MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pcopt2 Structured version   Visualization version   Unicode version

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
  Copyright terms: Public domain W3C validator