Step |
Hyp |
Ref |
Expression |
1 |
|
htpycc.1 |
|- N = ( x e. X , y e. ( 0 [,] 1 ) |-> if ( y <_ ( 1 / 2 ) , ( x L ( 2 x. y ) ) , ( x M ( ( 2 x. y ) - 1 ) ) ) ) |
2 |
|
htpycc.2 |
|- ( ph -> J e. ( TopOn ` X ) ) |
3 |
|
htpycc.4 |
|- ( ph -> F e. ( J Cn K ) ) |
4 |
|
htpycc.5 |
|- ( ph -> G e. ( J Cn K ) ) |
5 |
|
htpycc.6 |
|- ( ph -> H e. ( J Cn K ) ) |
6 |
|
htpycc.7 |
|- ( ph -> L e. ( F ( J Htpy K ) G ) ) |
7 |
|
htpycc.8 |
|- ( ph -> M e. ( G ( J Htpy K ) H ) ) |
8 |
|
iitopon |
|- II e. ( TopOn ` ( 0 [,] 1 ) ) |
9 |
8
|
a1i |
|- ( ph -> II e. ( TopOn ` ( 0 [,] 1 ) ) ) |
10 |
|
eqid |
|- ( topGen ` ran (,) ) = ( topGen ` ran (,) ) |
11 |
|
eqid |
|- ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) |
12 |
|
eqid |
|- ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) |
13 |
|
dfii2 |
|- II = ( ( topGen ` ran (,) ) |`t ( 0 [,] 1 ) ) |
14 |
|
0red |
|- ( ph -> 0 e. RR ) |
15 |
|
1red |
|- ( ph -> 1 e. RR ) |
16 |
|
halfre |
|- ( 1 / 2 ) e. RR |
17 |
|
halfge0 |
|- 0 <_ ( 1 / 2 ) |
18 |
|
1re |
|- 1 e. RR |
19 |
|
halflt1 |
|- ( 1 / 2 ) < 1 |
20 |
16 18 19
|
ltleii |
|- ( 1 / 2 ) <_ 1 |
21 |
|
elicc01 |
|- ( ( 1 / 2 ) e. ( 0 [,] 1 ) <-> ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ 1 ) ) |
22 |
16 17 20 21
|
mpbir3an |
|- ( 1 / 2 ) e. ( 0 [,] 1 ) |
23 |
22
|
a1i |
|- ( ph -> ( 1 / 2 ) e. ( 0 [,] 1 ) ) |
24 |
2 3 4 6
|
htpyi |
|- ( ( ph /\ s e. X ) -> ( ( s L 0 ) = ( F ` s ) /\ ( s L 1 ) = ( G ` s ) ) ) |
25 |
24
|
simprd |
|- ( ( ph /\ s e. X ) -> ( s L 1 ) = ( G ` s ) ) |
26 |
2 4 5 7
|
htpyi |
|- ( ( ph /\ s e. X ) -> ( ( s M 0 ) = ( G ` s ) /\ ( s M 1 ) = ( H ` s ) ) ) |
27 |
26
|
simpld |
|- ( ( ph /\ s e. X ) -> ( s M 0 ) = ( G ` s ) ) |
28 |
25 27
|
eqtr4d |
|- ( ( ph /\ s e. X ) -> ( s L 1 ) = ( s M 0 ) ) |
29 |
28
|
ralrimiva |
|- ( ph -> A. s e. X ( s L 1 ) = ( s M 0 ) ) |
30 |
|
oveq1 |
|- ( s = x -> ( s L 1 ) = ( x L 1 ) ) |
31 |
|
oveq1 |
|- ( s = x -> ( s M 0 ) = ( x M 0 ) ) |
32 |
30 31
|
eqeq12d |
|- ( s = x -> ( ( s L 1 ) = ( s M 0 ) <-> ( x L 1 ) = ( x M 0 ) ) ) |
33 |
32
|
rspccva |
|- ( ( A. s e. X ( s L 1 ) = ( s M 0 ) /\ x e. X ) -> ( x L 1 ) = ( x M 0 ) ) |
34 |
29 33
|
sylan |
|- ( ( ph /\ x e. X ) -> ( x L 1 ) = ( x M 0 ) ) |
35 |
34
|
adantrl |
|- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( x L 1 ) = ( x M 0 ) ) |
36 |
|
simprl |
|- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> y = ( 1 / 2 ) ) |
37 |
36
|
oveq2d |
|- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( 2 x. y ) = ( 2 x. ( 1 / 2 ) ) ) |
38 |
|
2cn |
|- 2 e. CC |
39 |
|
2ne0 |
|- 2 =/= 0 |
40 |
38 39
|
recidi |
|- ( 2 x. ( 1 / 2 ) ) = 1 |
41 |
37 40
|
eqtrdi |
|- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( 2 x. y ) = 1 ) |
42 |
41
|
oveq2d |
|- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( x L ( 2 x. y ) ) = ( x L 1 ) ) |
43 |
41
|
oveq1d |
|- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( ( 2 x. y ) - 1 ) = ( 1 - 1 ) ) |
44 |
|
1m1e0 |
|- ( 1 - 1 ) = 0 |
45 |
43 44
|
eqtrdi |
|- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( ( 2 x. y ) - 1 ) = 0 ) |
46 |
45
|
oveq2d |
|- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( x M ( ( 2 x. y ) - 1 ) ) = ( x M 0 ) ) |
47 |
35 42 46
|
3eqtr4d |
|- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( x L ( 2 x. y ) ) = ( x M ( ( 2 x. y ) - 1 ) ) ) |
48 |
|
retopon |
|- ( topGen ` ran (,) ) e. ( TopOn ` RR ) |
49 |
|
0re |
|- 0 e. RR |
50 |
|
iccssre |
|- ( ( 0 e. RR /\ ( 1 / 2 ) e. RR ) -> ( 0 [,] ( 1 / 2 ) ) C_ RR ) |
51 |
49 16 50
|
mp2an |
|- ( 0 [,] ( 1 / 2 ) ) C_ RR |
52 |
|
resttopon |
|- ( ( ( topGen ` ran (,) ) e. ( TopOn ` RR ) /\ ( 0 [,] ( 1 / 2 ) ) C_ RR ) -> ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) e. ( TopOn ` ( 0 [,] ( 1 / 2 ) ) ) ) |
53 |
48 51 52
|
mp2an |
|- ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) e. ( TopOn ` ( 0 [,] ( 1 / 2 ) ) ) |
54 |
53
|
a1i |
|- ( ph -> ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) e. ( TopOn ` ( 0 [,] ( 1 / 2 ) ) ) ) |
55 |
54 2
|
cnmpt2nd |
|- ( ph -> ( y e. ( 0 [,] ( 1 / 2 ) ) , x e. X |-> x ) e. ( ( ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) tX J ) Cn J ) ) |
56 |
54 2
|
cnmpt1st |
|- ( ph -> ( y e. ( 0 [,] ( 1 / 2 ) ) , x e. X |-> y ) e. ( ( ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) tX J ) Cn ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) ) ) |
57 |
11
|
iihalf1cn |
|- ( z e. ( 0 [,] ( 1 / 2 ) ) |-> ( 2 x. z ) ) e. ( ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) Cn II ) |
58 |
57
|
a1i |
|- ( ph -> ( z e. ( 0 [,] ( 1 / 2 ) ) |-> ( 2 x. z ) ) e. ( ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) Cn II ) ) |
59 |
|
oveq2 |
|- ( z = y -> ( 2 x. z ) = ( 2 x. y ) ) |
60 |
54 2 56 54 58 59
|
cnmpt21 |
|- ( ph -> ( y e. ( 0 [,] ( 1 / 2 ) ) , x e. X |-> ( 2 x. y ) ) e. ( ( ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) tX J ) Cn II ) ) |
61 |
2 3 4
|
htpycn |
|- ( ph -> ( F ( J Htpy K ) G ) C_ ( ( J tX II ) Cn K ) ) |
62 |
61 6
|
sseldd |
|- ( ph -> L e. ( ( J tX II ) Cn K ) ) |
63 |
54 2 55 60 62
|
cnmpt22f |
|- ( ph -> ( y e. ( 0 [,] ( 1 / 2 ) ) , x e. X |-> ( x L ( 2 x. y ) ) ) e. ( ( ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) tX J ) Cn K ) ) |
64 |
|
iccssre |
|- ( ( ( 1 / 2 ) e. RR /\ 1 e. RR ) -> ( ( 1 / 2 ) [,] 1 ) C_ RR ) |
65 |
16 18 64
|
mp2an |
|- ( ( 1 / 2 ) [,] 1 ) C_ RR |
66 |
|
resttopon |
|- ( ( ( topGen ` ran (,) ) e. ( TopOn ` RR ) /\ ( ( 1 / 2 ) [,] 1 ) C_ RR ) -> ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) e. ( TopOn ` ( ( 1 / 2 ) [,] 1 ) ) ) |
67 |
48 65 66
|
mp2an |
|- ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) e. ( TopOn ` ( ( 1 / 2 ) [,] 1 ) ) |
68 |
67
|
a1i |
|- ( ph -> ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) e. ( TopOn ` ( ( 1 / 2 ) [,] 1 ) ) ) |
69 |
68 2
|
cnmpt2nd |
|- ( ph -> ( y e. ( ( 1 / 2 ) [,] 1 ) , x e. X |-> x ) e. ( ( ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) tX J ) Cn J ) ) |
70 |
68 2
|
cnmpt1st |
|- ( ph -> ( y e. ( ( 1 / 2 ) [,] 1 ) , x e. X |-> y ) e. ( ( ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) tX J ) Cn ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) ) ) |
71 |
12
|
iihalf2cn |
|- ( z e. ( ( 1 / 2 ) [,] 1 ) |-> ( ( 2 x. z ) - 1 ) ) e. ( ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) Cn II ) |
72 |
71
|
a1i |
|- ( ph -> ( z e. ( ( 1 / 2 ) [,] 1 ) |-> ( ( 2 x. z ) - 1 ) ) e. ( ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) Cn II ) ) |
73 |
59
|
oveq1d |
|- ( z = y -> ( ( 2 x. z ) - 1 ) = ( ( 2 x. y ) - 1 ) ) |
74 |
68 2 70 68 72 73
|
cnmpt21 |
|- ( ph -> ( y e. ( ( 1 / 2 ) [,] 1 ) , x e. X |-> ( ( 2 x. y ) - 1 ) ) e. ( ( ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) tX J ) Cn II ) ) |
75 |
2 4 5
|
htpycn |
|- ( ph -> ( G ( J Htpy K ) H ) C_ ( ( J tX II ) Cn K ) ) |
76 |
75 7
|
sseldd |
|- ( ph -> M e. ( ( J tX II ) Cn K ) ) |
77 |
68 2 69 74 76
|
cnmpt22f |
|- ( ph -> ( y e. ( ( 1 / 2 ) [,] 1 ) , x e. X |-> ( x M ( ( 2 x. y ) - 1 ) ) ) e. ( ( ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) tX J ) Cn K ) ) |
78 |
10 11 12 13 14 15 23 2 47 63 77
|
cnmpopc |
|- ( ph -> ( y e. ( 0 [,] 1 ) , x e. X |-> if ( y <_ ( 1 / 2 ) , ( x L ( 2 x. y ) ) , ( x M ( ( 2 x. y ) - 1 ) ) ) ) e. ( ( II tX J ) Cn K ) ) |
79 |
9 2 78
|
cnmptcom |
|- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> if ( y <_ ( 1 / 2 ) , ( x L ( 2 x. y ) ) , ( x M ( ( 2 x. y ) - 1 ) ) ) ) e. ( ( J tX II ) Cn K ) ) |
80 |
1 79
|
eqeltrid |
|- ( ph -> N e. ( ( J tX II ) Cn K ) ) |
81 |
|
simpr |
|- ( ( ph /\ s e. X ) -> s e. X ) |
82 |
|
0elunit |
|- 0 e. ( 0 [,] 1 ) |
83 |
|
simpr |
|- ( ( x = s /\ y = 0 ) -> y = 0 ) |
84 |
83 17
|
eqbrtrdi |
|- ( ( x = s /\ y = 0 ) -> y <_ ( 1 / 2 ) ) |
85 |
84
|
iftrued |
|- ( ( x = s /\ y = 0 ) -> if ( y <_ ( 1 / 2 ) , ( x L ( 2 x. y ) ) , ( x M ( ( 2 x. y ) - 1 ) ) ) = ( x L ( 2 x. y ) ) ) |
86 |
|
simpl |
|- ( ( x = s /\ y = 0 ) -> x = s ) |
87 |
83
|
oveq2d |
|- ( ( x = s /\ y = 0 ) -> ( 2 x. y ) = ( 2 x. 0 ) ) |
88 |
|
2t0e0 |
|- ( 2 x. 0 ) = 0 |
89 |
87 88
|
eqtrdi |
|- ( ( x = s /\ y = 0 ) -> ( 2 x. y ) = 0 ) |
90 |
86 89
|
oveq12d |
|- ( ( x = s /\ y = 0 ) -> ( x L ( 2 x. y ) ) = ( s L 0 ) ) |
91 |
85 90
|
eqtrd |
|- ( ( x = s /\ y = 0 ) -> if ( y <_ ( 1 / 2 ) , ( x L ( 2 x. y ) ) , ( x M ( ( 2 x. y ) - 1 ) ) ) = ( s L 0 ) ) |
92 |
|
ovex |
|- ( s L 0 ) e. _V |
93 |
91 1 92
|
ovmpoa |
|- ( ( s e. X /\ 0 e. ( 0 [,] 1 ) ) -> ( s N 0 ) = ( s L 0 ) ) |
94 |
81 82 93
|
sylancl |
|- ( ( ph /\ s e. X ) -> ( s N 0 ) = ( s L 0 ) ) |
95 |
24
|
simpld |
|- ( ( ph /\ s e. X ) -> ( s L 0 ) = ( F ` s ) ) |
96 |
94 95
|
eqtrd |
|- ( ( ph /\ s e. X ) -> ( s N 0 ) = ( F ` s ) ) |
97 |
|
1elunit |
|- 1 e. ( 0 [,] 1 ) |
98 |
16 18
|
ltnlei |
|- ( ( 1 / 2 ) < 1 <-> -. 1 <_ ( 1 / 2 ) ) |
99 |
19 98
|
mpbi |
|- -. 1 <_ ( 1 / 2 ) |
100 |
|
simpr |
|- ( ( x = s /\ y = 1 ) -> y = 1 ) |
101 |
100
|
breq1d |
|- ( ( x = s /\ y = 1 ) -> ( y <_ ( 1 / 2 ) <-> 1 <_ ( 1 / 2 ) ) ) |
102 |
99 101
|
mtbiri |
|- ( ( x = s /\ y = 1 ) -> -. y <_ ( 1 / 2 ) ) |
103 |
102
|
iffalsed |
|- ( ( x = s /\ y = 1 ) -> if ( y <_ ( 1 / 2 ) , ( x L ( 2 x. y ) ) , ( x M ( ( 2 x. y ) - 1 ) ) ) = ( x M ( ( 2 x. y ) - 1 ) ) ) |
104 |
|
simpl |
|- ( ( x = s /\ y = 1 ) -> x = s ) |
105 |
100
|
oveq2d |
|- ( ( x = s /\ y = 1 ) -> ( 2 x. y ) = ( 2 x. 1 ) ) |
106 |
|
2t1e2 |
|- ( 2 x. 1 ) = 2 |
107 |
105 106
|
eqtrdi |
|- ( ( x = s /\ y = 1 ) -> ( 2 x. y ) = 2 ) |
108 |
107
|
oveq1d |
|- ( ( x = s /\ y = 1 ) -> ( ( 2 x. y ) - 1 ) = ( 2 - 1 ) ) |
109 |
|
2m1e1 |
|- ( 2 - 1 ) = 1 |
110 |
108 109
|
eqtrdi |
|- ( ( x = s /\ y = 1 ) -> ( ( 2 x. y ) - 1 ) = 1 ) |
111 |
104 110
|
oveq12d |
|- ( ( x = s /\ y = 1 ) -> ( x M ( ( 2 x. y ) - 1 ) ) = ( s M 1 ) ) |
112 |
103 111
|
eqtrd |
|- ( ( x = s /\ y = 1 ) -> if ( y <_ ( 1 / 2 ) , ( x L ( 2 x. y ) ) , ( x M ( ( 2 x. y ) - 1 ) ) ) = ( s M 1 ) ) |
113 |
|
ovex |
|- ( s M 1 ) e. _V |
114 |
112 1 113
|
ovmpoa |
|- ( ( s e. X /\ 1 e. ( 0 [,] 1 ) ) -> ( s N 1 ) = ( s M 1 ) ) |
115 |
81 97 114
|
sylancl |
|- ( ( ph /\ s e. X ) -> ( s N 1 ) = ( s M 1 ) ) |
116 |
26
|
simprd |
|- ( ( ph /\ s e. X ) -> ( s M 1 ) = ( H ` s ) ) |
117 |
115 116
|
eqtrd |
|- ( ( ph /\ s e. X ) -> ( s N 1 ) = ( H ` s ) ) |
118 |
2 3 5 80 96 117
|
ishtpyd |
|- ( ph -> N e. ( F ( J Htpy K ) H ) ) |