Step |
Hyp |
Ref |
Expression |
1 |
|
pcoass.2 |
⊢ ( 𝜑 → 𝐹 ∈ ( II Cn 𝐽 ) ) |
2 |
|
pcoass.3 |
⊢ ( 𝜑 → 𝐺 ∈ ( II Cn 𝐽 ) ) |
3 |
|
pcoass.4 |
⊢ ( 𝜑 → 𝐻 ∈ ( II Cn 𝐽 ) ) |
4 |
|
pcoass.5 |
⊢ ( 𝜑 → ( 𝐹 ‘ 1 ) = ( 𝐺 ‘ 0 ) ) |
5 |
|
pcoass.6 |
⊢ ( 𝜑 → ( 𝐺 ‘ 1 ) = ( 𝐻 ‘ 0 ) ) |
6 |
|
pcoass.7 |
⊢ 𝑃 = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) |
7 |
|
iftrue |
⊢ ( 𝑥 ≤ ( 1 / 4 ) → if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) = ( 2 · 𝑥 ) ) |
8 |
7
|
fveq2d |
⊢ ( 𝑥 ≤ ( 1 / 4 ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( 2 · 𝑥 ) ) ) |
9 |
8
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( 2 · 𝑥 ) ) ) |
10 |
|
2cn |
⊢ 2 ∈ ℂ |
11 |
|
elicc01 |
⊢ ( 𝑥 ∈ ( 0 [,] 1 ) ↔ ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 ≤ 1 ) ) |
12 |
11
|
simp1bi |
⊢ ( 𝑥 ∈ ( 0 [,] 1 ) → 𝑥 ∈ ℝ ) |
13 |
12
|
adantr |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 4 ) ) → 𝑥 ∈ ℝ ) |
14 |
13
|
recnd |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 4 ) ) → 𝑥 ∈ ℂ ) |
15 |
|
mulcom |
⊢ ( ( 2 ∈ ℂ ∧ 𝑥 ∈ ℂ ) → ( 2 · 𝑥 ) = ( 𝑥 · 2 ) ) |
16 |
10 14 15
|
sylancr |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 4 ) ) → ( 2 · 𝑥 ) = ( 𝑥 · 2 ) ) |
17 |
11
|
simp2bi |
⊢ ( 𝑥 ∈ ( 0 [,] 1 ) → 0 ≤ 𝑥 ) |
18 |
17
|
adantr |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 4 ) ) → 0 ≤ 𝑥 ) |
19 |
|
simpr |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 4 ) ) → 𝑥 ≤ ( 1 / 4 ) ) |
20 |
|
0re |
⊢ 0 ∈ ℝ |
21 |
|
4nn |
⊢ 4 ∈ ℕ |
22 |
|
nnrecre |
⊢ ( 4 ∈ ℕ → ( 1 / 4 ) ∈ ℝ ) |
23 |
21 22
|
ax-mp |
⊢ ( 1 / 4 ) ∈ ℝ |
24 |
20 23
|
elicc2i |
⊢ ( 𝑥 ∈ ( 0 [,] ( 1 / 4 ) ) ↔ ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 ≤ ( 1 / 4 ) ) ) |
25 |
13 18 19 24
|
syl3anbrc |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 4 ) ) → 𝑥 ∈ ( 0 [,] ( 1 / 4 ) ) ) |
26 |
|
2rp |
⊢ 2 ∈ ℝ+ |
27 |
10
|
mul02i |
⊢ ( 0 · 2 ) = 0 |
28 |
23
|
recni |
⊢ ( 1 / 4 ) ∈ ℂ |
29 |
28
|
2timesi |
⊢ ( 2 · ( 1 / 4 ) ) = ( ( 1 / 4 ) + ( 1 / 4 ) ) |
30 |
|
2ne0 |
⊢ 2 ≠ 0 |
31 |
|
recdiv2 |
⊢ ( ( ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ∧ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ) → ( ( 1 / 2 ) / 2 ) = ( 1 / ( 2 · 2 ) ) ) |
32 |
10 30 10 30 31
|
mp4an |
⊢ ( ( 1 / 2 ) / 2 ) = ( 1 / ( 2 · 2 ) ) |
33 |
|
2t2e4 |
⊢ ( 2 · 2 ) = 4 |
34 |
33
|
oveq2i |
⊢ ( 1 / ( 2 · 2 ) ) = ( 1 / 4 ) |
35 |
32 34
|
eqtri |
⊢ ( ( 1 / 2 ) / 2 ) = ( 1 / 4 ) |
36 |
35 35
|
oveq12i |
⊢ ( ( ( 1 / 2 ) / 2 ) + ( ( 1 / 2 ) / 2 ) ) = ( ( 1 / 4 ) + ( 1 / 4 ) ) |
37 |
|
halfcn |
⊢ ( 1 / 2 ) ∈ ℂ |
38 |
|
2halves |
⊢ ( ( 1 / 2 ) ∈ ℂ → ( ( ( 1 / 2 ) / 2 ) + ( ( 1 / 2 ) / 2 ) ) = ( 1 / 2 ) ) |
39 |
37 38
|
ax-mp |
⊢ ( ( ( 1 / 2 ) / 2 ) + ( ( 1 / 2 ) / 2 ) ) = ( 1 / 2 ) |
40 |
36 39
|
eqtr3i |
⊢ ( ( 1 / 4 ) + ( 1 / 4 ) ) = ( 1 / 2 ) |
41 |
29 40
|
eqtri |
⊢ ( 2 · ( 1 / 4 ) ) = ( 1 / 2 ) |
42 |
10 28 41
|
mulcomli |
⊢ ( ( 1 / 4 ) · 2 ) = ( 1 / 2 ) |
43 |
20 23 26 27 42
|
iccdili |
⊢ ( 𝑥 ∈ ( 0 [,] ( 1 / 4 ) ) → ( 𝑥 · 2 ) ∈ ( 0 [,] ( 1 / 2 ) ) ) |
44 |
25 43
|
syl |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 4 ) ) → ( 𝑥 · 2 ) ∈ ( 0 [,] ( 1 / 2 ) ) ) |
45 |
16 44
|
eqeltrd |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 4 ) ) → ( 2 · 𝑥 ) ∈ ( 0 [,] ( 1 / 2 ) ) ) |
46 |
2 3 5
|
pcocn |
⊢ ( 𝜑 → ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ∈ ( II Cn 𝐽 ) ) |
47 |
1 46
|
pcoval1 |
⊢ ( ( 𝜑 ∧ ( 2 · 𝑥 ) ∈ ( 0 [,] ( 1 / 2 ) ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( 2 · 𝑥 ) ) = ( 𝐹 ‘ ( 2 · ( 2 · 𝑥 ) ) ) ) |
48 |
1 2
|
pcoval1 |
⊢ ( ( 𝜑 ∧ ( 2 · 𝑥 ) ∈ ( 0 [,] ( 1 / 2 ) ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) = ( 𝐹 ‘ ( 2 · ( 2 · 𝑥 ) ) ) ) |
49 |
47 48
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 2 · 𝑥 ) ∈ ( 0 [,] ( 1 / 2 ) ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( 2 · 𝑥 ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) ) |
50 |
45 49
|
sylan2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 4 ) ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( 2 · 𝑥 ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) ) |
51 |
50
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( 2 · 𝑥 ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) ) |
52 |
9 51
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) ) |
53 |
52
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) ) |
54 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → 𝜑 ) |
55 |
12
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) → 𝑥 ∈ ℝ ) |
56 |
55
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → 𝑥 ∈ ℝ ) |
57 |
|
letric |
⊢ ( ( 𝑥 ∈ ℝ ∧ ( 1 / 4 ) ∈ ℝ ) → ( 𝑥 ≤ ( 1 / 4 ) ∨ ( 1 / 4 ) ≤ 𝑥 ) ) |
58 |
55 23 57
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) → ( 𝑥 ≤ ( 1 / 4 ) ∨ ( 1 / 4 ) ≤ 𝑥 ) ) |
59 |
58
|
orcanai |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( 1 / 4 ) ≤ 𝑥 ) |
60 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → 𝑥 ≤ ( 1 / 2 ) ) |
61 |
|
halfre |
⊢ ( 1 / 2 ) ∈ ℝ |
62 |
23 61
|
elicc2i |
⊢ ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ↔ ( 𝑥 ∈ ℝ ∧ ( 1 / 4 ) ≤ 𝑥 ∧ 𝑥 ≤ ( 1 / 2 ) ) ) |
63 |
56 59 60 62
|
syl3anbrc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ) |
64 |
62
|
simp1bi |
⊢ ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) → 𝑥 ∈ ℝ ) |
65 |
|
readdcl |
⊢ ( ( 𝑥 ∈ ℝ ∧ ( 1 / 4 ) ∈ ℝ ) → ( 𝑥 + ( 1 / 4 ) ) ∈ ℝ ) |
66 |
64 23 65
|
sylancl |
⊢ ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) → ( 𝑥 + ( 1 / 4 ) ) ∈ ℝ ) |
67 |
23
|
a1i |
⊢ ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) → ( 1 / 4 ) ∈ ℝ ) |
68 |
62
|
simp2bi |
⊢ ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) → ( 1 / 4 ) ≤ 𝑥 ) |
69 |
67 64 67 68
|
leadd1dd |
⊢ ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) → ( ( 1 / 4 ) + ( 1 / 4 ) ) ≤ ( 𝑥 + ( 1 / 4 ) ) ) |
70 |
40 69
|
eqbrtrrid |
⊢ ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) → ( 1 / 2 ) ≤ ( 𝑥 + ( 1 / 4 ) ) ) |
71 |
61
|
a1i |
⊢ ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) → ( 1 / 2 ) ∈ ℝ ) |
72 |
62
|
simp3bi |
⊢ ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) → 𝑥 ≤ ( 1 / 2 ) ) |
73 |
|
2lt4 |
⊢ 2 < 4 |
74 |
|
2re |
⊢ 2 ∈ ℝ |
75 |
|
4re |
⊢ 4 ∈ ℝ |
76 |
|
2pos |
⊢ 0 < 2 |
77 |
|
4pos |
⊢ 0 < 4 |
78 |
74 75 76 77
|
ltrecii |
⊢ ( 2 < 4 ↔ ( 1 / 4 ) < ( 1 / 2 ) ) |
79 |
73 78
|
mpbi |
⊢ ( 1 / 4 ) < ( 1 / 2 ) |
80 |
23 61 79
|
ltleii |
⊢ ( 1 / 4 ) ≤ ( 1 / 2 ) |
81 |
80
|
a1i |
⊢ ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) → ( 1 / 4 ) ≤ ( 1 / 2 ) ) |
82 |
64 67 71 71 72 81
|
le2addd |
⊢ ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) → ( 𝑥 + ( 1 / 4 ) ) ≤ ( ( 1 / 2 ) + ( 1 / 2 ) ) ) |
83 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
84 |
|
2halves |
⊢ ( 1 ∈ ℂ → ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1 ) |
85 |
83 84
|
ax-mp |
⊢ ( ( 1 / 2 ) + ( 1 / 2 ) ) = 1 |
86 |
82 85
|
breqtrdi |
⊢ ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) → ( 𝑥 + ( 1 / 4 ) ) ≤ 1 ) |
87 |
|
1re |
⊢ 1 ∈ ℝ |
88 |
61 87
|
elicc2i |
⊢ ( ( 𝑥 + ( 1 / 4 ) ) ∈ ( ( 1 / 2 ) [,] 1 ) ↔ ( ( 𝑥 + ( 1 / 4 ) ) ∈ ℝ ∧ ( 1 / 2 ) ≤ ( 𝑥 + ( 1 / 4 ) ) ∧ ( 𝑥 + ( 1 / 4 ) ) ≤ 1 ) ) |
89 |
66 70 86 88
|
syl3anbrc |
⊢ ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) → ( 𝑥 + ( 1 / 4 ) ) ∈ ( ( 1 / 2 ) [,] 1 ) ) |
90 |
63 89
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( 𝑥 + ( 1 / 4 ) ) ∈ ( ( 1 / 2 ) [,] 1 ) ) |
91 |
2 3
|
pco0 |
⊢ ( 𝜑 → ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ 0 ) = ( 𝐺 ‘ 0 ) ) |
92 |
4 91
|
eqtr4d |
⊢ ( 𝜑 → ( 𝐹 ‘ 1 ) = ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ 0 ) ) |
93 |
1 46 92
|
pcoval2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 + ( 1 / 4 ) ) ∈ ( ( 1 / 2 ) [,] 1 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( 𝑥 + ( 1 / 4 ) ) ) = ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ ( ( 2 · ( 𝑥 + ( 1 / 4 ) ) ) − 1 ) ) ) |
94 |
54 90 93
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( 𝑥 + ( 1 / 4 ) ) ) = ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ ( ( 2 · ( 𝑥 + ( 1 / 4 ) ) ) − 1 ) ) ) |
95 |
85
|
oveq2i |
⊢ ( ( 2 · ( 𝑥 + ( 1 / 4 ) ) ) − ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( ( 2 · ( 𝑥 + ( 1 / 4 ) ) ) − 1 ) |
96 |
|
2cnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → 2 ∈ ℂ ) |
97 |
56
|
recnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → 𝑥 ∈ ℂ ) |
98 |
28
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( 1 / 4 ) ∈ ℂ ) |
99 |
96 97 98
|
adddid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( 2 · ( 𝑥 + ( 1 / 4 ) ) ) = ( ( 2 · 𝑥 ) + ( 2 · ( 1 / 4 ) ) ) ) |
100 |
41
|
oveq2i |
⊢ ( ( 2 · 𝑥 ) + ( 2 · ( 1 / 4 ) ) ) = ( ( 2 · 𝑥 ) + ( 1 / 2 ) ) |
101 |
99 100
|
eqtrdi |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( 2 · ( 𝑥 + ( 1 / 4 ) ) ) = ( ( 2 · 𝑥 ) + ( 1 / 2 ) ) ) |
102 |
101
|
oveq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 2 · ( 𝑥 + ( 1 / 4 ) ) ) − ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( ( ( 2 · 𝑥 ) + ( 1 / 2 ) ) − ( ( 1 / 2 ) + ( 1 / 2 ) ) ) ) |
103 |
95 102
|
eqtr3id |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 2 · ( 𝑥 + ( 1 / 4 ) ) ) − 1 ) = ( ( ( 2 · 𝑥 ) + ( 1 / 2 ) ) − ( ( 1 / 2 ) + ( 1 / 2 ) ) ) ) |
104 |
|
remulcl |
⊢ ( ( 2 ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( 2 · 𝑥 ) ∈ ℝ ) |
105 |
74 56 104
|
sylancr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( 2 · 𝑥 ) ∈ ℝ ) |
106 |
105
|
recnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( 2 · 𝑥 ) ∈ ℂ ) |
107 |
37
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( 1 / 2 ) ∈ ℂ ) |
108 |
106 107 107
|
pnpcan2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( ( ( 2 · 𝑥 ) + ( 1 / 2 ) ) − ( ( 1 / 2 ) + ( 1 / 2 ) ) ) = ( ( 2 · 𝑥 ) − ( 1 / 2 ) ) ) |
109 |
103 108
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 2 · ( 𝑥 + ( 1 / 4 ) ) ) − 1 ) = ( ( 2 · 𝑥 ) − ( 1 / 2 ) ) ) |
110 |
109
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ ( ( 2 · ( 𝑥 + ( 1 / 4 ) ) ) − 1 ) ) = ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ ( ( 2 · 𝑥 ) − ( 1 / 2 ) ) ) ) |
111 |
10 97 15
|
sylancr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( 2 · 𝑥 ) = ( 𝑥 · 2 ) ) |
112 |
83 10 30
|
divcan1i |
⊢ ( ( 1 / 2 ) · 2 ) = 1 |
113 |
23 61 26 42 112
|
iccdili |
⊢ ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) → ( 𝑥 · 2 ) ∈ ( ( 1 / 2 ) [,] 1 ) ) |
114 |
63 113
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( 𝑥 · 2 ) ∈ ( ( 1 / 2 ) [,] 1 ) ) |
115 |
111 114
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( 2 · 𝑥 ) ∈ ( ( 1 / 2 ) [,] 1 ) ) |
116 |
37
|
subidi |
⊢ ( ( 1 / 2 ) − ( 1 / 2 ) ) = 0 |
117 |
|
1mhlfehlf |
⊢ ( 1 − ( 1 / 2 ) ) = ( 1 / 2 ) |
118 |
61 87 61 116 117
|
iccshftli |
⊢ ( ( 2 · 𝑥 ) ∈ ( ( 1 / 2 ) [,] 1 ) → ( ( 2 · 𝑥 ) − ( 1 / 2 ) ) ∈ ( 0 [,] ( 1 / 2 ) ) ) |
119 |
115 118
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 2 · 𝑥 ) − ( 1 / 2 ) ) ∈ ( 0 [,] ( 1 / 2 ) ) ) |
120 |
2 3
|
pcoval1 |
⊢ ( ( 𝜑 ∧ ( ( 2 · 𝑥 ) − ( 1 / 2 ) ) ∈ ( 0 [,] ( 1 / 2 ) ) ) → ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ ( ( 2 · 𝑥 ) − ( 1 / 2 ) ) ) = ( 𝐺 ‘ ( 2 · ( ( 2 · 𝑥 ) − ( 1 / 2 ) ) ) ) ) |
121 |
54 119 120
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ ( ( 2 · 𝑥 ) − ( 1 / 2 ) ) ) = ( 𝐺 ‘ ( 2 · ( ( 2 · 𝑥 ) − ( 1 / 2 ) ) ) ) ) |
122 |
96 106 107
|
subdid |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( 2 · ( ( 2 · 𝑥 ) − ( 1 / 2 ) ) ) = ( ( 2 · ( 2 · 𝑥 ) ) − ( 2 · ( 1 / 2 ) ) ) ) |
123 |
10 30
|
recidi |
⊢ ( 2 · ( 1 / 2 ) ) = 1 |
124 |
123
|
oveq2i |
⊢ ( ( 2 · ( 2 · 𝑥 ) ) − ( 2 · ( 1 / 2 ) ) ) = ( ( 2 · ( 2 · 𝑥 ) ) − 1 ) |
125 |
122 124
|
eqtrdi |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( 2 · ( ( 2 · 𝑥 ) − ( 1 / 2 ) ) ) = ( ( 2 · ( 2 · 𝑥 ) ) − 1 ) ) |
126 |
125
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( 𝐺 ‘ ( 2 · ( ( 2 · 𝑥 ) − ( 1 / 2 ) ) ) ) = ( 𝐺 ‘ ( ( 2 · ( 2 · 𝑥 ) ) − 1 ) ) ) |
127 |
121 126
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ ( ( 2 · 𝑥 ) − ( 1 / 2 ) ) ) = ( 𝐺 ‘ ( ( 2 · ( 2 · 𝑥 ) ) − 1 ) ) ) |
128 |
94 110 127
|
3eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( 𝑥 + ( 1 / 4 ) ) ) = ( 𝐺 ‘ ( ( 2 · ( 2 · 𝑥 ) ) − 1 ) ) ) |
129 |
|
iffalse |
⊢ ( ¬ 𝑥 ≤ ( 1 / 4 ) → if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) = ( 𝑥 + ( 1 / 4 ) ) ) |
130 |
129
|
fveq2d |
⊢ ( ¬ 𝑥 ≤ ( 1 / 4 ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( 𝑥 + ( 1 / 4 ) ) ) ) |
131 |
130
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( 𝑥 + ( 1 / 4 ) ) ) ) |
132 |
1 2 4
|
pcoval2 |
⊢ ( ( 𝜑 ∧ ( 2 · 𝑥 ) ∈ ( ( 1 / 2 ) [,] 1 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) = ( 𝐺 ‘ ( ( 2 · ( 2 · 𝑥 ) ) − 1 ) ) ) |
133 |
54 115 132
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) = ( 𝐺 ‘ ( ( 2 · ( 2 · 𝑥 ) ) − 1 ) ) ) |
134 |
128 131 133
|
3eqtr4d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 4 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) ) |
135 |
53 134
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) ) |
136 |
|
iftrue |
⊢ ( 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) = if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) ) |
137 |
136
|
fveq2d |
⊢ ( 𝑥 ≤ ( 1 / 2 ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) ) ) |
138 |
137
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) ) ) |
139 |
|
iftrue |
⊢ ( 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) , ( 𝐻 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) ) |
140 |
139
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) , ( 𝐻 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) ) |
141 |
135 138 140
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ 𝑥 ≤ ( 1 / 2 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) , ( 𝐻 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) |
142 |
|
elii2 |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) → 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) ) |
143 |
|
halfge0 |
⊢ 0 ≤ ( 1 / 2 ) |
144 |
|
halflt1 |
⊢ ( 1 / 2 ) < 1 |
145 |
61 87 144
|
ltleii |
⊢ ( 1 / 2 ) ≤ 1 |
146 |
|
elicc01 |
⊢ ( ( 1 / 2 ) ∈ ( 0 [,] 1 ) ↔ ( ( 1 / 2 ) ∈ ℝ ∧ 0 ≤ ( 1 / 2 ) ∧ ( 1 / 2 ) ≤ 1 ) ) |
147 |
61 143 145 146
|
mpbir3an |
⊢ ( 1 / 2 ) ∈ ( 0 [,] 1 ) |
148 |
|
1elunit |
⊢ 1 ∈ ( 0 [,] 1 ) |
149 |
|
iccss2 |
⊢ ( ( ( 1 / 2 ) ∈ ( 0 [,] 1 ) ∧ 1 ∈ ( 0 [,] 1 ) ) → ( ( 1 / 2 ) [,] 1 ) ⊆ ( 0 [,] 1 ) ) |
150 |
147 148 149
|
mp2an |
⊢ ( ( 1 / 2 ) [,] 1 ) ⊆ ( 0 [,] 1 ) |
151 |
150
|
sseli |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → 𝑥 ∈ ( 0 [,] 1 ) ) |
152 |
10 30
|
div0i |
⊢ ( 0 / 2 ) = 0 |
153 |
|
eqid |
⊢ ( 1 / 2 ) = ( 1 / 2 ) |
154 |
20 87 26 152 153
|
icccntri |
⊢ ( 𝑥 ∈ ( 0 [,] 1 ) → ( 𝑥 / 2 ) ∈ ( 0 [,] ( 1 / 2 ) ) ) |
155 |
37
|
addid2i |
⊢ ( 0 + ( 1 / 2 ) ) = ( 1 / 2 ) |
156 |
20 61 61 155 85
|
iccshftri |
⊢ ( ( 𝑥 / 2 ) ∈ ( 0 [,] ( 1 / 2 ) ) → ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ∈ ( ( 1 / 2 ) [,] 1 ) ) |
157 |
151 154 156
|
3syl |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ∈ ( ( 1 / 2 ) [,] 1 ) ) |
158 |
1 46 92
|
pcoval2 |
⊢ ( ( 𝜑 ∧ ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ∈ ( ( 1 / 2 ) [,] 1 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) = ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ ( ( 2 · ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) − 1 ) ) ) |
159 |
157 158
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) = ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ ( ( 2 · ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) − 1 ) ) ) |
160 |
61 87
|
elicc2i |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) ↔ ( 𝑥 ∈ ℝ ∧ ( 1 / 2 ) ≤ 𝑥 ∧ 𝑥 ≤ 1 ) ) |
161 |
160
|
simp1bi |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → 𝑥 ∈ ℝ ) |
162 |
161
|
recnd |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → 𝑥 ∈ ℂ ) |
163 |
|
1cnd |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → 1 ∈ ℂ ) |
164 |
|
2cnd |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → 2 ∈ ℂ ) |
165 |
30
|
a1i |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → 2 ≠ 0 ) |
166 |
162 163 164 165
|
divdird |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → ( ( 𝑥 + 1 ) / 2 ) = ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) |
167 |
166
|
oveq2d |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → ( 2 · ( ( 𝑥 + 1 ) / 2 ) ) = ( 2 · ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) |
168 |
|
peano2cn |
⊢ ( 𝑥 ∈ ℂ → ( 𝑥 + 1 ) ∈ ℂ ) |
169 |
162 168
|
syl |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → ( 𝑥 + 1 ) ∈ ℂ ) |
170 |
169 164 165
|
divcan2d |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → ( 2 · ( ( 𝑥 + 1 ) / 2 ) ) = ( 𝑥 + 1 ) ) |
171 |
167 170
|
eqtr3d |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → ( 2 · ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) = ( 𝑥 + 1 ) ) |
172 |
162 163 171
|
mvrraddd |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → ( ( 2 · ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) − 1 ) = 𝑥 ) |
173 |
172
|
fveq2d |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ ( ( 2 · ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) − 1 ) ) = ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ 𝑥 ) ) |
174 |
173
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) ) → ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ ( ( 2 · ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) − 1 ) ) = ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ 𝑥 ) ) |
175 |
2 3 5
|
pcoval2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) ) → ( ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ‘ 𝑥 ) = ( 𝐻 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) |
176 |
159 174 175
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) = ( 𝐻 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) |
177 |
142 176
|
sylan2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) = ( 𝐻 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) |
178 |
177
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) = ( 𝐻 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) |
179 |
|
iffalse |
⊢ ( ¬ 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) = ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) |
180 |
179
|
fveq2d |
⊢ ( ¬ 𝑥 ≤ ( 1 / 2 ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) |
181 |
180
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) |
182 |
|
iffalse |
⊢ ( ¬ 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) , ( 𝐻 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) = ( 𝐻 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) |
183 |
182
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) , ( 𝐻 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) = ( 𝐻 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) |
184 |
178 181 183
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) , ( 𝐻 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) |
185 |
141 184
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 0 [,] 1 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) , ( 𝐻 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) |
186 |
185
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) , ( 𝐻 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) ) |
187 |
|
iitopon |
⊢ II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) |
188 |
187
|
a1i |
⊢ ( 𝜑 → II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) ) |
189 |
188
|
cnmptid |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ 𝑥 ) ∈ ( II Cn II ) ) |
190 |
|
0elunit |
⊢ 0 ∈ ( 0 [,] 1 ) |
191 |
190
|
a1i |
⊢ ( 𝜑 → 0 ∈ ( 0 [,] 1 ) ) |
192 |
188 188 191
|
cnmptc |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ 0 ) ∈ ( II Cn II ) ) |
193 |
|
eqid |
⊢ ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) ) |
194 |
|
eqid |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) |
195 |
|
eqid |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) |
196 |
|
dfii2 |
⊢ II = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] 1 ) ) |
197 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
198 |
|
1red |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
199 |
147
|
a1i |
⊢ ( 𝜑 → ( 1 / 2 ) ∈ ( 0 [,] 1 ) ) |
200 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → 𝑦 = ( 1 / 2 ) ) |
201 |
200
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 𝑦 + ( 1 / 4 ) ) = ( ( 1 / 2 ) + ( 1 / 4 ) ) ) |
202 |
37 28
|
addcomi |
⊢ ( ( 1 / 2 ) + ( 1 / 4 ) ) = ( ( 1 / 4 ) + ( 1 / 2 ) ) |
203 |
201 202
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 𝑦 + ( 1 / 4 ) ) = ( ( 1 / 4 ) + ( 1 / 2 ) ) ) |
204 |
23 61
|
ltnlei |
⊢ ( ( 1 / 4 ) < ( 1 / 2 ) ↔ ¬ ( 1 / 2 ) ≤ ( 1 / 4 ) ) |
205 |
79 204
|
mpbi |
⊢ ¬ ( 1 / 2 ) ≤ ( 1 / 4 ) |
206 |
200
|
breq1d |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 𝑦 ≤ ( 1 / 4 ) ↔ ( 1 / 2 ) ≤ ( 1 / 4 ) ) ) |
207 |
205 206
|
mtbiri |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ¬ 𝑦 ≤ ( 1 / 4 ) ) |
208 |
207
|
iffalsed |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → if ( 𝑦 ≤ ( 1 / 4 ) , ( 2 · 𝑦 ) , ( 𝑦 + ( 1 / 4 ) ) ) = ( 𝑦 + ( 1 / 4 ) ) ) |
209 |
200
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 𝑦 / 2 ) = ( ( 1 / 2 ) / 2 ) ) |
210 |
209 35
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 𝑦 / 2 ) = ( 1 / 4 ) ) |
211 |
210
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( ( 𝑦 / 2 ) + ( 1 / 2 ) ) = ( ( 1 / 4 ) + ( 1 / 2 ) ) ) |
212 |
203 208 211
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → if ( 𝑦 ≤ ( 1 / 4 ) , ( 2 · 𝑦 ) , ( 𝑦 + ( 1 / 4 ) ) ) = ( ( 𝑦 / 2 ) + ( 1 / 2 ) ) ) |
213 |
|
eqid |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 4 ) ) ) = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 4 ) ) ) |
214 |
|
eqid |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ) = ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ) |
215 |
61
|
a1i |
⊢ ( 𝜑 → ( 1 / 2 ) ∈ ℝ ) |
216 |
75 77
|
recgt0ii |
⊢ 0 < ( 1 / 4 ) |
217 |
20 23 216
|
ltleii |
⊢ 0 ≤ ( 1 / 4 ) |
218 |
20 61
|
elicc2i |
⊢ ( ( 1 / 4 ) ∈ ( 0 [,] ( 1 / 2 ) ) ↔ ( ( 1 / 4 ) ∈ ℝ ∧ 0 ≤ ( 1 / 4 ) ∧ ( 1 / 4 ) ≤ ( 1 / 2 ) ) ) |
219 |
23 217 80 218
|
mpbir3an |
⊢ ( 1 / 4 ) ∈ ( 0 [,] ( 1 / 2 ) ) |
220 |
219
|
a1i |
⊢ ( 𝜑 → ( 1 / 4 ) ∈ ( 0 [,] ( 1 / 2 ) ) ) |
221 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 4 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → 𝑦 = ( 1 / 4 ) ) |
222 |
221
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 4 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 2 · 𝑦 ) = ( 2 · ( 1 / 4 ) ) ) |
223 |
221
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 4 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 𝑦 + ( 1 / 4 ) ) = ( ( 1 / 4 ) + ( 1 / 4 ) ) ) |
224 |
29 222 223
|
3eqtr4a |
⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 4 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 2 · 𝑦 ) = ( 𝑦 + ( 1 / 4 ) ) ) |
225 |
|
retopon |
⊢ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) |
226 |
|
0xr |
⊢ 0 ∈ ℝ* |
227 |
61
|
rexri |
⊢ ( 1 / 2 ) ∈ ℝ* |
228 |
|
lbicc2 |
⊢ ( ( 0 ∈ ℝ* ∧ ( 1 / 2 ) ∈ ℝ* ∧ 0 ≤ ( 1 / 2 ) ) → 0 ∈ ( 0 [,] ( 1 / 2 ) ) ) |
229 |
226 227 143 228
|
mp3an |
⊢ 0 ∈ ( 0 [,] ( 1 / 2 ) ) |
230 |
|
iccss2 |
⊢ ( ( 0 ∈ ( 0 [,] ( 1 / 2 ) ) ∧ ( 1 / 4 ) ∈ ( 0 [,] ( 1 / 2 ) ) ) → ( 0 [,] ( 1 / 4 ) ) ⊆ ( 0 [,] ( 1 / 2 ) ) ) |
231 |
229 219 230
|
mp2an |
⊢ ( 0 [,] ( 1 / 4 ) ) ⊆ ( 0 [,] ( 1 / 2 ) ) |
232 |
|
iccssre |
⊢ ( ( 0 ∈ ℝ ∧ ( 1 / 2 ) ∈ ℝ ) → ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ ) |
233 |
20 61 232
|
mp2an |
⊢ ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ |
234 |
231 233
|
sstri |
⊢ ( 0 [,] ( 1 / 4 ) ) ⊆ ℝ |
235 |
|
resttopon |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( 0 [,] ( 1 / 4 ) ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 4 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 4 ) ) ) ) |
236 |
225 234 235
|
mp2an |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 4 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 4 ) ) ) |
237 |
236
|
a1i |
⊢ ( 𝜑 → ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 4 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 4 ) ) ) ) |
238 |
237 188
|
cnmpt1st |
⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 [,] ( 1 / 4 ) ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ 𝑦 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 4 ) ) ) ×t II ) Cn ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 4 ) ) ) ) ) |
239 |
|
retop |
⊢ ( topGen ‘ ran (,) ) ∈ Top |
240 |
|
ovex |
⊢ ( 0 [,] ( 1 / 2 ) ) ∈ V |
241 |
|
restabs |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ Top ∧ ( 0 [,] ( 1 / 4 ) ) ⊆ ( 0 [,] ( 1 / 2 ) ) ∧ ( 0 [,] ( 1 / 2 ) ) ∈ V ) → ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ↾t ( 0 [,] ( 1 / 4 ) ) ) = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 4 ) ) ) ) |
242 |
239 231 240 241
|
mp3an |
⊢ ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ↾t ( 0 [,] ( 1 / 4 ) ) ) = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 4 ) ) ) |
243 |
242
|
eqcomi |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 4 ) ) ) = ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ↾t ( 0 [,] ( 1 / 4 ) ) ) |
244 |
|
resttopon |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) ) |
245 |
225 233 244
|
mp2an |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) |
246 |
245
|
a1i |
⊢ ( 𝜑 → ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) ) |
247 |
231
|
a1i |
⊢ ( 𝜑 → ( 0 [,] ( 1 / 4 ) ) ⊆ ( 0 [,] ( 1 / 2 ) ) ) |
248 |
194
|
iihalf1cn |
⊢ ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) ↦ ( 2 · 𝑥 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) Cn II ) |
249 |
248
|
a1i |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) ↦ ( 2 · 𝑥 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) Cn II ) ) |
250 |
243 246 247 249
|
cnmpt1res |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] ( 1 / 4 ) ) ↦ ( 2 · 𝑥 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 4 ) ) ) Cn II ) ) |
251 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 2 · 𝑥 ) = ( 2 · 𝑦 ) ) |
252 |
237 188 238 237 250 251
|
cnmpt21 |
⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 [,] ( 1 / 4 ) ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 2 · 𝑦 ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 4 ) ) ) ×t II ) Cn II ) ) |
253 |
|
iccssre |
⊢ ( ( ( 1 / 4 ) ∈ ℝ ∧ ( 1 / 2 ) ∈ ℝ ) → ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ⊆ ℝ ) |
254 |
23 61 253
|
mp2an |
⊢ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ⊆ ℝ |
255 |
|
resttopon |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ) ) |
256 |
225 254 255
|
mp2an |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ) |
257 |
256
|
a1i |
⊢ ( 𝜑 → ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ) ) |
258 |
257 188
|
cnmpt1st |
⊢ ( 𝜑 → ( 𝑦 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ 𝑦 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ) ×t II ) Cn ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ) ) ) |
259 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
260 |
254
|
a1i |
⊢ ( 𝜑 → ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ⊆ ℝ ) |
261 |
|
unitssre |
⊢ ( 0 [,] 1 ) ⊆ ℝ |
262 |
261
|
a1i |
⊢ ( 𝜑 → ( 0 [,] 1 ) ⊆ ℝ ) |
263 |
150 89
|
sselid |
⊢ ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) → ( 𝑥 + ( 1 / 4 ) ) ∈ ( 0 [,] 1 ) ) |
264 |
263
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ) → ( 𝑥 + ( 1 / 4 ) ) ∈ ( 0 [,] 1 ) ) |
265 |
259
|
cnfldtopon |
⊢ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) |
266 |
265
|
a1i |
⊢ ( 𝜑 → ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) ) |
267 |
266
|
cnmptid |
⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ 𝑥 ) ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
268 |
23
|
a1i |
⊢ ( 𝜑 → ( 1 / 4 ) ∈ ℝ ) |
269 |
268
|
recnd |
⊢ ( 𝜑 → ( 1 / 4 ) ∈ ℂ ) |
270 |
266 266 269
|
cnmptc |
⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ ( 1 / 4 ) ) ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
271 |
259
|
addcn |
⊢ + ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) ) |
272 |
271
|
a1i |
⊢ ( 𝜑 → + ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) ) ) |
273 |
266 267 270 272
|
cnmpt12f |
⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ ( 𝑥 + ( 1 / 4 ) ) ) ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
274 |
259 214 196 260 262 264 273
|
cnmptre |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ↦ ( 𝑥 + ( 1 / 4 ) ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ) Cn II ) ) |
275 |
|
oveq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 + ( 1 / 4 ) ) = ( 𝑦 + ( 1 / 4 ) ) ) |
276 |
257 188 258 257 274 275
|
cnmpt21 |
⊢ ( 𝜑 → ( 𝑦 ∈ ( ( 1 / 4 ) [,] ( 1 / 2 ) ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝑦 + ( 1 / 4 ) ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 4 ) [,] ( 1 / 2 ) ) ) ×t II ) Cn II ) ) |
277 |
193 213 214 194 197 215 220 188 224 252 276
|
cnmpopc |
⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ if ( 𝑦 ≤ ( 1 / 4 ) , ( 2 · 𝑦 ) , ( 𝑦 + ( 1 / 4 ) ) ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ×t II ) Cn II ) ) |
278 |
|
iccssre |
⊢ ( ( ( 1 / 2 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ ) |
279 |
61 87 278
|
mp2an |
⊢ ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ |
280 |
|
resttopon |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) ) |
281 |
225 279 280
|
mp2an |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) |
282 |
281
|
a1i |
⊢ ( 𝜑 → ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) ) |
283 |
282 188
|
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 ) ) ) ) |
284 |
279
|
a1i |
⊢ ( 𝜑 → ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ ) |
285 |
150 157
|
sselid |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ∈ ( 0 [,] 1 ) ) |
286 |
285
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) ) → ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ∈ ( 0 [,] 1 ) ) |
287 |
259
|
divccn |
⊢ ( ( 2 ∈ ℂ ∧ 2 ≠ 0 ) → ( 𝑥 ∈ ℂ ↦ ( 𝑥 / 2 ) ) ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
288 |
10 30 287
|
mp2an |
⊢ ( 𝑥 ∈ ℂ ↦ ( 𝑥 / 2 ) ) ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) |
289 |
288
|
a1i |
⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ ( 𝑥 / 2 ) ) ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
290 |
37
|
a1i |
⊢ ( 𝜑 → ( 1 / 2 ) ∈ ℂ ) |
291 |
266 266 290
|
cnmptc |
⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ ( 1 / 2 ) ) ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
292 |
266 289 291 272
|
cnmpt12f |
⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
293 |
259 195 196 284 262 286 292
|
cnmptre |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) ↦ ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) Cn II ) ) |
294 |
|
oveq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 / 2 ) = ( 𝑦 / 2 ) ) |
295 |
294
|
oveq1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) = ( ( 𝑦 / 2 ) + ( 1 / 2 ) ) ) |
296 |
282 188 283 282 293 295
|
cnmpt21 |
⊢ ( 𝜑 → ( 𝑦 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑦 / 2 ) + ( 1 / 2 ) ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ×t II ) Cn II ) ) |
297 |
193 194 195 196 197 198 199 188 212 277 296
|
cnmpopc |
⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 [,] 1 ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ if ( 𝑦 ≤ ( 1 / 2 ) , if ( 𝑦 ≤ ( 1 / 4 ) , ( 2 · 𝑦 ) , ( 𝑦 + ( 1 / 4 ) ) ) , ( ( 𝑦 / 2 ) + ( 1 / 2 ) ) ) ) ∈ ( ( II ×t II ) Cn II ) ) |
298 |
|
breq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ≤ ( 1 / 2 ) ↔ 𝑦 ≤ ( 1 / 2 ) ) ) |
299 |
|
breq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ≤ ( 1 / 4 ) ↔ 𝑦 ≤ ( 1 / 4 ) ) ) |
300 |
299 251 275
|
ifbieq12d |
⊢ ( 𝑥 = 𝑦 → if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) = if ( 𝑦 ≤ ( 1 / 4 ) , ( 2 · 𝑦 ) , ( 𝑦 + ( 1 / 4 ) ) ) ) |
301 |
298 300 295
|
ifbieq12d |
⊢ ( 𝑥 = 𝑦 → if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) = if ( 𝑦 ≤ ( 1 / 2 ) , if ( 𝑦 ≤ ( 1 / 4 ) , ( 2 · 𝑦 ) , ( 𝑦 + ( 1 / 4 ) ) ) , ( ( 𝑦 / 2 ) + ( 1 / 2 ) ) ) ) |
302 |
301
|
equcoms |
⊢ ( 𝑦 = 𝑥 → if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) = if ( 𝑦 ≤ ( 1 / 2 ) , if ( 𝑦 ≤ ( 1 / 4 ) , ( 2 · 𝑦 ) , ( 𝑦 + ( 1 / 4 ) ) ) , ( ( 𝑦 / 2 ) + ( 1 / 2 ) ) ) ) |
303 |
302
|
adantr |
⊢ ( ( 𝑦 = 𝑥 ∧ 𝑧 = 0 ) → if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) = if ( 𝑦 ≤ ( 1 / 2 ) , if ( 𝑦 ≤ ( 1 / 4 ) , ( 2 · 𝑦 ) , ( 𝑦 + ( 1 / 4 ) ) ) , ( ( 𝑦 / 2 ) + ( 1 / 2 ) ) ) ) |
304 |
303
|
eqcomd |
⊢ ( ( 𝑦 = 𝑥 ∧ 𝑧 = 0 ) → if ( 𝑦 ≤ ( 1 / 2 ) , if ( 𝑦 ≤ ( 1 / 4 ) , ( 2 · 𝑦 ) , ( 𝑦 + ( 1 / 4 ) ) ) , ( ( 𝑦 / 2 ) + ( 1 / 2 ) ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) |
305 |
188 189 192 188 188 297 304
|
cnmpt12 |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) ∈ ( II Cn II ) ) |
306 |
6 305
|
eqeltrid |
⊢ ( 𝜑 → 𝑃 ∈ ( II Cn II ) ) |
307 |
|
iiuni |
⊢ ( 0 [,] 1 ) = ∪ II |
308 |
307 307
|
cnf |
⊢ ( 𝑃 ∈ ( II Cn II ) → 𝑃 : ( 0 [,] 1 ) ⟶ ( 0 [,] 1 ) ) |
309 |
306 308
|
syl |
⊢ ( 𝜑 → 𝑃 : ( 0 [,] 1 ) ⟶ ( 0 [,] 1 ) ) |
310 |
6
|
fmpt |
⊢ ( ∀ 𝑥 ∈ ( 0 [,] 1 ) if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ∈ ( 0 [,] 1 ) ↔ 𝑃 : ( 0 [,] 1 ) ⟶ ( 0 [,] 1 ) ) |
311 |
309 310
|
sylibr |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ( 0 [,] 1 ) if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ∈ ( 0 [,] 1 ) ) |
312 |
6
|
a1i |
⊢ ( 𝜑 → 𝑃 = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) ) |
313 |
1 46 92
|
pcocn |
⊢ ( 𝜑 → ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ∈ ( II Cn 𝐽 ) ) |
314 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
315 |
307 314
|
cnf |
⊢ ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ∈ ( II Cn 𝐽 ) → ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) : ( 0 [,] 1 ) ⟶ ∪ 𝐽 ) |
316 |
313 315
|
syl |
⊢ ( 𝜑 → ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) : ( 0 [,] 1 ) ⟶ ∪ 𝐽 ) |
317 |
316
|
feqmptd |
⊢ ( 𝜑 → ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) = ( 𝑦 ∈ ( 0 [,] 1 ) ↦ ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ 𝑦 ) ) ) |
318 |
|
fveq2 |
⊢ ( 𝑦 = if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ 𝑦 ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) ) |
319 |
311 312 317 318
|
fmptcof |
⊢ ( 𝜑 → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ∘ 𝑃 ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ‘ if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) ) ) ) |
320 |
1 2 4
|
pcocn |
⊢ ( 𝜑 → ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ∈ ( II Cn 𝐽 ) ) |
321 |
320 3
|
pcoval |
⊢ ( 𝜑 → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ( *𝑝 ‘ 𝐽 ) 𝐻 ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ ( 2 · 𝑥 ) ) , ( 𝐻 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) ) |
322 |
186 319 321
|
3eqtr4rd |
⊢ ( 𝜑 → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ( *𝑝 ‘ 𝐽 ) 𝐻 ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ∘ 𝑃 ) ) |
323 |
|
id |
⊢ ( 𝑥 = 0 → 𝑥 = 0 ) |
324 |
323 143
|
eqbrtrdi |
⊢ ( 𝑥 = 0 → 𝑥 ≤ ( 1 / 2 ) ) |
325 |
324
|
iftrued |
⊢ ( 𝑥 = 0 → if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) = if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) ) |
326 |
323 217
|
eqbrtrdi |
⊢ ( 𝑥 = 0 → 𝑥 ≤ ( 1 / 4 ) ) |
327 |
326
|
iftrued |
⊢ ( 𝑥 = 0 → if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) = ( 2 · 𝑥 ) ) |
328 |
|
oveq2 |
⊢ ( 𝑥 = 0 → ( 2 · 𝑥 ) = ( 2 · 0 ) ) |
329 |
|
2t0e0 |
⊢ ( 2 · 0 ) = 0 |
330 |
328 329
|
eqtrdi |
⊢ ( 𝑥 = 0 → ( 2 · 𝑥 ) = 0 ) |
331 |
325 327 330
|
3eqtrd |
⊢ ( 𝑥 = 0 → if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) = 0 ) |
332 |
|
c0ex |
⊢ 0 ∈ V |
333 |
331 6 332
|
fvmpt |
⊢ ( 0 ∈ ( 0 [,] 1 ) → ( 𝑃 ‘ 0 ) = 0 ) |
334 |
191 333
|
syl |
⊢ ( 𝜑 → ( 𝑃 ‘ 0 ) = 0 ) |
335 |
148
|
a1i |
⊢ ( 𝜑 → 1 ∈ ( 0 [,] 1 ) ) |
336 |
61 87
|
ltnlei |
⊢ ( ( 1 / 2 ) < 1 ↔ ¬ 1 ≤ ( 1 / 2 ) ) |
337 |
144 336
|
mpbi |
⊢ ¬ 1 ≤ ( 1 / 2 ) |
338 |
|
breq1 |
⊢ ( 𝑥 = 1 → ( 𝑥 ≤ ( 1 / 2 ) ↔ 1 ≤ ( 1 / 2 ) ) ) |
339 |
337 338
|
mtbiri |
⊢ ( 𝑥 = 1 → ¬ 𝑥 ≤ ( 1 / 2 ) ) |
340 |
339
|
iffalsed |
⊢ ( 𝑥 = 1 → if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) = ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) |
341 |
|
oveq1 |
⊢ ( 𝑥 = 1 → ( 𝑥 / 2 ) = ( 1 / 2 ) ) |
342 |
341
|
oveq1d |
⊢ ( 𝑥 = 1 → ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) = ( ( 1 / 2 ) + ( 1 / 2 ) ) ) |
343 |
342 85
|
eqtrdi |
⊢ ( 𝑥 = 1 → ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) = 1 ) |
344 |
340 343
|
eqtrd |
⊢ ( 𝑥 = 1 → if ( 𝑥 ≤ ( 1 / 2 ) , if ( 𝑥 ≤ ( 1 / 4 ) , ( 2 · 𝑥 ) , ( 𝑥 + ( 1 / 4 ) ) ) , ( ( 𝑥 / 2 ) + ( 1 / 2 ) ) ) = 1 ) |
345 |
|
1ex |
⊢ 1 ∈ V |
346 |
344 6 345
|
fvmpt |
⊢ ( 1 ∈ ( 0 [,] 1 ) → ( 𝑃 ‘ 1 ) = 1 ) |
347 |
335 346
|
syl |
⊢ ( 𝜑 → ( 𝑃 ‘ 1 ) = 1 ) |
348 |
313 306 334 347
|
reparpht |
⊢ ( 𝜑 → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ∘ 𝑃 ) ( ≃ph ‘ 𝐽 ) ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ) |
349 |
322 348
|
eqbrtrd |
⊢ ( 𝜑 → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ( *𝑝 ‘ 𝐽 ) 𝐻 ) ( ≃ph ‘ 𝐽 ) ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( 𝐺 ( *𝑝 ‘ 𝐽 ) 𝐻 ) ) ) |