn II ) ) |
59 |
48
|
oveq1d |
⊢ ( 𝑥 = 𝑦 → ( ( 2 · 𝑥 ) − 1 ) = ( ( 2 · 𝑦 ) − 1 ) ) |
60 |
55 6 56 55 58 59
|
cnmpt21 |
⊢ ( 𝜑 → ( 𝑦 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ ( ( 2 · 𝑦 ) − 1 ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ×t II ) Cn II ) ) |
61 |
55 6 60 2
|
cnmpt21f |
⊢ ( 𝜑 → ( 𝑦 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐺 ‘ ( ( 2 · 𝑦 ) − 1 ) ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ×t II ) Cn 𝐽 ) ) |
62 |
11 12 13 14 16 18 25 6 38 50 61
|
cnmpopc |
⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 [,] 1 ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ if ( 𝑦 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑦 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑦 ) − 1 ) ) ) ) ∈ ( ( II ×t II ) Cn 𝐽 ) ) |
63 |
|
breq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ≤ ( 1 / 2 ) ↔ 𝑥 ≤ ( 1 / 2 ) ) ) |
64 |
|
oveq2 |
⊢ ( 𝑦 = 𝑥 → ( 2 · 𝑦 ) = ( 2 · 𝑥 ) ) |
65 |
64
|
fveq2d |
⊢ ( 𝑦 = 𝑥 → ( 𝐹 ‘ ( 2 · 𝑦 ) ) = ( 𝐹 ‘ ( 2 · 𝑥 ) ) ) |
66 |
64
|
oveq1d |
⊢ ( 𝑦 = 𝑥 → ( ( 2 · 𝑦 ) − 1 ) = ( ( 2 · 𝑥 ) − 1 ) ) |
67 |
66
|
fveq2d |
⊢ ( 𝑦 = 𝑥 → ( 𝐺 ‘ ( ( 2 · 𝑦 ) − 1 ) ) = ( 𝐺 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) |
68 |
63 65 67
|
ifbieq12d |
⊢ ( 𝑦 = 𝑥 → if ( 𝑦 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑦 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑦 ) − 1 ) ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) |
69 |
68
|
adantr |
⊢ ( ( 𝑦 = 𝑥 ∧ 𝑧 = 0 ) → if ( 𝑦 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑦 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑦 ) − 1 ) ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) |
70 |
6 7 10 6 6 62 69
|
cnmpt12 |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) ∈ ( II Cn 𝐽 ) ) |
71 |
4 70
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ∈ ( II Cn 𝐽 ) ) |