Step |
Hyp |
Ref |
Expression |
1 |
|
pcohtpy.4 |
|- ( ph -> ( F ` 1 ) = ( G ` 0 ) ) |
2 |
|
pcohtpy.5 |
|- ( ph -> F ( ~=ph ` J ) H ) |
3 |
|
pcohtpy.6 |
|- ( ph -> G ( ~=ph ` J ) K ) |
4 |
|
pcohtpylem.7 |
|- P = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( ( 2 x. x ) M y ) , ( ( ( 2 x. x ) - 1 ) N y ) ) ) |
5 |
|
pcohtpylem.8 |
|- ( ph -> M e. ( F ( PHtpy ` J ) H ) ) |
6 |
|
pcohtpylem.9 |
|- ( ph -> N e. ( G ( PHtpy ` J ) K ) ) |
7 |
|
isphtpc |
|- ( F ( ~=ph ` J ) H <-> ( F e. ( II Cn J ) /\ H e. ( II Cn J ) /\ ( F ( PHtpy ` J ) H ) =/= (/) ) ) |
8 |
2 7
|
sylib |
|- ( ph -> ( F e. ( II Cn J ) /\ H e. ( II Cn J ) /\ ( F ( PHtpy ` J ) H ) =/= (/) ) ) |
9 |
8
|
simp1d |
|- ( ph -> F e. ( II Cn J ) ) |
10 |
|
isphtpc |
|- ( G ( ~=ph ` J ) K <-> ( G e. ( II Cn J ) /\ K e. ( II Cn J ) /\ ( G ( PHtpy ` J ) K ) =/= (/) ) ) |
11 |
3 10
|
sylib |
|- ( ph -> ( G e. ( II Cn J ) /\ K e. ( II Cn J ) /\ ( G ( PHtpy ` J ) K ) =/= (/) ) ) |
12 |
11
|
simp1d |
|- ( ph -> G e. ( II Cn J ) ) |
13 |
9 12 1
|
pcocn |
|- ( ph -> ( F ( *p ` J ) G ) e. ( II Cn J ) ) |
14 |
8
|
simp2d |
|- ( ph -> H e. ( II Cn J ) ) |
15 |
11
|
simp2d |
|- ( ph -> K e. ( II Cn J ) ) |
16 |
9 14 5
|
phtpy01 |
|- ( ph -> ( ( F ` 0 ) = ( H ` 0 ) /\ ( F ` 1 ) = ( H ` 1 ) ) ) |
17 |
16
|
simprd |
|- ( ph -> ( F ` 1 ) = ( H ` 1 ) ) |
18 |
12 15 6
|
phtpy01 |
|- ( ph -> ( ( G ` 0 ) = ( K ` 0 ) /\ ( G ` 1 ) = ( K ` 1 ) ) ) |
19 |
18
|
simpld |
|- ( ph -> ( G ` 0 ) = ( K ` 0 ) ) |
20 |
1 17 19
|
3eqtr3d |
|- ( ph -> ( H ` 1 ) = ( K ` 0 ) ) |
21 |
14 15 20
|
pcocn |
|- ( ph -> ( H ( *p ` J ) K ) e. ( II Cn J ) ) |
22 |
|
eqid |
|- ( topGen ` ran (,) ) = ( topGen ` ran (,) ) |
23 |
|
eqid |
|- ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) |
24 |
|
eqid |
|- ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) |
25 |
|
dfii2 |
|- II = ( ( topGen ` ran (,) ) |`t ( 0 [,] 1 ) ) |
26 |
|
0red |
|- ( ph -> 0 e. RR ) |
27 |
|
1red |
|- ( ph -> 1 e. RR ) |
28 |
|
halfre |
|- ( 1 / 2 ) e. RR |
29 |
|
halfge0 |
|- 0 <_ ( 1 / 2 ) |
30 |
|
1re |
|- 1 e. RR |
31 |
|
halflt1 |
|- ( 1 / 2 ) < 1 |
32 |
28 30 31
|
ltleii |
|- ( 1 / 2 ) <_ 1 |
33 |
|
elicc01 |
|- ( ( 1 / 2 ) e. ( 0 [,] 1 ) <-> ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ 1 ) ) |
34 |
28 29 32 33
|
mpbir3an |
|- ( 1 / 2 ) e. ( 0 [,] 1 ) |
35 |
34
|
a1i |
|- ( ph -> ( 1 / 2 ) e. ( 0 [,] 1 ) ) |
36 |
|
iitopon |
|- II e. ( TopOn ` ( 0 [,] 1 ) ) |
37 |
36
|
a1i |
|- ( ph -> II e. ( TopOn ` ( 0 [,] 1 ) ) ) |
38 |
1
|
adantr |
|- ( ( ph /\ ( x = ( 1 / 2 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( F ` 1 ) = ( G ` 0 ) ) |
39 |
9 14 5
|
phtpyi |
|- ( ( ph /\ y e. ( 0 [,] 1 ) ) -> ( ( 0 M y ) = ( F ` 0 ) /\ ( 1 M y ) = ( F ` 1 ) ) ) |
40 |
39
|
simprd |
|- ( ( ph /\ y e. ( 0 [,] 1 ) ) -> ( 1 M y ) = ( F ` 1 ) ) |
41 |
40
|
adantrl |
|- ( ( ph /\ ( x = ( 1 / 2 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( 1 M y ) = ( F ` 1 ) ) |
42 |
12 15 6
|
phtpyi |
|- ( ( ph /\ y e. ( 0 [,] 1 ) ) -> ( ( 0 N y ) = ( G ` 0 ) /\ ( 1 N y ) = ( G ` 1 ) ) ) |
43 |
42
|
simpld |
|- ( ( ph /\ y e. ( 0 [,] 1 ) ) -> ( 0 N y ) = ( G ` 0 ) ) |
44 |
43
|
adantrl |
|- ( ( ph /\ ( x = ( 1 / 2 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( 0 N y ) = ( G ` 0 ) ) |
45 |
38 41 44
|
3eqtr4d |
|- ( ( ph /\ ( x = ( 1 / 2 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( 1 M y ) = ( 0 N y ) ) |
46 |
|
simprl |
|- ( ( ph /\ ( x = ( 1 / 2 ) /\ y e. ( 0 [,] 1 ) ) ) -> x = ( 1 / 2 ) ) |
47 |
46
|
oveq2d |
|- ( ( ph /\ ( x = ( 1 / 2 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( 2 x. x ) = ( 2 x. ( 1 / 2 ) ) ) |
48 |
|
2cn |
|- 2 e. CC |
49 |
|
2ne0 |
|- 2 =/= 0 |
50 |
48 49
|
recidi |
|- ( 2 x. ( 1 / 2 ) ) = 1 |
51 |
47 50
|
eqtrdi |
|- ( ( ph /\ ( x = ( 1 / 2 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( 2 x. x ) = 1 ) |
52 |
51
|
oveq1d |
|- ( ( ph /\ ( x = ( 1 / 2 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( ( 2 x. x ) M y ) = ( 1 M y ) ) |
53 |
51
|
oveq1d |
|- ( ( ph /\ ( x = ( 1 / 2 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( ( 2 x. x ) - 1 ) = ( 1 - 1 ) ) |
54 |
|
1m1e0 |
|- ( 1 - 1 ) = 0 |
55 |
53 54
|
eqtrdi |
|- ( ( ph /\ ( x = ( 1 / 2 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( ( 2 x. x ) - 1 ) = 0 ) |
56 |
55
|
oveq1d |
|- ( ( ph /\ ( x = ( 1 / 2 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( ( ( 2 x. x ) - 1 ) N y ) = ( 0 N y ) ) |
57 |
45 52 56
|
3eqtr4d |
|- ( ( ph /\ ( x = ( 1 / 2 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( ( 2 x. x ) M y ) = ( ( ( 2 x. x ) - 1 ) N y ) ) |
58 |
|
retopon |
|- ( topGen ` ran (,) ) e. ( TopOn ` RR ) |
59 |
|
0re |
|- 0 e. RR |
60 |
|
iccssre |
|- ( ( 0 e. RR /\ ( 1 / 2 ) e. RR ) -> ( 0 [,] ( 1 / 2 ) ) C_ RR ) |
61 |
59 28 60
|
mp2an |
|- ( 0 [,] ( 1 / 2 ) ) C_ RR |
62 |
|
resttopon |
|- ( ( ( topGen ` ran (,) ) e. ( TopOn ` RR ) /\ ( 0 [,] ( 1 / 2 ) ) C_ RR ) -> ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) e. ( TopOn ` ( 0 [,] ( 1 / 2 ) ) ) ) |
63 |
58 61 62
|
mp2an |
|- ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) e. ( TopOn ` ( 0 [,] ( 1 / 2 ) ) ) |
64 |
63
|
a1i |
|- ( ph -> ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) e. ( TopOn ` ( 0 [,] ( 1 / 2 ) ) ) ) |
65 |
64 37
|
cnmpt1st |
|- ( ph -> ( x e. ( 0 [,] ( 1 / 2 ) ) , y e. ( 0 [,] 1 ) |-> x ) e. ( ( ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) tX II ) Cn ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) ) ) |
66 |
23
|
iihalf1cn |
|- ( z e. ( 0 [,] ( 1 / 2 ) ) |-> ( 2 x. z ) ) e. ( ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) Cn II ) |
67 |
66
|
a1i |
|- ( ph -> ( z e. ( 0 [,] ( 1 / 2 ) ) |-> ( 2 x. z ) ) e. ( ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) Cn II ) ) |
68 |
|
oveq2 |
|- ( z = x -> ( 2 x. z ) = ( 2 x. x ) ) |
69 |
64 37 65 64 67 68
|
cnmpt21 |
|- ( ph -> ( x e. ( 0 [,] ( 1 / 2 ) ) , y e. ( 0 [,] 1 ) |-> ( 2 x. x ) ) e. ( ( ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) tX II ) Cn II ) ) |
70 |
64 37
|
cnmpt2nd |
|- ( ph -> ( x e. ( 0 [,] ( 1 / 2 ) ) , y e. ( 0 [,] 1 ) |-> y ) e. ( ( ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) tX II ) Cn II ) ) |
71 |
9 14
|
phtpycn |
|- ( ph -> ( F ( PHtpy ` J ) H ) C_ ( ( II tX II ) Cn J ) ) |
72 |
71 5
|
sseldd |
|- ( ph -> M e. ( ( II tX II ) Cn J ) ) |
73 |
64 37 69 70 72
|
cnmpt22f |
|- ( ph -> ( x e. ( 0 [,] ( 1 / 2 ) ) , y e. ( 0 [,] 1 ) |-> ( ( 2 x. x ) M y ) ) e. ( ( ( ( topGen ` ran (,) ) |`t ( 0 [,] ( 1 / 2 ) ) ) tX II ) Cn J ) ) |
74 |
|
iccssre |
|- ( ( ( 1 / 2 ) e. RR /\ 1 e. RR ) -> ( ( 1 / 2 ) [,] 1 ) C_ RR ) |
75 |
28 30 74
|
mp2an |
|- ( ( 1 / 2 ) [,] 1 ) C_ RR |
76 |
|
resttopon |
|- ( ( ( topGen ` ran (,) ) e. ( TopOn ` RR ) /\ ( ( 1 / 2 ) [,] 1 ) C_ RR ) -> ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) e. ( TopOn ` ( ( 1 / 2 ) [,] 1 ) ) ) |
77 |
58 75 76
|
mp2an |
|- ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) e. ( TopOn ` ( ( 1 / 2 ) [,] 1 ) ) |
78 |
77
|
a1i |
|- ( ph -> ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) e. ( TopOn ` ( ( 1 / 2 ) [,] 1 ) ) ) |
79 |
78 37
|
cnmpt1st |
|- ( ph -> ( x e. ( ( 1 / 2 ) [,] 1 ) , y e. ( 0 [,] 1 ) |-> x ) e. ( ( ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) tX II ) Cn ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) ) ) |
80 |
24
|
iihalf2cn |
|- ( z e. ( ( 1 / 2 ) [,] 1 ) |-> ( ( 2 x. z ) - 1 ) ) e. ( ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) Cn II ) |
81 |
80
|
a1i |
|- ( ph -> ( z e. ( ( 1 / 2 ) [,] 1 ) |-> ( ( 2 x. z ) - 1 ) ) e. ( ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) Cn II ) ) |
82 |
68
|
oveq1d |
|- ( z = x -> ( ( 2 x. z ) - 1 ) = ( ( 2 x. x ) - 1 ) ) |
83 |
78 37 79 78 81 82
|
cnmpt21 |
|- ( ph -> ( x e. ( ( 1 / 2 ) [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( 2 x. x ) - 1 ) ) e. ( ( ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) tX II ) Cn II ) ) |
84 |
78 37
|
cnmpt2nd |
|- ( ph -> ( x e. ( ( 1 / 2 ) [,] 1 ) , y e. ( 0 [,] 1 ) |-> y ) e. ( ( ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) tX II ) Cn II ) ) |
85 |
12 15
|
phtpycn |
|- ( ph -> ( G ( PHtpy ` J ) K ) C_ ( ( II tX II ) Cn J ) ) |
86 |
85 6
|
sseldd |
|- ( ph -> N e. ( ( II tX II ) Cn J ) ) |
87 |
78 37 83 84 86
|
cnmpt22f |
|- ( ph -> ( x e. ( ( 1 / 2 ) [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 2 x. x ) - 1 ) N y ) ) e. ( ( ( ( topGen ` ran (,) ) |`t ( ( 1 / 2 ) [,] 1 ) ) tX II ) Cn J ) ) |
88 |
22 23 24 25 26 27 35 37 57 73 87
|
cnmpopc |
|- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( ( 2 x. x ) M y ) , ( ( ( 2 x. x ) - 1 ) N y ) ) ) e. ( ( II tX II ) Cn J ) ) |
89 |
4 88
|
eqeltrid |
|- ( ph -> P e. ( ( II tX II ) Cn J ) ) |
90 |
|
simpll |
|- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ s <_ ( 1 / 2 ) ) -> ph ) |
91 |
|
elii1 |
|- ( s e. ( 0 [,] ( 1 / 2 ) ) <-> ( s e. ( 0 [,] 1 ) /\ s <_ ( 1 / 2 ) ) ) |
92 |
|
iihalf1 |
|- ( s e. ( 0 [,] ( 1 / 2 ) ) -> ( 2 x. s ) e. ( 0 [,] 1 ) ) |
93 |
91 92
|
sylbir |
|- ( ( s e. ( 0 [,] 1 ) /\ s <_ ( 1 / 2 ) ) -> ( 2 x. s ) e. ( 0 [,] 1 ) ) |
94 |
93
|
adantll |
|- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ s <_ ( 1 / 2 ) ) -> ( 2 x. s ) e. ( 0 [,] 1 ) ) |
95 |
9 14
|
phtpyhtpy |
|- ( ph -> ( F ( PHtpy ` J ) H ) C_ ( F ( II Htpy J ) H ) ) |
96 |
95 5
|
sseldd |
|- ( ph -> M e. ( F ( II Htpy J ) H ) ) |
97 |
37 9 14 96
|
htpyi |
|- ( ( ph /\ ( 2 x. s ) e. ( 0 [,] 1 ) ) -> ( ( ( 2 x. s ) M 0 ) = ( F ` ( 2 x. s ) ) /\ ( ( 2 x. s ) M 1 ) = ( H ` ( 2 x. s ) ) ) ) |
98 |
90 94 97
|
syl2anc |
|- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ s <_ ( 1 / 2 ) ) -> ( ( ( 2 x. s ) M 0 ) = ( F ` ( 2 x. s ) ) /\ ( ( 2 x. s ) M 1 ) = ( H ` ( 2 x. s ) ) ) ) |
99 |
98
|
simpld |
|- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ s <_ ( 1 / 2 ) ) -> ( ( 2 x. s ) M 0 ) = ( F ` ( 2 x. s ) ) ) |
100 |
|
simpll |
|- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> ph ) |
101 |
|
elii2 |
|- ( ( s e. ( 0 [,] 1 ) /\ -. s <_ ( 1 / 2 ) ) -> s e. ( ( 1 / 2 ) [,] 1 ) ) |
102 |
101
|
adantll |
|- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> s e. ( ( 1 / 2 ) [,] 1 ) ) |
103 |
|
iihalf2 |
|- ( s e. ( ( 1 / 2 ) [,] 1 ) -> ( ( 2 x. s ) - 1 ) e. ( 0 [,] 1 ) ) |
104 |
102 103
|
syl |
|- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> ( ( 2 x. s ) - 1 ) e. ( 0 [,] 1 ) ) |
105 |
12 15
|
phtpyhtpy |
|- ( ph -> ( G ( PHtpy ` J ) K ) C_ ( G ( II Htpy J ) K ) ) |
106 |
105 6
|
sseldd |
|- ( ph -> N e. ( G ( II Htpy J ) K ) ) |
107 |
37 12 15 106
|
htpyi |
|- ( ( ph /\ ( ( 2 x. s ) - 1 ) e. ( 0 [,] 1 ) ) -> ( ( ( ( 2 x. s ) - 1 ) N 0 ) = ( G ` ( ( 2 x. s ) - 1 ) ) /\ ( ( ( 2 x. s ) - 1 ) N 1 ) = ( K ` ( ( 2 x. s ) - 1 ) ) ) ) |
108 |
100 104 107
|
syl2anc |
|- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> ( ( ( ( 2 x. s ) - 1 ) N 0 ) = ( G ` ( ( 2 x. s ) - 1 ) ) /\ ( ( ( 2 x. s ) - 1 ) N 1 ) = ( K ` ( ( 2 x. s ) - 1 ) ) ) ) |
109 |
108
|
simpld |
|- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> ( ( ( 2 x. s ) - 1 ) N 0 ) = ( G ` ( ( 2 x. s ) - 1 ) ) ) |
110 |
99 109
|
ifeq12da |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> if ( s <_ ( 1 / 2 ) , ( ( 2 x. s ) M 0 ) , ( ( ( 2 x. s ) - 1 ) N 0 ) ) = if ( s <_ ( 1 / 2 ) , ( F ` ( 2 x. s ) ) , ( G ` ( ( 2 x. s ) - 1 ) ) ) ) |
111 |
|
simpr |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> s e. ( 0 [,] 1 ) ) |
112 |
|
0elunit |
|- 0 e. ( 0 [,] 1 ) |
113 |
|
simpl |
|- ( ( x = s /\ y = 0 ) -> x = s ) |
114 |
113
|
breq1d |
|- ( ( x = s /\ y = 0 ) -> ( x <_ ( 1 / 2 ) <-> s <_ ( 1 / 2 ) ) ) |
115 |
113
|
oveq2d |
|- ( ( x = s /\ y = 0 ) -> ( 2 x. x ) = ( 2 x. s ) ) |
116 |
|
simpr |
|- ( ( x = s /\ y = 0 ) -> y = 0 ) |
117 |
115 116
|
oveq12d |
|- ( ( x = s /\ y = 0 ) -> ( ( 2 x. x ) M y ) = ( ( 2 x. s ) M 0 ) ) |
118 |
115
|
oveq1d |
|- ( ( x = s /\ y = 0 ) -> ( ( 2 x. x ) - 1 ) = ( ( 2 x. s ) - 1 ) ) |
119 |
118 116
|
oveq12d |
|- ( ( x = s /\ y = 0 ) -> ( ( ( 2 x. x ) - 1 ) N y ) = ( ( ( 2 x. s ) - 1 ) N 0 ) ) |
120 |
114 117 119
|
ifbieq12d |
|- ( ( x = s /\ y = 0 ) -> if ( x <_ ( 1 / 2 ) , ( ( 2 x. x ) M y ) , ( ( ( 2 x. x ) - 1 ) N y ) ) = if ( s <_ ( 1 / 2 ) , ( ( 2 x. s ) M 0 ) , ( ( ( 2 x. s ) - 1 ) N 0 ) ) ) |
121 |
|
ovex |
|- ( ( 2 x. s ) M 0 ) e. _V |
122 |
|
ovex |
|- ( ( ( 2 x. s ) - 1 ) N 0 ) e. _V |
123 |
121 122
|
ifex |
|- if ( s <_ ( 1 / 2 ) , ( ( 2 x. s ) M 0 ) , ( ( ( 2 x. s ) - 1 ) N 0 ) ) e. _V |
124 |
120 4 123
|
ovmpoa |
|- ( ( s e. ( 0 [,] 1 ) /\ 0 e. ( 0 [,] 1 ) ) -> ( s P 0 ) = if ( s <_ ( 1 / 2 ) , ( ( 2 x. s ) M 0 ) , ( ( ( 2 x. s ) - 1 ) N 0 ) ) ) |
125 |
111 112 124
|
sylancl |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s P 0 ) = if ( s <_ ( 1 / 2 ) , ( ( 2 x. s ) M 0 ) , ( ( ( 2 x. s ) - 1 ) N 0 ) ) ) |
126 |
9 12
|
pcovalg |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F ( *p ` J ) G ) ` s ) = if ( s <_ ( 1 / 2 ) , ( F ` ( 2 x. s ) ) , ( G ` ( ( 2 x. s ) - 1 ) ) ) ) |
127 |
110 125 126
|
3eqtr4d |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s P 0 ) = ( ( F ( *p ` J ) G ) ` s ) ) |
128 |
98
|
simprd |
|- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ s <_ ( 1 / 2 ) ) -> ( ( 2 x. s ) M 1 ) = ( H ` ( 2 x. s ) ) ) |
129 |
108
|
simprd |
|- ( ( ( ph /\ s e. ( 0 [,] 1 ) ) /\ -. s <_ ( 1 / 2 ) ) -> ( ( ( 2 x. s ) - 1 ) N 1 ) = ( K ` ( ( 2 x. s ) - 1 ) ) ) |
130 |
128 129
|
ifeq12da |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> if ( s <_ ( 1 / 2 ) , ( ( 2 x. s ) M 1 ) , ( ( ( 2 x. s ) - 1 ) N 1 ) ) = if ( s <_ ( 1 / 2 ) , ( H ` ( 2 x. s ) ) , ( K ` ( ( 2 x. s ) - 1 ) ) ) ) |
131 |
|
1elunit |
|- 1 e. ( 0 [,] 1 ) |
132 |
|
simpl |
|- ( ( x = s /\ y = 1 ) -> x = s ) |
133 |
132
|
breq1d |
|- ( ( x = s /\ y = 1 ) -> ( x <_ ( 1 / 2 ) <-> s <_ ( 1 / 2 ) ) ) |
134 |
132
|
oveq2d |
|- ( ( x = s /\ y = 1 ) -> ( 2 x. x ) = ( 2 x. s ) ) |
135 |
|
simpr |
|- ( ( x = s /\ y = 1 ) -> y = 1 ) |
136 |
134 135
|
oveq12d |
|- ( ( x = s /\ y = 1 ) -> ( ( 2 x. x ) M y ) = ( ( 2 x. s ) M 1 ) ) |
137 |
134
|
oveq1d |
|- ( ( x = s /\ y = 1 ) -> ( ( 2 x. x ) - 1 ) = ( ( 2 x. s ) - 1 ) ) |
138 |
137 135
|
oveq12d |
|- ( ( x = s /\ y = 1 ) -> ( ( ( 2 x. x ) - 1 ) N y ) = ( ( ( 2 x. s ) - 1 ) N 1 ) ) |
139 |
133 136 138
|
ifbieq12d |
|- ( ( x = s /\ y = 1 ) -> if ( x <_ ( 1 / 2 ) , ( ( 2 x. x ) M y ) , ( ( ( 2 x. x ) - 1 ) N y ) ) = if ( s <_ ( 1 / 2 ) , ( ( 2 x. s ) M 1 ) , ( ( ( 2 x. s ) - 1 ) N 1 ) ) ) |
140 |
|
ovex |
|- ( ( 2 x. s ) M 1 ) e. _V |
141 |
|
ovex |
|- ( ( ( 2 x. s ) - 1 ) N 1 ) e. _V |
142 |
140 141
|
ifex |
|- if ( s <_ ( 1 / 2 ) , ( ( 2 x. s ) M 1 ) , ( ( ( 2 x. s ) - 1 ) N 1 ) ) e. _V |
143 |
139 4 142
|
ovmpoa |
|- ( ( s e. ( 0 [,] 1 ) /\ 1 e. ( 0 [,] 1 ) ) -> ( s P 1 ) = if ( s <_ ( 1 / 2 ) , ( ( 2 x. s ) M 1 ) , ( ( ( 2 x. s ) - 1 ) N 1 ) ) ) |
144 |
111 131 143
|
sylancl |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s P 1 ) = if ( s <_ ( 1 / 2 ) , ( ( 2 x. s ) M 1 ) , ( ( ( 2 x. s ) - 1 ) N 1 ) ) ) |
145 |
14 15
|
pcovalg |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( H ( *p ` J ) K ) ` s ) = if ( s <_ ( 1 / 2 ) , ( H ` ( 2 x. s ) ) , ( K ` ( ( 2 x. s ) - 1 ) ) ) ) |
146 |
130 144 145
|
3eqtr4d |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s P 1 ) = ( ( H ( *p ` J ) K ) ` s ) ) |
147 |
9 14 5
|
phtpyi |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 0 M s ) = ( F ` 0 ) /\ ( 1 M s ) = ( F ` 1 ) ) ) |
148 |
147
|
simpld |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 M s ) = ( F ` 0 ) ) |
149 |
|
simpl |
|- ( ( x = 0 /\ y = s ) -> x = 0 ) |
150 |
149 29
|
eqbrtrdi |
|- ( ( x = 0 /\ y = s ) -> x <_ ( 1 / 2 ) ) |
151 |
150
|
iftrued |
|- ( ( x = 0 /\ y = s ) -> if ( x <_ ( 1 / 2 ) , ( ( 2 x. x ) M y ) , ( ( ( 2 x. x ) - 1 ) N y ) ) = ( ( 2 x. x ) M y ) ) |
152 |
149
|
oveq2d |
|- ( ( x = 0 /\ y = s ) -> ( 2 x. x ) = ( 2 x. 0 ) ) |
153 |
|
2t0e0 |
|- ( 2 x. 0 ) = 0 |
154 |
152 153
|
eqtrdi |
|- ( ( x = 0 /\ y = s ) -> ( 2 x. x ) = 0 ) |
155 |
|
simpr |
|- ( ( x = 0 /\ y = s ) -> y = s ) |
156 |
154 155
|
oveq12d |
|- ( ( x = 0 /\ y = s ) -> ( ( 2 x. x ) M y ) = ( 0 M s ) ) |
157 |
151 156
|
eqtrd |
|- ( ( x = 0 /\ y = s ) -> if ( x <_ ( 1 / 2 ) , ( ( 2 x. x ) M y ) , ( ( ( 2 x. x ) - 1 ) N y ) ) = ( 0 M s ) ) |
158 |
|
ovex |
|- ( 0 M s ) e. _V |
159 |
157 4 158
|
ovmpoa |
|- ( ( 0 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> ( 0 P s ) = ( 0 M s ) ) |
160 |
112 111 159
|
sylancr |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 P s ) = ( 0 M s ) ) |
161 |
9 12
|
pco0 |
|- ( ph -> ( ( F ( *p ` J ) G ) ` 0 ) = ( F ` 0 ) ) |
162 |
161
|
adantr |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F ( *p ` J ) G ) ` 0 ) = ( F ` 0 ) ) |
163 |
148 160 162
|
3eqtr4d |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 P s ) = ( ( F ( *p ` J ) G ) ` 0 ) ) |
164 |
12 15 6
|
phtpyi |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 0 N s ) = ( G ` 0 ) /\ ( 1 N s ) = ( G ` 1 ) ) ) |
165 |
164
|
simprd |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 N s ) = ( G ` 1 ) ) |
166 |
28 30
|
ltnlei |
|- ( ( 1 / 2 ) < 1 <-> -. 1 <_ ( 1 / 2 ) ) |
167 |
31 166
|
mpbi |
|- -. 1 <_ ( 1 / 2 ) |
168 |
|
simpl |
|- ( ( x = 1 /\ y = s ) -> x = 1 ) |
169 |
168
|
breq1d |
|- ( ( x = 1 /\ y = s ) -> ( x <_ ( 1 / 2 ) <-> 1 <_ ( 1 / 2 ) ) ) |
170 |
167 169
|
mtbiri |
|- ( ( x = 1 /\ y = s ) -> -. x <_ ( 1 / 2 ) ) |
171 |
170
|
iffalsed |
|- ( ( x = 1 /\ y = s ) -> if ( x <_ ( 1 / 2 ) , ( ( 2 x. x ) M y ) , ( ( ( 2 x. x ) - 1 ) N y ) ) = ( ( ( 2 x. x ) - 1 ) N y ) ) |
172 |
168
|
oveq2d |
|- ( ( x = 1 /\ y = s ) -> ( 2 x. x ) = ( 2 x. 1 ) ) |
173 |
|
2t1e2 |
|- ( 2 x. 1 ) = 2 |
174 |
172 173
|
eqtrdi |
|- ( ( x = 1 /\ y = s ) -> ( 2 x. x ) = 2 ) |
175 |
174
|
oveq1d |
|- ( ( x = 1 /\ y = s ) -> ( ( 2 x. x ) - 1 ) = ( 2 - 1 ) ) |
176 |
|
2m1e1 |
|- ( 2 - 1 ) = 1 |
177 |
175 176
|
eqtrdi |
|- ( ( x = 1 /\ y = s ) -> ( ( 2 x. x ) - 1 ) = 1 ) |
178 |
|
simpr |
|- ( ( x = 1 /\ y = s ) -> y = s ) |
179 |
177 178
|
oveq12d |
|- ( ( x = 1 /\ y = s ) -> ( ( ( 2 x. x ) - 1 ) N y ) = ( 1 N s ) ) |
180 |
171 179
|
eqtrd |
|- ( ( x = 1 /\ y = s ) -> if ( x <_ ( 1 / 2 ) , ( ( 2 x. x ) M y ) , ( ( ( 2 x. x ) - 1 ) N y ) ) = ( 1 N s ) ) |
181 |
|
ovex |
|- ( 1 N s ) e. _V |
182 |
180 4 181
|
ovmpoa |
|- ( ( 1 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> ( 1 P s ) = ( 1 N s ) ) |
183 |
131 111 182
|
sylancr |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 P s ) = ( 1 N s ) ) |
184 |
9 12
|
pco1 |
|- ( ph -> ( ( F ( *p ` J ) G ) ` 1 ) = ( G ` 1 ) ) |
185 |
184
|
adantr |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F ( *p ` J ) G ) ` 1 ) = ( G ` 1 ) ) |
186 |
165 183 185
|
3eqtr4d |
|- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 P s ) = ( ( F ( *p ` J ) G ) ` 1 ) ) |
187 |
13 21 89 127 146 163 186
|
isphtpy2d |
|- ( ph -> P e. ( ( F ( *p ` J ) G ) ( PHtpy ` J ) ( H ( *p ` J ) K ) ) ) |