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Theorem pcopt2 21608
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 5775 . . . . . . . 8 |- ( P ` ( ( 2 x. x ) - 1 ) ) = ( ( ( 0 [,] 1 ) X. { Y } ) ` ( ( 2 x. x ) - 1 ) )
3 simpr 459 . . . . . . . . . 10 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( F ` 1 ) = Y )
4 iiuni 21470 . . . . . . . . . . . . 13 |- ( 0 [,] 1 ) = U. II
5 eqid 2382 . . . . . . . . . . . . 13 |- U. J = U. J
64, 5cnf 19833 . . . . . . . . . . . 12 |- ( F e. ( II Cn J ) -> F : ( 0 [,] 1 ) --> U. J )
76adantr 463 . . . . . . . . . . 11 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> F : ( 0 [,] 1 ) --> U. J )
8 1elunit 11560 . . . . . . . . . . 11 |- 1 e. ( 0 [,] 1 )
9 ffvelrn 5931 . . . . . . . . . . 11 |- ( ( F : ( 0 [,] 1 ) --> U. J /\ 1 e. ( 0 [,] 1 ) ) -> ( F ` 1 ) e. U. J )
107, 8, 9sylancl 660 . . . . . . . . . 10 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( F ` 1 ) e. U. J )
113, 10eqeltrrd 2471 . . . . . . . . 9 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> Y e. U. J )
12 elii2 21521 . . . . . . . . . 10 |- ( ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) -> x e. ( ( 1 / 2 ) [,] 1 ) )
13 iihalf2 21518 . . . . . . . . . 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 6027 . . . . . . . . 9 |- ( ( Y e. U. J /\ ( ( 2 x. x ) - 1 ) e. ( 0 [,] 1 ) ) -> ( ( ( 0 [,] 1 ) X. { Y } ) ` ( ( 2 x. x ) - 1 ) ) = Y )
1611, 14, 15syl2an 475 . . . . . . . 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 2435 . . . . . . 7 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) ) -> ( P ` ( ( 2 x. x ) - 1 ) ) = Y )
18 simplr 753 . . . . . . 7 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) ) -> ( F ` 1 ) = Y )
1917, 18eqtr4d 2426 . . . . . 6 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) ) -> ( P ` ( ( 2 x. x ) - 1 ) ) = ( F ` 1 ) )
2019anassrs 646 . . . . 5 |- ( ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ x e. ( 0 [,] 1 ) ) /\ -. x <_ ( 1 / 2 ) ) -> ( P ` ( ( 2 x. x ) - 1 ) ) = ( F ` 1 ) )
2120ifeq2da 3888 . . . 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 4453 . . 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 455 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> F e. ( II Cn J ) )
24 cntop2 19828 . . . . . . . 8 |- ( F e. ( II Cn J ) -> J e. Top )
2524adantr 463 . . . . . . 7 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> J e. Top )
265toptopon 19519 . . . . . . 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 21606 . . . . . 6 |- ( ( J e. (TopOn ` U. J ) /\ Y e. U. J ) -> ( P e. ( II Cn J ) /\ ( P ` 0 ) = Y /\ ( P ` 1 ) = Y ) )
2927, 11, 28syl2anc 659 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( P e. ( II Cn J ) /\ ( P ` 0 ) = Y /\ ( P ` 1 ) = Y ) )
3029simp1d 1006 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> P e. ( II Cn J ) )
3123, 30pcoval 21596 . . 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 3863 . . . . . . . . 9 |- ( x <_ ( 1 / 2 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = ( 2 x. x ) )
3332adantl 464 . . . . . . . 8 |- ( ( x e. ( 0 [,] 1 ) /\ x <_ ( 1 / 2 ) ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = ( 2 x. x ) )
34 elii1 21520 . . . . . . . . 9 |- ( x e. ( 0 [,] ( 1 / 2 ) ) <-> ( x e. ( 0 [,] 1 ) /\ x <_ ( 1 / 2 ) ) )
35 iihalf1 21516 . . . . . . . . 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 2470 . . . . . . 7 |- ( ( x e. ( 0 [,] 1 ) /\ x <_ ( 1 / 2 ) ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) e. ( 0 [,] 1 ) )
3837ex 432 . . . . . 6 |- ( x e. ( 0 [,] 1 ) -> ( x <_ ( 1 / 2 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) e. ( 0 [,] 1 ) ) )
39 iffalse 3866 . . . . . . 7 |- ( -. x <_ ( 1 / 2 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = 1 )
4039, 8syl6eqel 2478 . . . . . 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 464 . . . 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 2383 . . . 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 5827 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> F = ( y e. ( 0 [,] 1 ) |-> ( F ` y ) ) )
45 fveq2 5774 . . . . 5 |- ( y = if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) -> ( F ` y ) = ( F ` if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) )
46 fvif 5785 . . . . 5 |- ( F ` if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) = if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( F ` 1 ) )
4745, 46syl6eq 2439 . . . 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 5966 . . 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 2433 . 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 21468 . . . . 5 |- II e. (TopOn ` ( 0 [,] 1 ) )
5150a1i 11 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> II e. (TopOn ` ( 0 [,] 1 ) ) )
5251cnmptid 20247 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( x e. ( 0 [,] 1 ) |-> x ) e. ( II Cn II ) )
53 0elunit 11559 . . . . . 6 |- 0 e. ( 0 [,] 1 )
5453a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> 0 e. ( 0 [,] 1 ) )
5551, 51, 54cnmptc 20248 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( x e. ( 0 [,] 1 ) |-> 0 ) e. ( II Cn II ) )
56 eqid 2382 . . . . 5 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
57 eqid 2382 . . . . 5 |- ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) )
58 eqid 2382 . . . . 5 |- ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) )
59 dfii2 21471 . . . . 5 |- II = ( ( topGen ` ran (,) ) ↾t ( 0 [,] 1 ) )
60 0re 9507 . . . . . 6 |- 0 e. RR
6160a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> 0 e. RR )
62 1re 9506 . . . . . 6 |- 1 e. RR
6362a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> 1 e. RR )
64 halfre 10671 . . . . . . 7 |- ( 1 / 2 ) e. RR
65 halfgt0 10673 . . . . . . . 8 |- 0 < ( 1 / 2 )
6660, 64, 65ltleii 9618 . . . . . . 7 |- 0 <_ ( 1 / 2 )
67 halflt1 10674 . . . . . . . 8 |- ( 1 / 2 ) < 1
6864, 62, 67ltleii 9618 . . . . . . 7 |- ( 1 / 2 ) <_ 1
6960, 62elicc2i 11511 . . . . . . 7 |- ( ( 1 / 2 ) e. ( 0 [,] 1 ) <-> ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ 1 ) )
7064, 66, 68, 69mpbir3an 1176 . . . . . 6 |- ( 1 / 2 ) e. ( 0 [,] 1 )
7170a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( 1 / 2 ) e. ( 0 [,] 1 ) )
72 simprl 754 . . . . . . 7 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> y = ( 1 / 2 ) )
7372oveq2d 6212 . . . . . 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 10523 . . . . . . 7 |- 2 e. CC
75 2ne0 10545 . . . . . . 7 |- 2 =/= 0
7674, 75recidi 10192 . . . . . 6 |- ( 2 x. ( 1 / 2 ) ) = 1
7773, 76syl6eq 2439 . . . . 5 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( 2 x. y ) = 1 )
78 retopon 21355 . . . . . . . 8 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
79 iccssre 11527 . . . . . . . . 9 |- ( ( 0 e. RR /\ ( 1 / 2 ) e. RR ) -> ( 0 [,] ( 1 / 2 ) ) C_ RR )
8060, 64, 79mp2an 670 . . . . . . . 8 |- ( 0 [,] ( 1 / 2 ) ) C_ RR
81 resttopon 19748 . . . . . . . 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 670 . . . . . . 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 20254 . . . . . 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 21517 . . . . . . 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 6204 . . . . . 6 |- ( x = y -> ( 2 x. x ) = ( 2 x. y ) )
8883, 51, 84, 83, 86, 87cnmpt21 20257 . . . . 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 11527 . . . . . . . . 9 |- ( ( ( 1 / 2 ) e. RR /\ 1 e. RR ) -> ( ( 1 / 2 ) [,] 1 ) C_ RR )
9064, 62, 89mp2an 670 . . . . . . . 8 |- ( ( 1 / 2 ) [,] 1 ) C_ RR
91 resttopon 19748 . . . . . . . 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 670 . . . . . . 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 20256 . . . . 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 21513 . . . 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 4370 . . . . . 6 |- ( y = x -> ( y <_ ( 1 / 2 ) <-> x <_ ( 1 / 2 ) ) )
98 oveq2 6204 . . . . . 6 |- ( y = x -> ( 2 x. y ) = ( 2 x. x ) )
9997, 98ifbieq1d 3880 . . . . 5 |- ( y = x -> if ( y <_ ( 1 / 2 ) , ( 2 x. y ) , 1 ) = if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) )
10099adantr 463 . . . 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 20253 . . 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 4404 . . . . . . 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 6204 . . . . . . 7 |- ( x = 0 -> ( 2 x. x ) = ( 2 x. 0 ) )
106 2t0e0 10608 . . . . . . 7 |- ( 2 x. 0 ) = 0
107105, 106syl6eq 2439 . . . . . 6 |- ( x = 0 -> ( 2 x. x ) = 0 )
108104, 107eqtrd 2423 . . . . 5 |- ( x = 0 -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = 0 )
109 eqid 2382 . . . . 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 9501 . . . . 5 |- 0 e. _V
111108, 109, 110fvmpt 5857 . . . 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 9616 . . . . . . . 8 |- ( ( 1 / 2 ) < 1 <-> -. 1 <_ ( 1 / 2 ) )
11467, 113mpbi 208 . . . . . . 7 |- -. 1 <_ ( 1 / 2 )
115 breq1 4370 . . . . . . 7 |- ( x = 1 -> ( x <_ ( 1 / 2 ) <-> 1 <_ ( 1 / 2 ) ) )
116114, 115mtbiri 301 . . . . . 6 |- ( x = 1 -> -. x <_ ( 1 / 2 ) )
117116, 39syl 16 . . . . 5 |- ( x = 1 -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = 1 )
118 1ex 9502 . . . . 5 |- 1 e. _V
119117, 109, 118fvmpt 5857 . . . 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 21583 . 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 4387 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 367   /\ w3a 971   = wceq 1399   e. wcel 1826   C_ wss 3389   ifcif 3857   {csn 3944   U.cuni 4163   class class class wbr 4367   |-> cmpt 4425   X. cxp 4911   ran crn 4914   o. ccom 4917   -->wf 5492   ` cfv 5496  (class class class)co 6196   RRcr 9402   0cc0 9403   1c1 9404   x. cmul 9408   < clt 9539   <_ cle 9540   - cmin 9718   / cdiv 10123   2c2 10502   (,)cioo 11450   [,]cicc 11453   ↾t crest 14828   topGenctg 14845   Topctop 19479  TopOnctopon 19480   Cn ccn 19811   IIcii 21464   ~=ph cphtpc 21554   *pcpco 21585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482  ax-mulf 9483
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-map 7340  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-fi 7786  df-sup 7816  df-oi 7850  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ioo 11454  df-icc 11457  df-fz 11594  df-fzo 11718  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-sca 14718  df-vsca 14719  df-ip 14720  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-hom 14726  df-cco 14727  df-rest 14830  df-topn 14831  df-0g 14849  df-gsum 14850  df-topgen 14851  df-pt 14852  df-prds 14855  df-xrs 14909  df-qtop 14914  df-imas 14915  df-xps 14917  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-mulg 16177  df-cntz 16472  df-cmn 16917  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-cnfld 18534  df-top 19484  df-bases 19486  df-topon 19487  df-topsp 19488  df-cld 19605  df-cn 19814  df-cnp 19815  df-tx 20148  df-hmeo 20341  df-xms 20908  df-ms 20909  df-tms 20910  df-ii 21466  df-htpy 21555  df-phtpy 21556  df-phtpc 21577  df-pco 21590
This theorem is referenced by:  pcophtb  21614  pi1xfrcnvlem  21641
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