Step |
Hyp |
Ref |
Expression |
1 |
|
pcopt.1 |
⊢ 𝑃 = ( ( 0 [,] 1 ) × { 𝑌 } ) |
2 |
1
|
fveq1i |
⊢ ( 𝑃 ‘ ( ( 2 · 𝑥 ) − 1 ) ) = ( ( ( 0 [,] 1 ) × { 𝑌 } ) ‘ ( ( 2 · 𝑥 ) − 1 ) ) |
3 |
|
simpr |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝐹 ‘ 1 ) = 𝑌 ) |
4 |
|
iiuni |
⊢ ( 0 [,] 1 ) = ∪ II |
5 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
6 |
4 5
|
cnf |
⊢ ( 𝐹 ∈ ( II Cn 𝐽 ) → 𝐹 : ( 0 [,] 1 ) ⟶ ∪ 𝐽 ) |
7 |
6
|
adantr |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 𝐹 : ( 0 [,] 1 ) ⟶ ∪ 𝐽 ) |
8 |
|
1elunit |
⊢ 1 ∈ ( 0 [,] 1 ) |
9 |
|
ffvelrn |
⊢ ( ( 𝐹 : ( 0 [,] 1 ) ⟶ ∪ 𝐽 ∧ 1 ∈ ( 0 [,] 1 ) ) → ( 𝐹 ‘ 1 ) ∈ ∪ 𝐽 ) |
10 |
7 8 9
|
sylancl |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝐹 ‘ 1 ) ∈ ∪ 𝐽 ) |
11 |
3 10
|
eqeltrrd |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 𝑌 ∈ ∪ 𝐽 ) |
12 |
|
elii2 |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) → 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) ) |
13 |
|
iihalf2 |
⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → ( ( 2 · 𝑥 ) − 1 ) ∈ ( 0 [,] 1 ) ) |
14 |
12 13
|
syl |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) → ( ( 2 · 𝑥 ) − 1 ) ∈ ( 0 [,] 1 ) ) |
15 |
|
fvconst2g |
⊢ ( ( 𝑌 ∈ ∪ 𝐽 ∧ ( ( 2 · 𝑥 ) − 1 ) ∈ ( 0 [,] 1 ) ) → ( ( ( 0 [,] 1 ) × { 𝑌 } ) ‘ ( ( 2 · 𝑥 ) − 1 ) ) = 𝑌 ) |
16 |
11 14 15
|
syl2an |
⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) ) → ( ( ( 0 [,] 1 ) × { 𝑌 } ) ‘ ( ( 2 · 𝑥 ) − 1 ) ) = 𝑌 ) |
17 |
2 16
|
eqtrid |
⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) ) → ( 𝑃 ‘ ( ( 2 · 𝑥 ) − 1 ) ) = 𝑌 ) |
18 |
|
simplr |
⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) ) → ( 𝐹 ‘ 1 ) = 𝑌 ) |
19 |
17 18
|
eqtr4d |
⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) ) → ( 𝑃 ‘ ( ( 2 · 𝑥 ) − 1 ) ) = ( 𝐹 ‘ 1 ) ) |
20 |
19
|
anassrs |
⊢ ( ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) → ( 𝑃 ‘ ( ( 2 · 𝑥 ) − 1 ) ) = ( 𝐹 ‘ 1 ) ) |
21 |
20
|
ifeq2da |
⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ 𝑥 ∈ ( 0 [,] 1 ) ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝑃 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ 1 ) ) ) |
22 |
21
|
mpteq2dva |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝑃 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ 1 ) ) ) ) |
23 |
|
simpl |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 𝐹 ∈ ( II Cn 𝐽 ) ) |
24 |
|
cntop2 |
⊢ ( 𝐹 ∈ ( II Cn 𝐽 ) → 𝐽 ∈ Top ) |
25 |
24
|
adantr |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 𝐽 ∈ Top ) |
26 |
|
toptopon2 |
⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) ) |
27 |
25 26
|
sylib |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) ) |
28 |
1
|
pcoptcl |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) ∧ 𝑌 ∈ ∪ 𝐽 ) → ( 𝑃 ∈ ( II Cn 𝐽 ) ∧ ( 𝑃 ‘ 0 ) = 𝑌 ∧ ( 𝑃 ‘ 1 ) = 𝑌 ) ) |
29 |
27 11 28
|
syl2anc |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑃 ∈ ( II Cn 𝐽 ) ∧ ( 𝑃 ‘ 0 ) = 𝑌 ∧ ( 𝑃 ‘ 1 ) = 𝑌 ) ) |
30 |
29
|
simp1d |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 𝑃 ∈ ( II Cn 𝐽 ) ) |
31 |
23 30
|
pcoval |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝑃 ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝑃 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) ) |
32 |
|
iftrue |
⊢ ( 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) = ( 2 · 𝑥 ) ) |
33 |
32
|
adantl |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 2 ) ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) = ( 2 · 𝑥 ) ) |
34 |
|
elii1 |
⊢ ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) ↔ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ) |
35 |
|
iihalf1 |
⊢ ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) → ( 2 · 𝑥 ) ∈ ( 0 [,] 1 ) ) |
36 |
34 35
|
sylbir |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 2 ) ) → ( 2 · 𝑥 ) ∈ ( 0 [,] 1 ) ) |
37 |
33 36
|
eqeltrd |
⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 2 ) ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ∈ ( 0 [,] 1 ) ) |
38 |
37
|
ex |
⊢ ( 𝑥 ∈ ( 0 [,] 1 ) → ( 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ∈ ( 0 [,] 1 ) ) ) |
39 |
|
iffalse |
⊢ ( ¬ 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) = 1 ) |
40 |
39 8
|
eqeltrdi |
⊢ ( ¬ 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ∈ ( 0 [,] 1 ) ) |
41 |
38 40
|
pm2.61d1 |
⊢ ( 𝑥 ∈ ( 0 [,] 1 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ∈ ( 0 [,] 1 ) ) |
42 |
41
|
adantl |
⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ 𝑥 ∈ ( 0 [,] 1 ) ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ∈ ( 0 [,] 1 ) ) |
43 |
|
eqidd |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ) |
44 |
7
|
feqmptd |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 𝐹 = ( 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝐹 ‘ 𝑦 ) ) ) |
45 |
|
fveq2 |
⊢ ( 𝑦 = if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ) |
46 |
|
fvif |
⊢ ( 𝐹 ‘ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ 1 ) ) |
47 |
45 46
|
eqtrdi |
⊢ ( 𝑦 = if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) → ( 𝐹 ‘ 𝑦 ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ 1 ) ) ) |
48 |
42 43 44 47
|
fmptco |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝐹 ∘ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ 1 ) ) ) ) |
49 |
22 31 48
|
3eqtr4d |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝑃 ) = ( 𝐹 ∘ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ) ) |
50 |
|
iitopon |
⊢ II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) |
51 |
50
|
a1i |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) ) |
52 |
51
|
cnmptid |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ 𝑥 ) ∈ ( II Cn II ) ) |
53 |
|
0elunit |
⊢ 0 ∈ ( 0 [,] 1 ) |
54 |
53
|
a1i |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 0 ∈ ( 0 [,] 1 ) ) |
55 |
51 51 54
|
cnmptc |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ 0 ) ∈ ( II Cn II ) ) |
56 |
|
eqid |
⊢ ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) ) |
57 |
|
eqid |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) |
58 |
|
eqid |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) |
59 |
|
dfii2 |
⊢ II = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] 1 ) ) |
60 |
|
0re |
⊢ 0 ∈ ℝ |
61 |
60
|
a1i |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 0 ∈ ℝ ) |
62 |
|
1re |
⊢ 1 ∈ ℝ |
63 |
62
|
a1i |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 1 ∈ ℝ ) |
64 |
|
halfre |
⊢ ( 1 / 2 ) ∈ ℝ |
65 |
|
halfge0 |
⊢ 0 ≤ ( 1 / 2 ) |
66 |
|
halflt1 |
⊢ ( 1 / 2 ) < 1 |
67 |
64 62 66
|
ltleii |
⊢ ( 1 / 2 ) ≤ 1 |
68 |
|
elicc01 |
⊢ ( ( 1 / 2 ) ∈ ( 0 [,] 1 ) ↔ ( ( 1 / 2 ) ∈ ℝ ∧ 0 ≤ ( 1 / 2 ) ∧ ( 1 / 2 ) ≤ 1 ) ) |
69 |
64 65 67 68
|
mpbir3an |
⊢ ( 1 / 2 ) ∈ ( 0 [,] 1 ) |
70 |
69
|
a1i |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 1 / 2 ) ∈ ( 0 [,] 1 ) ) |
71 |
|
simprl |
⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → 𝑦 = ( 1 / 2 ) ) |
72 |
71
|
oveq2d |
⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 2 · 𝑦 ) = ( 2 · ( 1 / 2 ) ) ) |
73 |
|
2cn |
⊢ 2 ∈ ℂ |
74 |
|
2ne0 |
⊢ 2 ≠ 0 |
75 |
73 74
|
recidi |
⊢ ( 2 · ( 1 / 2 ) ) = 1 |
76 |
72 75
|
eqtrdi |
⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 2 · 𝑦 ) = 1 ) |
77 |
|
retopon |
⊢ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) |
78 |
|
iccssre |
⊢ ( ( 0 ∈ ℝ ∧ ( 1 / 2 ) ∈ ℝ ) → ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ ) |
79 |
60 64 78
|
mp2an |
⊢ ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ |
80 |
|
resttopon |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) ) |
81 |
77 79 80
|
mp2an |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) |
82 |
81
|
a1i |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) ) |
83 |
82 51
|
cnmpt1st |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑦 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ 𝑦 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ×t II ) Cn ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ) ) |
84 |
57
|
iihalf1cn |
⊢ ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) ↦ ( 2 · 𝑥 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) Cn II ) |
85 |
84
|
a1i |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) ↦ ( 2 · 𝑥 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) Cn II ) ) |
86 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 2 · 𝑥 ) = ( 2 · 𝑦 ) ) |
87 |
82 51 83 82 85 86
|
cnmpt21 |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑦 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 2 · 𝑦 ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ×t II ) Cn II ) ) |
88 |
|
iccssre |
⊢ ( ( ( 1 / 2 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ ) |
89 |
64 62 88
|
mp2an |
⊢ ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ |
90 |
|
resttopon |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) ) |
91 |
77 89 90
|
mp2an |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) |
92 |
91
|
a1i |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) ) |
93 |
8
|
a1i |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 1 ∈ ( 0 [,] 1 ) ) |
94 |
92 51 51 93
|
cnmpt2c |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑦 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ 1 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ×t II ) Cn II ) ) |
95 |
56 57 58 59 61 63 70 51 76 87 94
|
cnmpopc |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑦 ∈ ( 0 [,] 1 ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ if ( 𝑦 ≤ ( 1 / 2 ) , ( 2 · 𝑦 ) , 1 ) ) ∈ ( ( II ×t II ) Cn II ) ) |
96 |
|
breq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ≤ ( 1 / 2 ) ↔ 𝑥 ≤ ( 1 / 2 ) ) ) |
97 |
|
oveq2 |
⊢ ( 𝑦 = 𝑥 → ( 2 · 𝑦 ) = ( 2 · 𝑥 ) ) |
98 |
96 97
|
ifbieq1d |
⊢ ( 𝑦 = 𝑥 → if ( 𝑦 ≤ ( 1 / 2 ) , ( 2 · 𝑦 ) , 1 ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) |
99 |
98
|
adantr |
⊢ ( ( 𝑦 = 𝑥 ∧ 𝑧 = 0 ) → if ( 𝑦 ≤ ( 1 / 2 ) , ( 2 · 𝑦 ) , 1 ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) |
100 |
51 52 55 51 51 95 99
|
cnmpt12 |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ∈ ( II Cn II ) ) |
101 |
|
id |
⊢ ( 𝑥 = 0 → 𝑥 = 0 ) |
102 |
101 65
|
eqbrtrdi |
⊢ ( 𝑥 = 0 → 𝑥 ≤ ( 1 / 2 ) ) |
103 |
102 32
|
syl |
⊢ ( 𝑥 = 0 → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) = ( 2 · 𝑥 ) ) |
104 |
|
oveq2 |
⊢ ( 𝑥 = 0 → ( 2 · 𝑥 ) = ( 2 · 0 ) ) |
105 |
|
2t0e0 |
⊢ ( 2 · 0 ) = 0 |
106 |
104 105
|
eqtrdi |
⊢ ( 𝑥 = 0 → ( 2 · 𝑥 ) = 0 ) |
107 |
103 106
|
eqtrd |
⊢ ( 𝑥 = 0 → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) = 0 ) |
108 |
|
eqid |
⊢ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) |
109 |
|
c0ex |
⊢ 0 ∈ V |
110 |
107 108 109
|
fvmpt |
⊢ ( 0 ∈ ( 0 [,] 1 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ‘ 0 ) = 0 ) |
111 |
53 110
|
mp1i |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ‘ 0 ) = 0 ) |
112 |
64 62
|
ltnlei |
⊢ ( ( 1 / 2 ) < 1 ↔ ¬ 1 ≤ ( 1 / 2 ) ) |
113 |
66 112
|
mpbi |
⊢ ¬ 1 ≤ ( 1 / 2 ) |
114 |
|
breq1 |
⊢ ( 𝑥 = 1 → ( 𝑥 ≤ ( 1 / 2 ) ↔ 1 ≤ ( 1 / 2 ) ) ) |
115 |
113 114
|
mtbiri |
⊢ ( 𝑥 = 1 → ¬ 𝑥 ≤ ( 1 / 2 ) ) |
116 |
115 39
|
syl |
⊢ ( 𝑥 = 1 → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) = 1 ) |
117 |
|
1ex |
⊢ 1 ∈ V |
118 |
116 108 117
|
fvmpt |
⊢ ( 1 ∈ ( 0 [,] 1 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ‘ 1 ) = 1 ) |
119 |
8 118
|
mp1i |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ‘ 1 ) = 1 ) |
120 |
23 100 111 119
|
reparpht |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝐹 ∘ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ) ( ≃ph ‘ 𝐽 ) 𝐹 ) |
121 |
49 120
|
eqbrtrd |
⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝑃 ) ( ≃ph ‘ 𝐽 ) 𝐹 ) |