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Theorem pcopt2 21500
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 5857 . . . . . . . 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 21362 . . . . . . . . . . . . 13  |-  ( 0 [,] 1 )  = 
U. II
5 eqid 2443 . . . . . . . . . . . . 13  |-  U. J  =  U. J
64, 5cnf 19724 . . . . . . . . . . . 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 11649 . . . . . . . . . . 11  |-  1  e.  ( 0 [,] 1
)
9 ffvelrn 6014 . . . . . . . . . . 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 2532 . . . . . . . . 9  |-  ( ( F  e.  ( II 
Cn  J )  /\  ( F `  1 )  =  Y )  ->  Y  e.  U. J )
12 elii2 21413 . . . . . . . . . 10  |-  ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  <_  ( 1  /  2 ) )  ->  x  e.  ( ( 1  /  2
) [,] 1 ) )
13 iihalf2 21410 . . . . . . . . . 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 6109 . . . . . . . . 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 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 755 . . . . . . 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 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 3957 . . . 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 4523 . . 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 19719 . . . . . . . 8  |-  ( F  e.  ( II  Cn  J )  ->  J  e.  Top )
2524adantr 465 . . . . . . 7  |-  ( ( F  e.  ( II 
Cn  J )  /\  ( F `  1 )  =  Y )  ->  J  e.  Top )
265toptopon 19411 . . . . . . 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 21498 . . . . . 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 1009 . . . 4  |-  ( ( F  e.  ( II 
Cn  J )  /\  ( F `  1 )  =  Y )  ->  P  e.  ( II  Cn  J ) )
3123, 30pcoval 21488 . . 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 3932 . . . . . . . . 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 21412 . . . . . . . . 9  |-  ( x  e.  ( 0 [,] ( 1  /  2
) )  <->  ( x  e.  ( 0 [,] 1
)  /\  x  <_  ( 1  /  2 ) ) )
35 iihalf1 21408 . . . . . . . . 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 2531 . . . . . . 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 3935 . . . . . . 7  |-  ( -.  x  <_  ( 1  /  2 )  ->  if ( x  <_  (
1  /  2 ) ,  ( 2  x.  x ) ,  1 )  =  1 )
4039, 8syl6eqel 2539 . . . . . 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 2444 . . . 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 5856 . . . . 5  |-  ( y  =  if ( x  <_  ( 1  / 
2 ) ,  ( 2  x.  x ) ,  1 )  -> 
( F `  y
)  =  ( F `
 if ( x  <_  ( 1  / 
2 ) ,  ( 2  x.  x ) ,  1 ) ) )
46 fvif 5867 . . . . 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 6049 . . 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 21360 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
5150a1i 11 . . . 4  |-  ( ( F  e.  ( II 
Cn  J )  /\  ( F `  1 )  =  Y )  ->  II  e.  (TopOn `  (
0 [,] 1 ) ) )
5251cnmptid 20139 . . . 4  |-  ( ( F  e.  ( II 
Cn  J )  /\  ( F `  1 )  =  Y )  -> 
( x  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( II  Cn  II ) )
53 0elunit 11648 . . . . . 6  |-  0  e.  ( 0 [,] 1
)
5453a1i 11 . . . . 5  |-  ( ( F  e.  ( II 
Cn  J )  /\  ( F `  1 )  =  Y )  -> 
0  e.  ( 0 [,] 1 ) )
5551, 51, 54cnmptc 20140 . . . 4  |-  ( ( F  e.  ( II 
Cn  J )  /\  ( F `  1 )  =  Y )  -> 
( x  e.  ( 0 [,] 1 ) 
|->  0 )  e.  ( II  Cn  II ) )
56 eqid 2443 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
57 eqid 2443 . . . . 5  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )
58 eqid 2443 . . . . 5  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )
59 dfii2 21363 . . . . 5  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
60 0re 9599 . . . . . 6  |-  0  e.  RR
6160a1i 11 . . . . 5  |-  ( ( F  e.  ( II 
Cn  J )  /\  ( F `  1 )  =  Y )  -> 
0  e.  RR )
62 1re 9598 . . . . . 6  |-  1  e.  RR
6362a1i 11 . . . . 5  |-  ( ( F  e.  ( II 
Cn  J )  /\  ( F `  1 )  =  Y )  -> 
1  e.  RR )
64 halfre 10761 . . . . . . 7  |-  ( 1  /  2 )  e.  RR
65 halfgt0 10763 . . . . . . . 8  |-  0  <  ( 1  /  2
)
6660, 64, 65ltleii 9710 . . . . . . 7  |-  0  <_  ( 1  /  2
)
67 halflt1 10764 . . . . . . . 8  |-  ( 1  /  2 )  <  1
6864, 62, 67ltleii 9710 . . . . . . 7  |-  ( 1  /  2 )  <_ 
1
6960, 62elicc2i 11600 . . . . . . 7  |-  ( ( 1  /  2 )  e.  ( 0 [,] 1 )  <->  ( (
1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_ 
1 ) )
7064, 66, 68, 69mpbir3an 1179 . . . . . 6  |-  ( 1  /  2 )  e.  ( 0 [,] 1
)
7170a1i 11 . . . . 5  |-  ( ( F  e.  ( II 
Cn  J )  /\  ( F `  1 )  =  Y )  -> 
( 1  /  2
)  e.  ( 0 [,] 1 ) )
72 simprl 756 . . . . . . 7  |-  ( ( ( F  e.  ( II  Cn  J )  /\  ( F ` 
1 )  =  Y )  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
y  =  ( 1  /  2 ) )
7372oveq2d 6297 . . . . . 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 10613 . . . . . . 7  |-  2  e.  CC
75 2ne0 10635 . . . . . . 7  |-  2  =/=  0
7674, 75recidi 10282 . . . . . 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 21247 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
79 iccssre 11616 . . . . . . . . 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 19639 . . . . . . . 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 20146 . . . . . 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 21409 . . . . . . 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 6289 . . . . . 6  |-  ( x  =  y  ->  (
2  x.  x )  =  ( 2  x.  y ) )
8883, 51, 84, 83, 86, 87cnmpt21 20149 . . . . 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 11616 . . . . . . . . 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 19639 . . . . . . . 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 20148 . . . . 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 21405 . . . 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 4440 . . . . . 6  |-  ( y  =  x  ->  (
y  <_  ( 1  /  2 )  <->  x  <_  ( 1  /  2 ) ) )
98 oveq2 6289 . . . . . 6  |-  ( y  =  x  ->  (
2  x.  y )  =  ( 2  x.  x ) )
9997, 98ifbieq1d 3949 . . . . 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 20145 . . 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 4474 . . . . . . 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 6289 . . . . . . 7  |-  ( x  =  0  ->  (
2  x.  x )  =  ( 2  x.  0 ) )
106 2t0e0 10698 . . . . . . 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 2443 . . . . 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 9593 . . . . 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 9708 . . . . . . . 8  |-  ( ( 1  /  2 )  <  1  <->  -.  1  <_  ( 1  /  2
) )
11467, 113mpbi 208 . . . . . . 7  |-  -.  1  <_  ( 1  /  2
)
115 breq1 4440 . . . . . . 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 9594 . . . . 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 21475 . 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 4457 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 974    = wceq 1383    e. wcel 1804    C_ wss 3461   ifcif 3926   {csn 4014   U.cuni 4234   class class class wbr 4437    |-> cmpt 4495    X. cxp 4987   ran crn 4990    o. ccom 4993   -->wf 5574   ` cfv 5578  (class class class)co 6281   RRcr 9494   0cc0 9495   1c1 9496    x. cmul 9500    < clt 9631    <_ cle 9632    - cmin 9810    / cdiv 10213   2c2 10592   (,)cioo 11539   [,]cicc 11542   ↾t crest 14799   topGenctg 14816   Topctop 19371  TopOnctopon 19372    Cn ccn 19702   IIcii 21356    ~=ph cphtpc 21446   *pcpco 21477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-uz 11092  df-q 11193  df-rp 11231  df-xneg 11328  df-xadd 11329  df-xmul 11330  df-ioo 11543  df-icc 11546  df-fz 11683  df-fzo 11806  df-seq 12089  df-exp 12148  df-hash 12387  df-cj 12913  df-re 12914  df-im 12915  df-sqrt 13049  df-abs 13050  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-mulr 14692  df-starv 14693  df-sca 14694  df-vsca 14695  df-ip 14696  df-tset 14697  df-ple 14698  df-ds 14700  df-unif 14701  df-hom 14702  df-cco 14703  df-rest 14801  df-topn 14802  df-0g 14820  df-gsum 14821  df-topgen 14822  df-pt 14823  df-prds 14826  df-xrs 14880  df-qtop 14885  df-imas 14886  df-xps 14888  df-mre 14964  df-mrc 14965  df-acs 14967  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-submnd 15945  df-mulg 16038  df-cntz 16333  df-cmn 16778  df-psmet 18389  df-xmet 18390  df-met 18391  df-bl 18392  df-mopn 18393  df-cnfld 18399  df-top 19376  df-bases 19378  df-topon 19379  df-topsp 19380  df-cld 19497  df-cn 19705  df-cnp 19706  df-tx 20040  df-hmeo 20233  df-xms 20800  df-ms 20801  df-tms 20802  df-ii 21358  df-htpy 21447  df-phtpy 21448  df-phtpc 21469  df-pco 21482
This theorem is referenced by:  pcophtb  21506  pi1xfrcnvlem  21533
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