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Theorem pcopt2 21251
Description: Concatenation with a point does not affect homotopy class. (Contributed by Mario Carneiro, 12-Feb-2015.)
Hypothesis
Ref Expression
pcopt.1 |- P = ( ( 0 [,] 1 ) X. { Y } )
Assertion
Ref Expression
pcopt2 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( F ( *p ` J ) P ) ( ~=ph ` J ) F )
Proof of Theorem pcopt2
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pcopt.1 . . . . . . . . 9 |- P = ( ( 0 [,] 1 ) X. { Y } )
21fveq1i 5858 . . . . . . . 8 |- ( P ` ( ( 2 x. x ) - 1 ) ) = ( ( ( 0 [,] 1 ) X. { Y } ) ` ( ( 2 x. x ) - 1 ) )
3 simpr 461 . . . . . . . . . 10 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( F ` 1 ) = Y )
4 iiuni 21113 . . . . . . . . . . . . 13 |- ( 0 [,] 1 ) = U. II
5 eqid 2460 . . . . . . . . . . . . 13 |- U. J = U. J
64, 5cnf 19506 . . . . . . . . . . . 12 |- ( F e. ( II Cn J ) -> F : ( 0 [,] 1 ) --> U. J )
76adantr 465 . . . . . . . . . . 11 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> F : ( 0 [,] 1 ) --> U. J )
8 1elunit 11628 . . . . . . . . . . 11 |- 1 e. ( 0 [,] 1 )
9 ffvelrn 6010 . . . . . . . . . . 11 |- ( ( F : ( 0 [,] 1 ) --> U. J /\ 1 e. ( 0 [,] 1 ) ) -> ( F ` 1 ) e. U. J )
107, 8, 9sylancl 662 . . . . . . . . . 10 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( F ` 1 ) e. U. J )
113, 10eqeltrrd 2549 . . . . . . . . 9 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> Y e. U. J )
12 elii2 21164 . . . . . . . . . 10 |- ( ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) -> x e. ( ( 1 / 2 ) [,] 1 ) )
13 iihalf2 21161 . . . . . . . . . 10 |- ( x e. ( ( 1 / 2 ) [,] 1 ) -> ( ( 2 x. x ) - 1 ) e. ( 0 [,] 1 ) )
1412, 13syl 16 . . . . . . . . 9 |- ( ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) -> ( ( 2 x. x ) - 1 ) e. ( 0 [,] 1 ) )
15 fvconst2g 6105 . . . . . . . . 9 |- ( ( Y e. U. J /\ ( ( 2 x. x ) - 1 ) e. ( 0 [,] 1 ) ) -> ( ( ( 0 [,] 1 ) X. { Y } ) ` ( ( 2 x. x ) - 1 ) ) = Y )
1611, 14, 15syl2an 477 . . . . . . . 8 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) ) -> ( ( ( 0 [,] 1 ) X. { Y } ) ` ( ( 2 x. x ) - 1 ) ) = Y )
172, 16syl5eq 2513 . . . . . . 7 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) ) -> ( P ` ( ( 2 x. x ) - 1 ) ) = Y )
18 simplr 754 . . . . . . 7 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) ) -> ( F ` 1 ) = Y )
1917, 18eqtr4d 2504 . . . . . 6 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) ) -> ( P ` ( ( 2 x. x ) - 1 ) ) = ( F ` 1 ) )
2019anassrs 648 . . . . 5 |- ( ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ x e. ( 0 [,] 1 ) ) /\ -. x <_ ( 1 / 2 ) ) -> ( P ` ( ( 2 x. x ) - 1 ) ) = ( F ` 1 ) )
2120ifeq2da 3963 . . . 4 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ x e. ( 0 [,] 1 ) ) -> if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( P ` ( ( 2 x. x ) - 1 ) ) ) = if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( F ` 1 ) ) )
2221mpteq2dva 4526 . . 3 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( P ` ( ( 2 x. x ) - 1 ) ) ) ) = ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( F ` 1 ) ) ) )
23 simpl 457 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> F e. ( II Cn J ) )
24 cntop2 19501 . . . . . . . 8 |- ( F e. ( II Cn J ) -> J e. Top )
2524adantr 465 . . . . . . 7 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> J e. Top )
265toptopon 19194 . . . . . . 7 |- ( J e. Top <-> J e. (TopOn ` U. J ) )
2725, 26sylib 196 . . . . . 6 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> J e. (TopOn ` U. J ) )
281pcoptcl 21249 . . . . . 6 |- ( ( J e. (TopOn ` U. J ) /\ Y e. U. J ) -> ( P e. ( II Cn J ) /\ ( P ` 0 ) = Y /\ ( P ` 1 ) = Y ) )
2927, 11, 28syl2anc 661 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( P e. ( II Cn J ) /\ ( P ` 0 ) = Y /\ ( P ` 1 ) = Y ) )
3029simp1d 1003 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> P e. ( II Cn J ) )
3123, 30pcoval 21239 . . 3 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( F ( *p ` J ) P ) = ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( P ` ( ( 2 x. x ) - 1 ) ) ) ) )
32 iftrue 3938 . . . . . . . . 9 |- ( x <_ ( 1 / 2 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = ( 2 x. x ) )
3332adantl 466 . . . . . . . 8 |- ( ( x e. ( 0 [,] 1 ) /\ x <_ ( 1 / 2 ) ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = ( 2 x. x ) )
34 elii1 21163 . . . . . . . . 9 |- ( x e. ( 0 [,] ( 1 / 2 ) ) <-> ( x e. ( 0 [,] 1 ) /\ x <_ ( 1 / 2 ) ) )
35 iihalf1 21159 . . . . . . . . 9 |- ( x e. ( 0 [,] ( 1 / 2 ) ) -> ( 2 x. x ) e. ( 0 [,] 1 ) )
3634, 35sylbir 213 . . . . . . . 8 |- ( ( x e. ( 0 [,] 1 ) /\ x <_ ( 1 / 2 ) ) -> ( 2 x. x ) e. ( 0 [,] 1 ) )
3733, 36eqeltrd 2548 . . . . . . 7 |- ( ( x e. ( 0 [,] 1 ) /\ x <_ ( 1 / 2 ) ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) e. ( 0 [,] 1 ) )
3837ex 434 . . . . . 6 |- ( x e. ( 0 [,] 1 ) -> ( x <_ ( 1 / 2 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) e. ( 0 [,] 1 ) ) )
39 iffalse 3941 . . . . . . 7 |- ( -. x <_ ( 1 / 2 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = 1 )
4039, 8syl6eqel 2556 . . . . . 6 |- ( -. x <_ ( 1 / 2 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) e. ( 0 [,] 1 ) )
4138, 40pm2.61d1 159 . . . . 5 |- ( x e. ( 0 [,] 1 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) e. ( 0 [,] 1 ) )
4241adantl 466 . . . 4 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ x e. ( 0 [,] 1 ) ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) e. ( 0 [,] 1 ) )
43 eqidd 2461 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) = ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) )
447feqmptd 5911 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> F = ( y e. ( 0 [,] 1 ) |-> ( F ` y ) ) )
45 fveq2 5857 . . . . 5 |- ( y = if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) -> ( F ` y ) = ( F ` if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) )
46 fvif 5868 . . . . 5 |- ( F ` if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) = if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( F ` 1 ) )
4745, 46syl6eq 2517 . . . 4 |- ( y = if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) -> ( F ` y ) = if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( F ` 1 ) ) )
4842, 43, 44, 47fmptco 6045 . . 3 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( F o. ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) ) = ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( F ` 1 ) ) ) )
4922, 31, 483eqtr4d 2511 . 2 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( F ( *p ` J ) P ) = ( F o. ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) ) )
50 iitopon 21111 . . . . 5 |- II e. (TopOn ` ( 0 [,] 1 ) )
5150a1i 11 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> II e. (TopOn ` ( 0 [,] 1 ) ) )
5251cnmptid 19890 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( x e. ( 0 [,] 1 ) |-> x ) e. ( II Cn II ) )
53 0elunit 11627 . . . . . 6 |- 0 e. ( 0 [,] 1 )
5453a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> 0 e. ( 0 [,] 1 ) )
5551, 51, 54cnmptc 19891 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( x e. ( 0 [,] 1 ) |-> 0 ) e. ( II Cn II ) )
56 eqid 2460 . . . . 5 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
57 eqid 2460 . . . . 5 |- ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) )
58 eqid 2460 . . . . 5 |- ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) )
59 dfii2 21114 . . . . 5 |- II = ( ( topGen ` ran (,) ) ↾t ( 0 [,] 1 ) )
60 0re 9585 . . . . . 6 |- 0 e. RR
6160a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> 0 e. RR )
62 1re 9584 . . . . . 6 |- 1 e. RR
6362a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> 1 e. RR )
64 halfre 10743 . . . . . . 7 |- ( 1 / 2 ) e. RR
65 halfgt0 10745 . . . . . . . 8 |- 0 < ( 1 / 2 )
6660, 64, 65ltleii 9696 . . . . . . 7 |- 0 <_ ( 1 / 2 )
67 halflt1 10746 . . . . . . . 8 |- ( 1 / 2 ) < 1
6864, 62, 67ltleii 9696 . . . . . . 7 |- ( 1 / 2 ) <_ 1
6960, 62elicc2i 11579 . . . . . . 7 |- ( ( 1 / 2 ) e. ( 0 [,] 1 ) <-> ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ 1 ) )
7064, 66, 68, 69mpbir3an 1173 . . . . . 6 |- ( 1 / 2 ) e. ( 0 [,] 1 )
7170a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( 1 / 2 ) e. ( 0 [,] 1 ) )
72 simprl 755 . . . . . . 7 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> y = ( 1 / 2 ) )
7372oveq2d 6291 . . . . . 6 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( 2 x. y ) = ( 2 x. ( 1 / 2 ) ) )
74 2cn 10595 . . . . . . 7 |- 2 e. CC
75 2ne0 10617 . . . . . . 7 |- 2 =/= 0
7674, 75recidi 10264 . . . . . 6 |- ( 2 x. ( 1 / 2 ) ) = 1
7773, 76syl6eq 2517 . . . . 5 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( 2 x. y ) = 1 )
78 retopon 20998 . . . . . . . 8 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
79 iccssre 11595 . . . . . . . . 9 |- ( ( 0 e. RR /\ ( 1 / 2 ) e. RR ) -> ( 0 [,] ( 1 / 2 ) ) C_ RR )
8060, 64, 79mp2an 672 . . . . . . . 8 |- ( 0 [,] ( 1 / 2 ) ) C_ RR
81 resttopon 19421 . . . . . . . 8 |- ( ( ( topGen ` ran (,) ) e. (TopOn ` RR ) /\ ( 0 [,] ( 1 / 2 ) ) C_ RR ) -> ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) e. (TopOn ` ( 0 [,] ( 1 / 2 ) ) ) )
8278, 80, 81mp2an 672 . . . . . . 7 |- ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) e. (TopOn ` ( 0 [,] ( 1 / 2 ) ) )
8382a1i 11 . . . . . 6 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) e. (TopOn ` ( 0 [,] ( 1 / 2 ) ) ) )
8483, 51cnmpt1st 19897 . . . . . 6 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( y e. ( 0 [,] ( 1 / 2 ) ) , z e. ( 0 [,] 1 ) |-> y ) e. ( ( ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) tX II ) Cn ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ) )
8557iihalf1cn 21160 . . . . . . 7 |- ( x e. ( 0 [,] ( 1 / 2 ) ) |-> ( 2 x. x ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) Cn II )
8685a1i 11 . . . . . 6 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( x e. ( 0 [,] ( 1 / 2 ) ) |-> ( 2 x. x ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) Cn II ) )
87 oveq2 6283 . . . . . 6 |- ( x = y -> ( 2 x. x ) = ( 2 x. y ) )
8883, 51, 84, 83, 86, 87cnmpt21 19900 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( y e. ( 0 [,] ( 1 / 2 ) ) , z e. ( 0 [,] 1 ) |-> ( 2 x. y ) ) e. ( ( ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) tX II ) Cn II ) )
89 iccssre 11595 . . . . . . . . 9 |- ( ( ( 1 / 2 ) e. RR /\ 1 e. RR ) -> ( ( 1 / 2 ) [,] 1 ) C_ RR )
9064, 62, 89mp2an 672 . . . . . . . 8 |- ( ( 1 / 2 ) [,] 1 ) C_ RR
91 resttopon 19421 . . . . . . . 8 |- ( ( ( topGen ` ran (,) ) e. (TopOn ` RR ) /\ ( ( 1 / 2 ) [,] 1 ) C_ RR ) -> ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) e. (TopOn ` ( ( 1 / 2 ) [,] 1 ) ) )
9278, 90, 91mp2an 672 . . . . . . 7 |- ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) e. (TopOn ` ( ( 1 / 2 ) [,] 1 ) )
9392a1i 11 . . . . . 6 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) e. (TopOn ` ( ( 1 / 2 ) [,] 1 ) ) )
948a1i 11 . . . . . 6 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> 1 e. ( 0 [,] 1 ) )
9593, 51, 51, 94cnmpt2c 19899 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( y e. ( ( 1 / 2 ) [,] 1 ) , z e. ( 0 [,] 1 ) |-> 1 ) e. ( ( ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) tX II ) Cn II ) )
9656, 57, 58, 59, 61, 63, 71, 51, 77, 88, 95cnmpt2pc 21156 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( y e. ( 0 [,] 1 ) , z e. ( 0 [,] 1 ) |-> if ( y <_ ( 1 / 2 ) , ( 2 x. y ) , 1 ) ) e. ( ( II tX II ) Cn II ) )
97 breq1 4443 . . . . . 6 |- ( y = x -> ( y <_ ( 1 / 2 ) <-> x <_ ( 1 / 2 ) ) )
98 oveq2 6283 . . . . . 6 |- ( y = x -> ( 2 x. y ) = ( 2 x. x ) )
9997, 98ifbieq1d 3955 . . . . 5 |- ( y = x -> if ( y <_ ( 1 / 2 ) , ( 2 x. y ) , 1 ) = if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) )
10099adantr 465 . . . 4 |- ( ( y = x /\ z = 0 ) -> if ( y <_ ( 1 / 2 ) , ( 2 x. y ) , 1 ) = if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) )
10151, 52, 55, 51, 51, 96, 100cnmpt12 19896 . . 3 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) e. ( II Cn II ) )
102 id 22 . . . . . . . 8 |- ( x = 0 -> x = 0 )
103102, 66syl6eqbr 4477 . . . . . . 7 |- ( x = 0 -> x <_ ( 1 / 2 ) )
104103, 32syl 16 . . . . . 6 |- ( x = 0 -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = ( 2 x. x ) )
105 oveq2 6283 . . . . . . 7 |- ( x = 0 -> ( 2 x. x ) = ( 2 x. 0 ) )
106 2t0e0 10680 . . . . . . 7 |- ( 2 x. 0 ) = 0
107105, 106syl6eq 2517 . . . . . 6 |- ( x = 0 -> ( 2 x. x ) = 0 )
108104, 107eqtrd 2501 . . . . 5 |- ( x = 0 -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = 0 )
109 eqid 2460 . . . . 5 |- ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) = ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) )
110 c0ex 9579 . . . . 5 |- 0 e. _V
111108, 109, 110fvmpt 5941 . . . 4 |- ( 0 e. ( 0 [,] 1 ) -> ( ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) ` 0 ) = 0 )
11253, 111mp1i 12 . . 3 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) ` 0 ) = 0 )
11364, 62ltnlei 9694 . . . . . . . 8 |- ( ( 1 / 2 ) < 1 <-> -. 1 <_ ( 1 / 2 ) )
11467, 113mpbi 208 . . . . . . 7 |- -. 1 <_ ( 1 / 2 )
115 breq1 4443 . . . . . . 7 |- ( x = 1 -> ( x <_ ( 1 / 2 ) <-> 1 <_ ( 1 / 2 ) ) )
116114, 115mtbiri 303 . . . . . 6 |- ( x = 1 -> -. x <_ ( 1 / 2 ) )
117116, 39syl 16 . . . . 5 |- ( x = 1 -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = 1 )
118 1ex 9580 . . . . 5 |- 1 e. _V
119117, 109, 118fvmpt 5941 . . . 4 |- ( 1 e. ( 0 [,] 1 ) -> ( ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) ` 1 ) = 1 )
1208, 119mp1i 12 . . 3 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) ` 1 ) = 1 )
12123, 101, 112, 120reparpht 21226 . 2 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( F o. ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) ) ( ~=ph ` J ) F )
12249, 121eqbrtrd 4460 1 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( F ( *p ` J ) P ) ( ~=ph ` J ) F )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   C_ wss 3469   ifcif 3932   {csn 4020   U.cuni 4238   class class class wbr 4440   |-> cmpt 4498   X. cxp 4990   ran crn 4993   o. ccom 4996   -->wf 5575   ` cfv 5579  (class class class)co 6275   RRcr 9480   0cc0 9481   1c1 9482   x. cmul 9486   < clt 9617   <_ cle 9618   - cmin 9794   / cdiv 10195   2c2 10574   (,)cioo 11518   [,]cicc 11521   ↾t crest 14665   topGenctg 14682   Topctop 19154  TopOnctopon 19155   Cn ccn 19484   IIcii 21107   ~=ph cphtpc 21197   *pcpco 21228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-cn 19487  df-cnp 19488  df-tx 19791  df-hmeo 19984  df-xms 20551  df-ms 20552  df-tms 20553  df-ii 21109  df-htpy 21198  df-phtpy 21199  df-phtpc 21220  df-pco 21233
This theorem is referenced by:  pcophtb  21257  pi1xfrcnvlem  21284
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