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Theorem pcopt2 22047
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 5864 . . . . . . . 8 |- ( P ` ( ( 2 x. x ) - 1 ) ) = ( ( ( 0 [,] 1 ) X. { Y } ) ` ( ( 2 x. x ) - 1 ) )
3 simpr 463 . . . . . . . . . 10 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( F ` 1 ) = Y )
4 iiuni 21906 . . . . . . . . . . . . 13 |- ( 0 [,] 1 ) = U. II
5 eqid 2450 . . . . . . . . . . . . 13 |- U. J = U. J
64, 5cnf 20255 . . . . . . . . . . . 12 |- ( F e. ( II Cn J ) -> F : ( 0 [,] 1 ) --> U. J )
76adantr 467 . . . . . . . . . . 11 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> F : ( 0 [,] 1 ) --> U. J )
8 1elunit 11748 . . . . . . . . . . 11 |- 1 e. ( 0 [,] 1 )
9 ffvelrn 6018 . . . . . . . . . . 11 |- ( ( F : ( 0 [,] 1 ) --> U. J /\ 1 e. ( 0 [,] 1 ) ) -> ( F ` 1 ) e. U. J )
107, 8, 9sylancl 667 . . . . . . . . . 10 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( F ` 1 ) e. U. J )
113, 10eqeltrrd 2529 . . . . . . . . 9 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> Y e. U. J )
12 elii2 21957 . . . . . . . . . 10 |- ( ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) -> x e. ( ( 1 / 2 ) [,] 1 ) )
13 iihalf2 21954 . . . . . . . . . 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 6116 . . . . . . . . 9 |- ( ( Y e. U. J /\ ( ( 2 x. x ) - 1 ) e. ( 0 [,] 1 ) ) -> ( ( ( 0 [,] 1 ) X. { Y } ) ` ( ( 2 x. x ) - 1 ) ) = Y )
1611, 14, 15syl2an 480 . . . . . . . 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 2496 . . . . . . 7 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) ) -> ( P ` ( ( 2 x. x ) - 1 ) ) = Y )
18 simplr 761 . . . . . . 7 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) ) -> ( F ` 1 ) = Y )
1917, 18eqtr4d 2487 . . . . . 6 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( x e. ( 0 [,] 1 ) /\ -. x <_ ( 1 / 2 ) ) ) -> ( P ` ( ( 2 x. x ) - 1 ) ) = ( F ` 1 ) )
2019anassrs 653 . . . . 5 |- ( ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ x e. ( 0 [,] 1 ) ) /\ -. x <_ ( 1 / 2 ) ) -> ( P ` ( ( 2 x. x ) - 1 ) ) = ( F ` 1 ) )
2120ifeq2da 3911 . . . 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 4488 . . 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 459 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> F e. ( II Cn J ) )
24 cntop2 20250 . . . . . . . 8 |- ( F e. ( II Cn J ) -> J e. Top )
2524adantr 467 . . . . . . 7 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> J e. Top )
265toptopon 19941 . . . . . . 7 |- ( J e. Top <-> J e. (TopOn ` U. J ) )
2725, 26sylib 200 . . . . . 6 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> J e. (TopOn ` U. J ) )
281pcoptcl 22045 . . . . . 6 |- ( ( J e. (TopOn ` U. J ) /\ Y e. U. J ) -> ( P e. ( II Cn J ) /\ ( P ` 0 ) = Y /\ ( P ` 1 ) = Y ) )
2927, 11, 28syl2anc 666 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( P e. ( II Cn J ) /\ ( P ` 0 ) = Y /\ ( P ` 1 ) = Y ) )
3029simp1d 1019 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> P e. ( II Cn J ) )
3123, 30pcoval 22035 . . 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 3886 . . . . . . . . 9 |- ( x <_ ( 1 / 2 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = ( 2 x. x ) )
3332adantl 468 . . . . . . . 8 |- ( ( x e. ( 0 [,] 1 ) /\ x <_ ( 1 / 2 ) ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = ( 2 x. x ) )
34 elii1 21956 . . . . . . . . 9 |- ( x e. ( 0 [,] ( 1 / 2 ) ) <-> ( x e. ( 0 [,] 1 ) /\ x <_ ( 1 / 2 ) ) )
35 iihalf1 21952 . . . . . . . . 9 |- ( x e. ( 0 [,] ( 1 / 2 ) ) -> ( 2 x. x ) e. ( 0 [,] 1 ) )
3634, 35sylbir 217 . . . . . . . 8 |- ( ( x e. ( 0 [,] 1 ) /\ x <_ ( 1 / 2 ) ) -> ( 2 x. x ) e. ( 0 [,] 1 ) )
3733, 36eqeltrd 2528 . . . . . . 7 |- ( ( x e. ( 0 [,] 1 ) /\ x <_ ( 1 / 2 ) ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) e. ( 0 [,] 1 ) )
3837ex 436 . . . . . 6 |- ( x e. ( 0 [,] 1 ) -> ( x <_ ( 1 / 2 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) e. ( 0 [,] 1 ) ) )
39 iffalse 3889 . . . . . . 7 |- ( -. x <_ ( 1 / 2 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = 1 )
4039, 8syl6eqel 2536 . . . . . 6 |- ( -. x <_ ( 1 / 2 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) e. ( 0 [,] 1 ) )
4138, 40pm2.61d1 163 . . . . 5 |- ( x e. ( 0 [,] 1 ) -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) e. ( 0 [,] 1 ) )
4241adantl 468 . . . 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 2451 . . . 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 5916 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> F = ( y e. ( 0 [,] 1 ) |-> ( F ` y ) ) )
45 fveq2 5863 . . . . 5 |- ( y = if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) -> ( F ` y ) = ( F ` if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) )
46 fvif 5874 . . . . 5 |- ( F ` if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) ) = if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( F ` 1 ) )
4745, 46syl6eq 2500 . . . 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 6054 . . 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 2494 . 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 21904 . . . . 5 |- II e. (TopOn ` ( 0 [,] 1 ) )
5150a1i 11 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> II e. (TopOn ` ( 0 [,] 1 ) ) )
5251cnmptid 20669 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( x e. ( 0 [,] 1 ) |-> x ) e. ( II Cn II ) )
53 0elunit 11747 . . . . . 6 |- 0 e. ( 0 [,] 1 )
5453a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> 0 e. ( 0 [,] 1 ) )
5551, 51, 54cnmptc 20670 . . . 4 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( x e. ( 0 [,] 1 ) |-> 0 ) e. ( II Cn II ) )
56 eqid 2450 . . . . 5 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
57 eqid 2450 . . . . 5 |- ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) )
58 eqid 2450 . . . . 5 |- ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) )
59 dfii2 21907 . . . . 5 |- II = ( ( topGen ` ran (,) ) ↾t ( 0 [,] 1 ) )
60 0re 9640 . . . . . 6 |- 0 e. RR
6160a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> 0 e. RR )
62 1re 9639 . . . . . 6 |- 1 e. RR
6362a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> 1 e. RR )
64 halfre 10825 . . . . . . 7 |- ( 1 / 2 ) e. RR
65 halfgt0 10827 . . . . . . . 8 |- 0 < ( 1 / 2 )
6660, 64, 65ltleii 9754 . . . . . . 7 |- 0 <_ ( 1 / 2 )
67 halflt1 10828 . . . . . . . 8 |- ( 1 / 2 ) < 1
6864, 62, 67ltleii 9754 . . . . . . 7 |- ( 1 / 2 ) <_ 1
6960, 62elicc2i 11697 . . . . . . 7 |- ( ( 1 / 2 ) e. ( 0 [,] 1 ) <-> ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ 1 ) )
7064, 66, 68, 69mpbir3an 1189 . . . . . 6 |- ( 1 / 2 ) e. ( 0 [,] 1 )
7170a1i 11 . . . . 5 |- ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) -> ( 1 / 2 ) e. ( 0 [,] 1 ) )
72 simprl 763 . . . . . . 7 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> y = ( 1 / 2 ) )
7372oveq2d 6304 . . . . . 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 10677 . . . . . . 7 |- 2 e. CC
75 2ne0 10699 . . . . . . 7 |- 2 =/= 0
7674, 75recidi 10335 . . . . . 6 |- ( 2 x. ( 1 / 2 ) ) = 1
7773, 76syl6eq 2500 . . . . 5 |- ( ( ( F e. ( II Cn J ) /\ ( F ` 1 ) = Y ) /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( 2 x. y ) = 1 )
78 retopon 21777 . . . . . . . 8 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
79 iccssre 11713 . . . . . . . . 9 |- ( ( 0 e. RR /\ ( 1 / 2 ) e. RR ) -> ( 0 [,] ( 1 / 2 ) ) C_ RR )
8060, 64, 79mp2an 677 . . . . . . . 8 |- ( 0 [,] ( 1 / 2 ) ) C_ RR
81 resttopon 20170 . . . . . . . 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 677 . . . . . . 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 20676 . . . . . 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 21953 . . . . . . 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 6296 . . . . . 6 |- ( x = y -> ( 2 x. x ) = ( 2 x. y ) )
8883, 51, 84, 83, 86, 87cnmpt21 20679 . . . . 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 11713 . . . . . . . . 9 |- ( ( ( 1 / 2 ) e. RR /\ 1 e. RR ) -> ( ( 1 / 2 ) [,] 1 ) C_ RR )
9064, 62, 89mp2an 677 . . . . . . . 8 |- ( ( 1 / 2 ) [,] 1 ) C_ RR
91 resttopon 20170 . . . . . . . 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 677 . . . . . . 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 20678 . . . . 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 21949 . . . 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 4404 . . . . . 6 |- ( y = x -> ( y <_ ( 1 / 2 ) <-> x <_ ( 1 / 2 ) ) )
98 oveq2 6296 . . . . . 6 |- ( y = x -> ( 2 x. y ) = ( 2 x. x ) )
9997, 98ifbieq1d 3903 . . . . 5 |- ( y = x -> if ( y <_ ( 1 / 2 ) , ( 2 x. y ) , 1 ) = if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) )
10099adantr 467 . . . 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 20675 . . 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 4439 . . . . . . 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 6296 . . . . . . 7 |- ( x = 0 -> ( 2 x. x ) = ( 2 x. 0 ) )
106 2t0e0 10762 . . . . . . 7 |- ( 2 x. 0 ) = 0
107105, 106syl6eq 2500 . . . . . 6 |- ( x = 0 -> ( 2 x. x ) = 0 )
108104, 107eqtrd 2484 . . . . 5 |- ( x = 0 -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = 0 )
109 eqid 2450 . . . . 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 9634 . . . . 5 |- 0 e. _V
111108, 109, 110fvmpt 5946 . . . 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 9752 . . . . . . . 8 |- ( ( 1 / 2 ) < 1 <-> -. 1 <_ ( 1 / 2 ) )
11467, 113mpbi 212 . . . . . . 7 |- -. 1 <_ ( 1 / 2 )
115 breq1 4404 . . . . . . 7 |- ( x = 1 -> ( x <_ ( 1 / 2 ) <-> 1 <_ ( 1 / 2 ) ) )
116114, 115mtbiri 305 . . . . . 6 |- ( x = 1 -> -. x <_ ( 1 / 2 ) )
117116, 39syl 17 . . . . 5 |- ( x = 1 -> if ( x <_ ( 1 / 2 ) , ( 2 x. x ) , 1 ) = 1 )
118 1ex 9635 . . . . 5 |- 1 e. _V
119117, 109, 118fvmpt 5946 . . . 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 22022 . 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 4422 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 371   /\ w3a 984   = wceq 1443   e. wcel 1886   C_ wss 3403   ifcif 3880   {csn 3967   U.cuni 4197   class class class wbr 4401   |-> cmpt 4460   X. cxp 4831   ran crn 4834   o. ccom 4837   -->wf 5577   ` cfv 5581  (class class class)co 6288   RRcr 9535   0cc0 9536   1c1 9537   x. cmul 9541   < clt 9672   <_ cle 9673   - cmin 9857   / cdiv 10266   2c2 10656   (,)cioo 11632   [,]cicc 11635   ↾t crest 15312   topGenctg 15329   Topctop 19910  TopOnctopon 19911   Cn ccn 20233   IIcii 21900   ~=ph cphtpc 21993   *pcpco 22024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-icc 11639  df-fz 11782  df-fzo 11913  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-cn 20236  df-cnp 20237  df-tx 20570  df-hmeo 20763  df-xms 21328  df-ms 21329  df-tms 21330  df-ii 21902  df-htpy 21994  df-phtpy 21995  df-phtpc 22016  df-pco 22029
This theorem is referenced by:  pcophtb  22053  pi1xfrcnvlem  22080
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