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Theorem reparphti 18495
Description: Lemma for reparpht 18496. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)
Hypotheses
Ref Expression
reparpht.2  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
reparpht.3  |-  ( ph  ->  G  e.  ( II 
Cn  II ) )
reparpht.4  |-  ( ph  ->  ( G `  0
)  =  0 )
reparpht.5  |-  ( ph  ->  ( G `  1
)  =  1 )
reparphti.6  |-  H  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( F `  (
( ( 1  -  y )  x.  ( G `  x )
)  +  ( y  x.  x ) ) ) )
Assertion
Ref Expression
reparphti  |-  ( ph  ->  H  e.  ( ( F  o.  G ) ( PHtpy `  J ) F ) )
Distinct variable groups:    x, y, F    x, G, y    x, J, y    ph, x, y
Allowed substitution hints:    H( x, y)

Proof of Theorem reparphti
Dummy variables  s 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reparpht.3 . . 3  |-  ( ph  ->  G  e.  ( II 
Cn  II ) )
2 reparpht.2 . . 3  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
3 cnco 16995 . . 3  |-  ( ( G  e.  ( II 
Cn  II )  /\  F  e.  ( II  Cn  J ) )  -> 
( F  o.  G
)  e.  ( II 
Cn  J ) )
41, 2, 3syl2anc 642 . 2  |-  ( ph  ->  ( F  o.  G
)  e.  ( II 
Cn  J ) )
5 reparphti.6 . . 3  |-  H  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( F `  (
( ( 1  -  y )  x.  ( G `  x )
)  +  ( y  x.  x ) ) ) )
6 iitopon 18383 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
76a1i 10 . . . 4  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
8 eqid 2283 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
98cnfldtop 18293 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  Top
10 cnrest2r 17015 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) )  C_  (
( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
119, 10mp1i 11 . . . . . . . . 9  |-  ( ph  ->  ( ( II  tX  II )  Cn  (
( TopOpen ` fld )t  ( 0 [,] 1 ) ) ) 
C_  ( ( II 
tX  II )  Cn  ( TopOpen ` fld ) ) )
127, 7cnmpt2nd 17363 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( II  tX  II )  Cn  II ) )
13 iirevcn 18428 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 [,] 1 )  |->  ( 1  -  z ) )  e.  ( II  Cn  II )
1413a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  ( z  e.  ( 0 [,] 1 ) 
|->  ( 1  -  z
) )  e.  ( II  Cn  II ) )
15 oveq2 5866 . . . . . . . . . . 11  |-  ( z  =  y  ->  (
1  -  z )  =  ( 1  -  y ) )
167, 7, 12, 7, 14, 15cnmpt21 17365 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( 1  -  y
) )  e.  ( ( II  tX  II )  Cn  II ) )
178dfii3 18387 . . . . . . . . . . 11  |-  II  =  ( ( TopOpen ` fld )t  ( 0 [,] 1 ) )
1817oveq2i 5869 . . . . . . . . . 10  |-  ( ( II  tX  II )  Cn  II )  =  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) )
1916, 18syl6eleq 2373 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( 1  -  y
) )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
2011, 19sseldd 3181 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( 1  -  y
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
217, 7cnmpt1st 17362 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( ( II  tX  II )  Cn  II ) )
227, 7, 21, 1cnmpt21f 17366 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( G `  x
) )  e.  ( ( II  tX  II )  Cn  II ) )
2322, 18syl6eleq 2373 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( G `  x
) )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
2411, 23sseldd 3181 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( G `  x
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
258mulcn 18371 . . . . . . . . 9  |-  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
2625a1i 10 . . . . . . . 8  |-  ( ph  ->  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
277, 7, 20, 24, 26cnmpt22f 17369 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( 1  -  y )  x.  ( G `  x )
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
2812, 18syl6eleq 2373 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
2911, 28sseldd 3181 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
3021, 18syl6eleq 2373 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
3111, 30sseldd 3181 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
327, 7, 29, 31, 26cnmpt22f 17369 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( y  x.  x
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
338addcn 18369 . . . . . . . 8  |-  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
3433a1i 10 . . . . . . 7  |-  ( ph  ->  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
357, 7, 27, 32, 34cnmpt22f 17369 . . . . . 6  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
368cnfldtopon 18292 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
3736a1i 10 . . . . . . 7  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
38 iiuni 18385 . . . . . . . . . . . . . . 15  |-  ( 0 [,] 1 )  = 
U. II
3938, 38cnf 16976 . . . . . . . . . . . . . 14  |-  ( G  e.  ( II  Cn  II )  ->  G :
( 0 [,] 1
) --> ( 0 [,] 1 ) )
401, 39syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  G : ( 0 [,] 1 ) --> ( 0 [,] 1 ) )
41 ffvelrn 5663 . . . . . . . . . . . . 13  |-  ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 )  /\  x  e.  ( 0 [,] 1 ) )  ->  ( G `  x )  e.  ( 0 [,] 1 ) )
4240, 41sylan 457 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 0 [,] 1
) )  ->  ( G `  x )  e.  ( 0 [,] 1
) )
4342adantrr 697 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( G `  x
)  e.  ( 0 [,] 1 ) )
44 simprl 732 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  ->  x  e.  ( 0 [,] 1 ) )
45 simprr 733 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
y  e.  ( 0 [,] 1 ) )
46 0re 8838 . . . . . . . . . . . 12  |-  0  e.  RR
47 1re 8837 . . . . . . . . . . . 12  |-  1  e.  RR
48 icccvx 18448 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  1  e.  RR )  ->  ( ( ( G `
 x )  e.  ( 0 [,] 1
)  /\  x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  y )  x.  ( G `  x )
)  +  ( y  x.  x ) )  e.  ( 0 [,] 1 ) ) )
4946, 47, 48mp2an 653 . . . . . . . . . . 11  |-  ( ( ( G `  x
)  e.  ( 0 [,] 1 )  /\  x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  e.  ( 0 [,] 1 ) )
5043, 44, 45, 49syl3anc 1182 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) )  e.  ( 0 [,] 1 ) )
5150ralrimivva 2635 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  ( 0 [,] 1 ) A. y  e.  ( 0 [,] 1 ) ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) )  e.  ( 0 [,] 1 ) )
52 eqid 2283 . . . . . . . . . 10  |-  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( ( ( 1  -  y
)  x.  ( G `
 x ) )  +  ( y  x.  x ) ) )  =  ( x  e.  ( 0 [,] 1
) ,  y  e.  ( 0 [,] 1
)  |->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) ) )
5352fmpt2 6191 . . . . . . . . 9  |-  ( A. x  e.  ( 0 [,] 1 ) A. y  e.  ( 0 [,] 1 ) ( ( ( 1  -  y )  x.  ( G `  x )
)  +  ( y  x.  x ) )  e.  ( 0 [,] 1 )  <->  ( x  e.  ( 0 [,] 1
) ,  y  e.  ( 0 [,] 1
)  |->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) ) ) : ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) --> ( 0 [,] 1 ) )
5451, 53sylib 188 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) ) : ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) --> ( 0 [,] 1
) )
55 frn 5395 . . . . . . . 8  |-  ( ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( ( ( 1  -  y )  x.  ( G `  x )
)  +  ( y  x.  x ) ) ) : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> ( 0 [,] 1 )  ->  ran  ( x  e.  ( 0 [,] 1
) ,  y  e.  ( 0 [,] 1
)  |->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) ) )  C_  ( 0 [,] 1
) )
5654, 55syl 15 . . . . . . 7  |-  ( ph  ->  ran  ( x  e.  ( 0 [,] 1
) ,  y  e.  ( 0 [,] 1
)  |->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) ) )  C_  ( 0 [,] 1
) )
57 iccssre 10731 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  1  e.  RR )  ->  ( 0 [,] 1
)  C_  RR )
5846, 47, 57mp2an 653 . . . . . . . . 9  |-  ( 0 [,] 1 )  C_  RR
59 ax-resscn 8794 . . . . . . . . 9  |-  RR  C_  CC
6058, 59sstri 3188 . . . . . . . 8  |-  ( 0 [,] 1 )  C_  CC
6160a1i 10 . . . . . . 7  |-  ( ph  ->  ( 0 [,] 1
)  C_  CC )
62 cnrest2 17014 . . . . . . 7  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ran  ( x  e.  (
0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  C_  (
0 [,] 1 )  /\  ( 0 [,] 1 )  C_  CC )  ->  ( ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( ( ( 1  -  y
)  x.  ( G `
 x ) )  +  ( y  x.  x ) ) )  e.  ( ( II 
tX  II )  Cn  ( TopOpen ` fld ) )  <->  ( x  e.  ( 0 [,] 1
) ,  y  e.  ( 0 [,] 1
)  |->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) ) )  e.  ( ( II  tX  II )  Cn  (
( TopOpen ` fld )t  ( 0 [,] 1 ) ) ) ) )
6337, 56, 61, 62syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( x  e.  ( 0 [,] 1
) ,  y  e.  ( 0 [,] 1
)  |->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) ) )  e.  ( ( II  tX  II )  Cn  ( TopOpen
` fld
) )  <->  ( x  e.  ( 0 [,] 1
) ,  y  e.  ( 0 [,] 1
)  |->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) ) )  e.  ( ( II  tX  II )  Cn  (
( TopOpen ` fld )t  ( 0 [,] 1 ) ) ) ) )
6435, 63mpbid 201 . . . . 5  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
6564, 18syl6eleqr 2374 . . . 4  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  e.  ( ( II  tX  II )  Cn  II ) )
667, 7, 65, 2cnmpt21f 17366 . . 3  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( F `  (
( ( 1  -  y )  x.  ( G `  x )
)  +  ( y  x.  x ) ) ) )  e.  ( ( II  tX  II )  Cn  J ) )
675, 66syl5eqel 2367 . 2  |-  ( ph  ->  H  e.  ( ( II  tX  II )  Cn  J ) )
68 ffvelrn 5663 . . . . . . . . 9  |-  ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( G `  s )  e.  ( 0 [,] 1 ) )
6940, 68sylan 457 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  s )  e.  ( 0 [,] 1
) )
7060, 69sseldi 3178 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  s )  e.  CC )
7170mulid2d 8853 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  x.  ( G `
 s ) )  =  ( G `  s ) )
7260sseli 3176 . . . . . . . 8  |-  ( s  e.  ( 0 [,] 1 )  ->  s  e.  CC )
7372adantl 452 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  CC )
7473mul02d 9010 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  x.  s )  =  0 )
7571, 74oveq12d 5876 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  x.  ( G `  s )
)  +  ( 0  x.  s ) )  =  ( ( G `
 s )  +  0 ) )
7670addid1d 9012 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( G `  s
)  +  0 )  =  ( G `  s ) )
7775, 76eqtrd 2315 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  x.  ( G `  s )
)  +  ( 0  x.  s ) )  =  ( G `  s ) )
7877fveq2d 5529 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
1  x.  ( G `
 s ) )  +  ( 0  x.  s ) ) )  =  ( F `  ( G `  s ) ) )
79 simpr 447 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
80 0elunit 10754 . . . 4  |-  0  e.  ( 0 [,] 1
)
81 simpr 447 . . . . . . . . . 10  |-  ( ( x  =  s  /\  y  =  0 )  ->  y  =  0 )
8281oveq2d 5874 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 1  -  y )  =  ( 1  -  0 ) )
83 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
8483subid1i 9118 . . . . . . . . 9  |-  ( 1  -  0 )  =  1
8582, 84syl6eq 2331 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 1  -  y )  =  1 )
86 simpl 443 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  0 )  ->  x  =  s )
8786fveq2d 5529 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( G `  x )  =  ( G `  s ) )
8885, 87oveq12d 5876 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( 1  x.  ( G `
 s ) ) )
8981, 86oveq12d 5876 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( y  x.  x )  =  ( 0  x.  s ) )
9088, 89oveq12d 5876 . . . . . 6  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( 1  x.  ( G `  s )
)  +  ( 0  x.  s ) ) )
9190fveq2d 5529 . . . . 5  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( F `  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  =  ( F `  ( ( 1  x.  ( G `
 s ) )  +  ( 0  x.  s ) ) ) )
92 fvex 5539 . . . . 5  |-  ( F `
 ( ( 1  x.  ( G `  s ) )  +  ( 0  x.  s
) ) )  e. 
_V
9391, 5, 92ovmpt2a 5978 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s H 0 )  =  ( F `  ( ( 1  x.  ( G `
 s ) )  +  ( 0  x.  s ) ) ) )
9479, 80, 93sylancl 643 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 0 )  =  ( F `  ( ( 1  x.  ( G `  s
) )  +  ( 0  x.  s ) ) ) )
95 fvco3 5596 . . . 4  |-  ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  G ) `  s )  =  ( F `  ( G `
 s ) ) )
9640, 95sylan 457 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  G
) `  s )  =  ( F `  ( G `  s ) ) )
9778, 94, 963eqtr4d 2325 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 0 )  =  ( ( F  o.  G ) `  s ) )
98 1elunit 10755 . . . 4  |-  1  e.  ( 0 [,] 1
)
99 simpr 447 . . . . . . . . . 10  |-  ( ( x  =  s  /\  y  =  1 )  ->  y  =  1 )
10099oveq2d 5874 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 1  -  y )  =  ( 1  -  1 ) )
101 1m1e0 9814 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
102100, 101syl6eq 2331 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 1  -  y )  =  0 )
103 simpl 443 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  x  =  s )
104103fveq2d 5529 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( G `  x )  =  ( G `  s ) )
105102, 104oveq12d 5876 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( 0  x.  ( G `
 s ) ) )
10699, 103oveq12d 5876 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( y  x.  x )  =  ( 1  x.  s ) )
107105, 106oveq12d 5876 . . . . . 6  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( 0  x.  ( G `  s )
)  +  ( 1  x.  s ) ) )
108107fveq2d 5529 . . . . 5  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( F `  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  =  ( F `  ( ( 0  x.  ( G `
 s ) )  +  ( 1  x.  s ) ) ) )
109 fvex 5539 . . . . 5  |-  ( F `
 ( ( 0  x.  ( G `  s ) )  +  ( 1  x.  s
) ) )  e. 
_V
110108, 5, 109ovmpt2a 5978 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s H 1 )  =  ( F `  ( ( 0  x.  ( G `
 s ) )  +  ( 1  x.  s ) ) ) )
11179, 98, 110sylancl 643 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 1 )  =  ( F `  ( ( 0  x.  ( G `  s
) )  +  ( 1  x.  s ) ) ) )
11270mul02d 9010 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  x.  ( G `
 s ) )  =  0 )
11373mulid2d 8853 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  x.  s )  =  s )
114112, 113oveq12d 5876 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0  x.  ( G `  s )
)  +  ( 1  x.  s ) )  =  ( 0  +  s ) )
11573addid2d 9013 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  +  s )  =  s )
116114, 115eqtrd 2315 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0  x.  ( G `  s )
)  +  ( 1  x.  s ) )  =  s )
117116fveq2d 5529 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
0  x.  ( G `
 s ) )  +  ( 1  x.  s ) ) )  =  ( F `  s ) )
118111, 117eqtrd 2315 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 1 )  =  ( F `  s ) )
119 reparpht.4 . . . . . . . . 9  |-  ( ph  ->  ( G `  0
)  =  0 )
120119adantr 451 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  0 )  =  0 )
121120oveq2d 5874 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 0 ) )  =  ( ( 1  -  s )  x.  0 ) )
122 subcl 9051 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  s  e.  CC )  ->  ( 1  -  s
)  e.  CC )
12383, 73, 122sylancr 644 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  -  s )  e.  CC )
124123mul01d 9011 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  0 )  =  0 )
125121, 124eqtrd 2315 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 0 ) )  =  0 )
12673mul01d 9011 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s  x.  0 )  =  0 )
127125, 126oveq12d 5876 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  0 )
)  +  ( s  x.  0 ) )  =  ( 0  +  0 ) )
128 00id 8987 . . . . 5  |-  ( 0  +  0 )  =  0
129127, 128syl6eq 2331 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  0 )
)  +  ( s  x.  0 ) )  =  0 )
130129fveq2d 5529 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
( 1  -  s
)  x.  ( G `
 0 ) )  +  ( s  x.  0 ) ) )  =  ( F ` 
0 ) )
131 simpr 447 . . . . . . . . 9  |-  ( ( x  =  0  /\  y  =  s )  ->  y  =  s )
132131oveq2d 5874 . . . . . . . 8  |-  ( ( x  =  0  /\  y  =  s )  ->  ( 1  -  y )  =  ( 1  -  s ) )
133 simpl 443 . . . . . . . . 9  |-  ( ( x  =  0  /\  y  =  s )  ->  x  =  0 )
134133fveq2d 5529 . . . . . . . 8  |-  ( ( x  =  0  /\  y  =  s )  ->  ( G `  x )  =  ( G `  0 ) )
135132, 134oveq12d 5876 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( ( 1  -  s
)  x.  ( G `
 0 ) ) )
136131, 133oveq12d 5876 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( y  x.  x )  =  ( s  x.  0 ) )
137135, 136oveq12d 5876 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( ( 1  -  s )  x.  ( G `  0 )
)  +  ( s  x.  0 ) ) )
138137fveq2d 5529 . . . . 5  |-  ( ( x  =  0  /\  y  =  s )  ->  ( F `  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  =  ( F `  ( ( ( 1  -  s
)  x.  ( G `
 0 ) )  +  ( s  x.  0 ) ) ) )
139 fvex 5539 . . . . 5  |-  ( F `
 ( ( ( 1  -  s )  x.  ( G ` 
0 ) )  +  ( s  x.  0 ) ) )  e. 
_V
140138, 5, 139ovmpt2a 5978 . . . 4  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 0 H s )  =  ( F `  ( ( ( 1  -  s
)  x.  ( G `
 0 ) )  +  ( s  x.  0 ) ) ) )
14180, 79, 140sylancr 644 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 H s )  =  ( F `  ( ( ( 1  -  s )  x.  ( G `  0
) )  +  ( s  x.  0 ) ) ) )
142 fvco3 5596 . . . . . 6  |-  ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  G ) `  0 )  =  ( F `  ( G `  0 )
) )
14340, 80, 142sylancl 643 . . . . 5  |-  ( ph  ->  ( ( F  o.  G ) `  0
)  =  ( F `
 ( G ` 
0 ) ) )
144119fveq2d 5529 . . . . 5  |-  ( ph  ->  ( F `  ( G `  0 )
)  =  ( F `
 0 ) )
145143, 144eqtrd 2315 . . . 4  |-  ( ph  ->  ( ( F  o.  G ) `  0
)  =  ( F `
 0 ) )
146145adantr 451 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  G
) `  0 )  =  ( F ` 
0 ) )
147130, 141, 1463eqtr4d 2325 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 H s )  =  ( ( F  o.  G ) ` 
0 ) )
148 reparpht.5 . . . . . . . . 9  |-  ( ph  ->  ( G `  1
)  =  1 )
149148adantr 451 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  1 )  =  1 )
150149oveq2d 5874 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 1 ) )  =  ( ( 1  -  s )  x.  1 ) )
151123mulid1d 8852 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  1 )  =  ( 1  -  s ) )
152150, 151eqtrd 2315 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 1 ) )  =  ( 1  -  s ) )
15373mulid1d 8852 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s  x.  1 )  =  s )
154152, 153oveq12d 5876 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  1 )
)  +  ( s  x.  1 ) )  =  ( ( 1  -  s )  +  s ) )
155 npcan 9060 . . . . . 6  |-  ( ( 1  e.  CC  /\  s  e.  CC )  ->  ( ( 1  -  s )  +  s )  =  1 )
15683, 73, 155sylancr 644 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  +  s )  =  1 )
157154, 156eqtrd 2315 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  1 )
)  +  ( s  x.  1 ) )  =  1 )
158157fveq2d 5529 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
( 1  -  s
)  x.  ( G `
 1 ) )  +  ( s  x.  1 ) ) )  =  ( F ` 
1 ) )
159 simpr 447 . . . . . . . . 9  |-  ( ( x  =  1  /\  y  =  s )  ->  y  =  s )
160159oveq2d 5874 . . . . . . . 8  |-  ( ( x  =  1  /\  y  =  s )  ->  ( 1  -  y )  =  ( 1  -  s ) )
161 simpl 443 . . . . . . . . 9  |-  ( ( x  =  1  /\  y  =  s )  ->  x  =  1 )
162161fveq2d 5529 . . . . . . . 8  |-  ( ( x  =  1  /\  y  =  s )  ->  ( G `  x )  =  ( G `  1 ) )
163160, 162oveq12d 5876 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( ( 1  -  s
)  x.  ( G `
 1 ) ) )
164159, 161oveq12d 5876 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( y  x.  x )  =  ( s  x.  1 ) )
165163, 164oveq12d 5876 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( ( 1  -  s )  x.  ( G `  1 )
)  +  ( s  x.  1 ) ) )
166165fveq2d 5529 . . . . 5  |-  ( ( x  =  1  /\  y  =  s )  ->  ( F `  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  =  ( F `  ( ( ( 1  -  s
)  x.  ( G `
 1 ) )  +  ( s  x.  1 ) ) ) )
167 fvex 5539 . . . . 5  |-  ( F `
 ( ( ( 1  -  s )  x.  ( G ` 
1 ) )  +  ( s  x.  1 ) ) )  e. 
_V
168166, 5, 167ovmpt2a 5978 . . . 4  |-  ( ( 1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 1 H s )  =  ( F `  ( ( ( 1  -  s
)  x.  ( G `
 1 ) )  +  ( s  x.  1 ) ) ) )
16998, 79, 168sylancr 644 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 H s )  =  ( F `  ( ( ( 1  -  s )  x.  ( G `  1
) )  +  ( s  x.  1 ) ) ) )
170 fvco3 5596 . . . . . 6  |-  ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  G ) `  1 )  =  ( F `  ( G `  1 )
) )
17140, 98, 170sylancl 643 . . . . 5  |-  ( ph  ->  ( ( F  o.  G ) `  1
)  =  ( F `
 ( G ` 
1 ) ) )
172148fveq2d 5529 . . . . 5  |-  ( ph  ->  ( F `  ( G `  1 )
)  =  ( F `
 1 ) )
173171, 172eqtrd 2315 . . . 4  |-  ( ph  ->  ( ( F  o.  G ) `  1
)  =  ( F `
 1 ) )
174173adantr 451 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  G
) `  1 )  =  ( F ` 
1 ) )
175158, 169, 1743eqtr4d 2325 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 H s )  =  ( ( F  o.  G ) ` 
1 ) )
1764, 2, 67, 97, 118, 147, 175isphtpy2d 18485 1  |-  ( ph  ->  H  e.  ( ( F  o.  G ) ( PHtpy `  J ) F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152    e. cmpt 4077    X. cxp 4687   ran crn 4690    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   [,]cicc 10659   ↾t crest 13325   TopOpenctopn 13326  ℂfldccnfld 16377   Topctop 16631  TopOnctopon 16632    Cn ccn 16954    tX ctx 17255   IIcii 18379   PHtpycphtpy 18466
This theorem is referenced by:  reparpht  18496
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-cnp 16958  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887  df-ii 18381  df-htpy 18468  df-phtpy 18469
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