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Theorem pcopt2 22132
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 5880 . . . . . . . 8 |- ( P ` ( ( 2 x. x ) - 1 ) ) = ( ( ( 0 [,] 1 ) X. { Y } ) ` ( ( 2 x. x ) - 1 ) )
3 simpr 468 . . . . . . . . . 10 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( F ` 1 ) = Y )
4 iiuni 21991 . . . . . . . . . . . . 13 |- ( 0 [,] 1 ) = U. II
5 eqid 2471 . . . . . . . . . . . . 13 |- U. J = U. J
64, 5cnf 20339 . . . . . . . . . . . 12 |- ( F e. ( II Cn J ) -> F : ( 0 [,] 1 ) --> U. J )
76adantr 472 . . . . . . . . . . 11 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> F : ( 0 [,] 1 ) --> U. J )
8 1elunit 11777 . . . . . . . . . . 11 |- 1 e. ( 0 [,] 1 )
9 ffvelrn 6035 . . . . . . . . . . 11 |- ( ( F : ( 0 [,] 1 ) --> U. J /\ 1 e. ( 0 [,] 1 ) ) -> ( F ` 1 ) e. U. J )
107, 8, 9sylancl 675 . . . . . . . . . 10 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( F ` 1 ) e. U. J )
113, 10eqeltrrd 2550 . . . . . . . . 9 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> Y e. U. J )
12 elii2 22042 . . . . . . . . . 10 |- ( ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) -> x e. ( ( 1 / 2 ) [,] 1 ) )
13 iihalf2 22039 . . . . . . . . . 10 |- ( x e. ( ( 1 / 2 ) [,] 1 ) -> ( ( 2 x. x ) - 1 ) e. ( 0 [,] 1 ) )
1412, 13syl 17 . . . . . . . . 9 |- ( ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) -> ( ( 2 x. x ) - 1 ) e. ( 0 [,] 1 ) )
15 fvconst2g 6134 . . . . . . . . 9 |- ( ( Y e. U. J /\ ( ( 2 x. x ) - 1 ) e. ( 0 [,] 1 ) ) -> ( ( ( 0 [,] 1 ) X. { Y } ) ` ( ( 2 x. x ) - 1 ) ) = Y )
1611, 14, 15syl2an 485 . . . . . . . 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 2517 . . . . . . 7 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) ) -> ( P ` ( ( 2 x. x ) - 1 ) ) = Y )
18 simplr 770 . . . . . . 7 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) ) -> ( F ` 1 ) = Y )
1917, 18eqtr4d 2508 . . . . . 6 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) ) -> ( P ` ( ( 2 x. x ) - 1 ) ) = ( F ` 1 ) )
2019anassrs 660 . . . . 5 |- ( ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ x e. ( 0 [,] 1 ) ) /\ -. x <_ ( 1 / 2 ) ) -> ( P ` ( ( 2 x. x ) - 1 ) ) = ( F ` 1 ) )
2120ifeq2da 3903 . . . 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 4482 . . 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 464 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> F e. ( II Cn J ) )
24 cntop2 20334 . . . . . . . 8 |- ( F e. ( II Cn J ) -> J e. Top )
2524adantr 472 . . . . . . 7 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> J e. Top )
265toptopon 20025 . . . . . . 7 |- ( J e. Top <-> J e. (TopOn ` U. J ) )
2725, 26sylib 201 . . . . . 6 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> J e. (TopOn ` U. J ) )
281pcoptcl 22130 . . . . . 6 |- ( ( J e. (TopOn ` U. J ) /\ Y e. U. J ) -> ( P e. ( II Cn J ) /\ ( P ` 0 ) = Y /\ ( P ` 1 ) = Y ) )
2927, 11, 28syl2anc 673 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( P e. ( II Cn J ) /\ ( P ` 0 ) = Y /\ ( P ` 1 ) = Y ) )
3029simp1d 1042 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> P e. ( II Cn J ) )
3123, 30pcoval 22120 . . 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 3878 . . . . . . . . 9 |- ( x <_ ( 1 / 2 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = ( 2 x. x ) )
3332adantl 473 . . . . . . . 8 |- ( ( x e. ( 0 [,] 1 ) /\ x <_ ( 1 / 2 ) ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = ( 2 x. x ) )
34 elii1 22041 . . . . . . . . 9 |- ( x e. ( 0 [,] ( 1 / 2 ) ) <-> ( x e. ( 0 [,] 1 ) /\ x <_ ( 1 / 2 ) ) )
35 iihalf1 22037 . . . . . . . . 9 |- ( x e. ( 0 [,] ( 1 / 2 ) ) -> ( 2 x. x ) e. ( 0 [,] 1 ) )
3634, 35sylbir 218 . . . . . . . 8 |- ( ( x e. ( 0 [,] 1 ) /\ x <_ ( 1 / 2 ) ) -> ( 2 x. x ) e. ( 0 [,] 1 ) )
3733, 36eqeltrd 2549 . . . . . . 7 |- ( ( x e. ( 0 [,] 1 ) /\ x <_ ( 1 / 2 ) ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) e. ( 0 [,] 1 ) )
3837ex 441 . . . . . 6 |- ( x e. ( 0 [,] 1 ) -> ( x <_ ( 1 / 2 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) e. ( 0 [,] 1 ) ) )
39 iffalse 3881 . . . . . . 7 |- ( -. x <_ ( 1 / 2 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = 1 )
4039, 8syl6eqel 2557 . . . . . 6 |- ( -. x <_ ( 1 / 2 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) e. ( 0 [,] 1 ) )
4138, 40pm2.61d1 164 . . . . 5 |- ( x e. ( 0 [,] 1 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) e. ( 0 [,] 1 ) )
4241adantl 473 . . . 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 2472 . . . 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 5932 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> F = ( y e. ( 0 [,] 1 ) |-> ( F ` y ) ) )
45 fveq2 5879 . . . . 5 |- ( y = if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) -> ( F ` y ) = ( F ` if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) )
46 fvif 5890 . . . . 5 |- ( F ` if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) = if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( F ` 1 ) )
4745, 46syl6eq 2521 . . . 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 6072 . . 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 2515 . 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 21989 . . . . 5 |- II e. (TopOn ` ( 0 [,] 1 ) )
5150a1i 11 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> II e. (TopOn ` ( 0 [,] 1 ) ) )
5251cnmptid 20753 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( x e. ( 0 [,] 1 ) |-> x ) e. ( II Cn II ) )
53 0elunit 11776 . . . . . 6 |- 0 e. ( 0 [,] 1 )
5453a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> 0 e. ( 0 [,] 1 ) )
5551, 51, 54cnmptc 20754 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( x e. ( 0 [,] 1 ) |-> 0 ) e. ( II Cn II ) )
56 eqid 2471 . . . . 5 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
57 eqid 2471 . . . . 5 |- ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) )
58 eqid 2471 . . . . 5 |- ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) )
59 dfii2 21992 . . . . 5 |- II = ( ( topGen ` ran (,) ) ↾t ( 0 [,] 1 ) )
60 0re 9661 . . . . . 6 |- 0 e. RR
6160a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> 0 e. RR )
62 1re 9660 . . . . . 6 |- 1 e. RR
6362a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> 1 e. RR )
64 halfre 10851 . . . . . . 7 |- ( 1 / 2 ) e. RR
65 halfgt0 10853 . . . . . . . 8 |- 0 < ( 1 / 2 )
6660, 64, 65ltleii 9775 . . . . . . 7 |- 0 <_ ( 1 / 2 )
67 halflt1 10854 . . . . . . . 8 |- ( 1 / 2 ) < 1
6864, 62, 67ltleii 9775 . . . . . . 7 |- ( 1 / 2 ) <_ 1
6960, 62elicc2i 11725 . . . . . . 7 |- ( ( 1 / 2 ) e. ( 0 [,] 1 ) <-> ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ 1 ) )
7064, 66, 68, 69mpbir3an 1212 . . . . . 6 |- ( 1 / 2 ) e. ( 0 [,] 1 )
7170a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( 1 / 2 ) e. ( 0 [,] 1 ) )
72 simprl 772 . . . . . . 7 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> y = ( 1 / 2 ) )
7372oveq2d 6324 . . . . . 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 10702 . . . . . . 7 |- 2 e. CC
75 2ne0 10724 . . . . . . 7 |- 2 =/= 0
7674, 75recidi 10360 . . . . . 6 |- ( 2 x. ( 1 / 2 ) ) = 1
7773, 76syl6eq 2521 . . . . 5 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( 2 x. y ) = 1 )
78 retopon 21862 . . . . . . . 8 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
79 iccssre 11741 . . . . . . . . 9 |- ( ( 0 e. RR /\ ( 1 / 2 ) e. RR ) -> ( 0 [,] ( 1 / 2 ) ) C_ RR )
8060, 64, 79mp2an 686 . . . . . . . 8 |- ( 0 [,] ( 1 / 2 ) ) C_ RR
81 resttopon 20254 . . . . . . . 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 686 . . . . . . 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 20760 . . . . . 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 22038 . . . . . . 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 6316 . . . . . 6 |- ( x = y -> ( 2 x. x ) = ( 2 x. y ) )
8883, 51, 84, 83, 86, 87cnmpt21 20763 . . . . 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 11741 . . . . . . . . 9 |- ( ( ( 1 / 2 ) e. RR /\ 1 e. RR ) -> ( ( 1 / 2 ) [,] 1 ) C_ RR )
9064, 62, 89mp2an 686 . . . . . . . 8 |- ( ( 1 / 2 ) [,] 1 ) C_ RR
91 resttopon 20254 . . . . . . . 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 686 . . . . . . 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 20762 . . . . 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 22034 . . . 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 4398 . . . . . 6 |- ( y = x -> ( y <_ ( 1 / 2 ) <-> x <_ ( 1 / 2 ) ) )
98 oveq2 6316 . . . . . 6 |- ( y = x -> ( 2 x. y ) = ( 2 x. x ) )
9997, 98ifbieq1d 3895 . . . . 5 |- ( y = x -> if ( y <_ ( 1 / 2 ) , ( 2 x. y ) , 1 ) = if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) )
10099adantr 472 . . . 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 20759 . . 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 4433 . . . . . . 7 |- ( x = 0 -> x <_ ( 1 / 2 ) )
104103, 32syl 17 . . . . . 6 |- ( x = 0 -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = ( 2 x. x ) )
105 oveq2 6316 . . . . . . 7 |- ( x = 0 -> ( 2 x. x ) = ( 2 x. 0 ) )
106 2t0e0 10788 . . . . . . 7 |- ( 2 x. 0 ) = 0
107105, 106syl6eq 2521 . . . . . 6 |- ( x = 0 -> ( 2 x. x ) = 0 )
108104, 107eqtrd 2505 . . . . 5 |- ( x = 0 -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = 0 )
109 eqid 2471 . . . . 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 9655 . . . . 5 |- 0 e. _V
111108, 109, 110fvmpt 5963 . . . 4 |- ( 0 e. ( 0 [,] 1 ) -> ( ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) ` 0 ) = 0 )
11253, 111mp1i 13 . . 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 9773 . . . . . . . 8 |- ( ( 1 / 2 ) < 1 <-> -. 1 <_ ( 1 / 2 ) )
11467, 113mpbi 213 . . . . . . 7 |- -. 1 <_ ( 1 / 2 )
115 breq1 4398 . . . . . . 7 |- ( x = 1 -> ( x <_ ( 1 / 2 ) <-> 1 <_ ( 1 / 2 ) ) )
116114, 115mtbiri 310 . . . . . 6 |- ( x = 1 -> -. x <_ ( 1 / 2 ) )
117116, 39syl 17 . . . . 5 |- ( x = 1 -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = 1 )
118 1ex 9656 . . . . 5 |- 1 e. _V
119117, 109, 118fvmpt 5963 . . . 4 |- ( 1 e. ( 0 [,] 1 ) -> ( ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) ` 1 ) = 1 )
1208, 119mp1i 13 . . 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 22107 . 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 4416 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 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   C_ wss 3390   ifcif 3872   {csn 3959   U.cuni 4190   class class class wbr 4395   |-> cmpt 4454   X. cxp 4837   ran crn 4840   o. ccom 4843   -->wf 5585   ` cfv 5589  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558   x. cmul 9562   < clt 9693   <_ cle 9694   - cmin 9880   / cdiv 10291   2c2 10681   (,)cioo 11660   [,]cicc 11663   ↾t crest 15397   topGenctg 15414   Topctop 19994  TopOnctopon 19995   Cn ccn 20317   IIcii 21985   ~=ph cphtpc 22078   *pcpco 22109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-cn 20320  df-cnp 20321  df-tx 20654  df-hmeo 20847  df-xms 21413  df-ms 21414  df-tms 21415  df-ii 21987  df-htpy 22079  df-phtpy 22080  df-phtpc 22101  df-pco 22114
This theorem is referenced by:  pcophtb  22138  pi1xfrcnvlem  22165
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