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Theorem reparphti 18975
Description: Lemma for reparpht 18976. (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 17284 . . 3  |-  ( ( G  e.  ( II 
Cn  II )  /\  F  e.  ( II  Cn  J ) )  -> 
( F  o.  G
)  e.  ( II 
Cn  J ) )
41, 2, 3syl2anc 643 . 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 18862 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
76a1i 11 . . . 4  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
8 eqid 2404 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
98cnfldtop 18771 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  Top
10 cnrest2r 17305 . . . . . . . . . 10  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) )  C_  (
( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
119, 10mp1i 12 . . . . . . . . 9  |-  ( ph  ->  ( ( II  tX  II )  Cn  (
( TopOpen ` fld )t  ( 0 [,] 1 ) ) ) 
C_  ( ( II 
tX  II )  Cn  ( TopOpen ` fld ) ) )
127, 7cnmpt2nd 17654 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( II  tX  II )  Cn  II ) )
13 iirevcn 18908 . . . . . . . . . . . 12  |-  ( z  e.  ( 0 [,] 1 )  |->  ( 1  -  z ) )  e.  ( II  Cn  II )
1413a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( z  e.  ( 0 [,] 1 ) 
|->  ( 1  -  z
) )  e.  ( II  Cn  II ) )
15 oveq2 6048 . . . . . . . . . . 11  |-  ( z  =  y  ->  (
1  -  z )  =  ( 1  -  y ) )
167, 7, 12, 7, 14, 15cnmpt21 17656 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( 1  -  y
) )  e.  ( ( II  tX  II )  Cn  II ) )
178dfii3 18866 . . . . . . . . . . 11  |-  II  =  ( ( TopOpen ` fld )t  ( 0 [,] 1 ) )
1817oveq2i 6051 . . . . . . . . . 10  |-  ( ( II  tX  II )  Cn  II )  =  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) )
1916, 18syl6eleq 2494 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( 1  -  y
) )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
2011, 19sseldd 3309 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( 1  -  y
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
217, 7cnmpt1st 17653 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( ( II  tX  II )  Cn  II ) )
227, 7, 21, 1cnmpt21f 17657 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( G `  x
) )  e.  ( ( II  tX  II )  Cn  II ) )
2322, 18syl6eleq 2494 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( G `  x
) )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
2411, 23sseldd 3309 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( G `  x
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
258mulcn 18850 . . . . . . . . 9  |-  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
2625a1i 11 . . . . . . . 8  |-  ( ph  ->  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
277, 7, 20, 24, 26cnmpt22f 17660 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( 1  -  y )  x.  ( G `  x )
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
2812, 18syl6eleq 2494 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
2911, 28sseldd 3309 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
3021, 18syl6eleq 2494 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
3111, 30sseldd 3309 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
327, 7, 29, 31, 26cnmpt22f 17660 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( y  x.  x
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
338addcn 18848 . . . . . . . 8  |-  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
3433a1i 11 . . . . . . 7  |-  ( ph  ->  +  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld )
)  Cn  ( TopOpen ` fld )
) )
357, 7, 27, 32, 34cnmpt22f 17660 . . . . . 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 18770 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
3736a1i 11 . . . . . . 7  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
38 iiuni 18864 . . . . . . . . . . . . . . 15  |-  ( 0 [,] 1 )  = 
U. II
3938, 38cnf 17264 . . . . . . . . . . . . . 14  |-  ( G  e.  ( II  Cn  II )  ->  G :
( 0 [,] 1
) --> ( 0 [,] 1 ) )
401, 39syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  G : ( 0 [,] 1 ) --> ( 0 [,] 1 ) )
4140ffvelrnda 5829 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 0 [,] 1
) )  ->  ( G `  x )  e.  ( 0 [,] 1
) )
4241adantrr 698 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( G `  x
)  e.  ( 0 [,] 1 ) )
43 simprl 733 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  ->  x  e.  ( 0 [,] 1 ) )
44 simprr 734 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
y  e.  ( 0 [,] 1 ) )
45 0re 9047 . . . . . . . . . . . 12  |-  0  e.  RR
46 1re 9046 . . . . . . . . . . . 12  |-  1  e.  RR
47 icccvx 18928 . . . . . . . . . . . 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 ) ) )
4845, 46, 47mp2an 654 . . . . . . . . . . 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 ) )
4942, 43, 44, 48syl3anc 1184 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) )  e.  ( 0 [,] 1 ) )
5049ralrimivva 2758 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  ( 0 [,] 1 ) A. y  e.  ( 0 [,] 1 ) ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) )  e.  ( 0 [,] 1 ) )
51 eqid 2404 . . . . . . . . . 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
) ) )
5251fmpt2 6377 . . . . . . . . 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 ) )
5350, 52sylib 189 . . . . . . . 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
) )
54 frn 5556 . . . . . . . 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
) )
5553, 54syl 16 . . . . . . 7  |-  ( ph  ->  ran  ( x  e.  ( 0 [,] 1
) ,  y  e.  ( 0 [,] 1
)  |->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) ) )  C_  ( 0 [,] 1
) )
56 unitssre 10998 . . . . . . . . 9  |-  ( 0 [,] 1 )  C_  RR
57 ax-resscn 9003 . . . . . . . . 9  |-  RR  C_  CC
5856, 57sstri 3317 . . . . . . . 8  |-  ( 0 [,] 1 )  C_  CC
5958a1i 11 . . . . . . 7  |-  ( ph  ->  ( 0 [,] 1
)  C_  CC )
60 cnrest2 17304 . . . . . . 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 ) ) ) ) )
6137, 55, 59, 60syl3anc 1184 . . . . . 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 ) ) ) ) )
6235, 61mpbid 202 . . . . 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 ) ) ) )
6362, 18syl6eleqr 2495 . . . 4  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  e.  ( ( II  tX  II )  Cn  II ) )
647, 7, 63, 2cnmpt21f 17657 . . 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 ) )
655, 64syl5eqel 2488 . 2  |-  ( ph  ->  H  e.  ( ( II  tX  II )  Cn  J ) )
6640ffvelrnda 5829 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  s )  e.  ( 0 [,] 1
) )
6758, 66sseldi 3306 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  s )  e.  CC )
6867mulid2d 9062 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  x.  ( G `
 s ) )  =  ( G `  s ) )
6958sseli 3304 . . . . . . . 8  |-  ( s  e.  ( 0 [,] 1 )  ->  s  e.  CC )
7069adantl 453 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  CC )
7170mul02d 9220 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  x.  s )  =  0 )
7268, 71oveq12d 6058 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  x.  ( G `  s )
)  +  ( 0  x.  s ) )  =  ( ( G `
 s )  +  0 ) )
7367addid1d 9222 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( G `  s
)  +  0 )  =  ( G `  s ) )
7472, 73eqtrd 2436 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  x.  ( G `  s )
)  +  ( 0  x.  s ) )  =  ( G `  s ) )
7574fveq2d 5691 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
1  x.  ( G `
 s ) )  +  ( 0  x.  s ) ) )  =  ( F `  ( G `  s ) ) )
76 simpr 448 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
77 0elunit 10971 . . . 4  |-  0  e.  ( 0 [,] 1
)
78 simpr 448 . . . . . . . . . 10  |-  ( ( x  =  s  /\  y  =  0 )  ->  y  =  0 )
7978oveq2d 6056 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 1  -  y )  =  ( 1  -  0 ) )
80 ax-1cn 9004 . . . . . . . . . 10  |-  1  e.  CC
8180subid1i 9328 . . . . . . . . 9  |-  ( 1  -  0 )  =  1
8279, 81syl6eq 2452 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 1  -  y )  =  1 )
83 simpl 444 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  0 )  ->  x  =  s )
8483fveq2d 5691 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( G `  x )  =  ( G `  s ) )
8582, 84oveq12d 6058 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( 1  x.  ( G `
 s ) ) )
8678, 83oveq12d 6058 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( y  x.  x )  =  ( 0  x.  s ) )
8785, 86oveq12d 6058 . . . . . 6  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( 1  x.  ( G `  s )
)  +  ( 0  x.  s ) ) )
8887fveq2d 5691 . . . . 5  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( F `  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  =  ( F `  ( ( 1  x.  ( G `
 s ) )  +  ( 0  x.  s ) ) ) )
89 fvex 5701 . . . . 5  |-  ( F `
 ( ( 1  x.  ( G `  s ) )  +  ( 0  x.  s
) ) )  e. 
_V
9088, 5, 89ovmpt2a 6163 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s H 0 )  =  ( F `  ( ( 1  x.  ( G `
 s ) )  +  ( 0  x.  s ) ) ) )
9176, 77, 90sylancl 644 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 0 )  =  ( F `  ( ( 1  x.  ( G `  s
) )  +  ( 0  x.  s ) ) ) )
92 fvco3 5759 . . . 4  |-  ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  G ) `  s )  =  ( F `  ( G `
 s ) ) )
9340, 92sylan 458 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  G
) `  s )  =  ( F `  ( G `  s ) ) )
9475, 91, 933eqtr4d 2446 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 0 )  =  ( ( F  o.  G ) `  s ) )
95 1elunit 10972 . . . 4  |-  1  e.  ( 0 [,] 1
)
96 simpr 448 . . . . . . . . . 10  |-  ( ( x  =  s  /\  y  =  1 )  ->  y  =  1 )
9796oveq2d 6056 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 1  -  y )  =  ( 1  -  1 ) )
98 1m1e0 10024 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
9997, 98syl6eq 2452 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 1  -  y )  =  0 )
100 simpl 444 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  x  =  s )
101100fveq2d 5691 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( G `  x )  =  ( G `  s ) )
10299, 101oveq12d 6058 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( 0  x.  ( G `
 s ) ) )
10396, 100oveq12d 6058 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( y  x.  x )  =  ( 1  x.  s ) )
104102, 103oveq12d 6058 . . . . . 6  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( 0  x.  ( G `  s )
)  +  ( 1  x.  s ) ) )
105104fveq2d 5691 . . . . 5  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( F `  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  =  ( F `  ( ( 0  x.  ( G `
 s ) )  +  ( 1  x.  s ) ) ) )
106 fvex 5701 . . . . 5  |-  ( F `
 ( ( 0  x.  ( G `  s ) )  +  ( 1  x.  s
) ) )  e. 
_V
107105, 5, 106ovmpt2a 6163 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s H 1 )  =  ( F `  ( ( 0  x.  ( G `
 s ) )  +  ( 1  x.  s ) ) ) )
10876, 95, 107sylancl 644 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 1 )  =  ( F `  ( ( 0  x.  ( G `  s
) )  +  ( 1  x.  s ) ) ) )
10967mul02d 9220 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  x.  ( G `
 s ) )  =  0 )
11070mulid2d 9062 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  x.  s )  =  s )
111109, 110oveq12d 6058 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0  x.  ( G `  s )
)  +  ( 1  x.  s ) )  =  ( 0  +  s ) )
11270addid2d 9223 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  +  s )  =  s )
113111, 112eqtrd 2436 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0  x.  ( G `  s )
)  +  ( 1  x.  s ) )  =  s )
114113fveq2d 5691 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
0  x.  ( G `
 s ) )  +  ( 1  x.  s ) ) )  =  ( F `  s ) )
115108, 114eqtrd 2436 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 1 )  =  ( F `  s ) )
116 reparpht.4 . . . . . . . . 9  |-  ( ph  ->  ( G `  0
)  =  0 )
117116adantr 452 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  0 )  =  0 )
118117oveq2d 6056 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 0 ) )  =  ( ( 1  -  s )  x.  0 ) )
119 subcl 9261 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  s  e.  CC )  ->  ( 1  -  s
)  e.  CC )
12080, 70, 119sylancr 645 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  -  s )  e.  CC )
121120mul01d 9221 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  0 )  =  0 )
122118, 121eqtrd 2436 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 0 ) )  =  0 )
12370mul01d 9221 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s  x.  0 )  =  0 )
124122, 123oveq12d 6058 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  0 )
)  +  ( s  x.  0 ) )  =  ( 0  +  0 ) )
125 00id 9197 . . . . 5  |-  ( 0  +  0 )  =  0
126124, 125syl6eq 2452 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  0 )
)  +  ( s  x.  0 ) )  =  0 )
127126fveq2d 5691 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
( 1  -  s
)  x.  ( G `
 0 ) )  +  ( s  x.  0 ) ) )  =  ( F ` 
0 ) )
128 simpr 448 . . . . . . . . 9  |-  ( ( x  =  0  /\  y  =  s )  ->  y  =  s )
129128oveq2d 6056 . . . . . . . 8  |-  ( ( x  =  0  /\  y  =  s )  ->  ( 1  -  y )  =  ( 1  -  s ) )
130 simpl 444 . . . . . . . . 9  |-  ( ( x  =  0  /\  y  =  s )  ->  x  =  0 )
131130fveq2d 5691 . . . . . . . 8  |-  ( ( x  =  0  /\  y  =  s )  ->  ( G `  x )  =  ( G `  0 ) )
132129, 131oveq12d 6058 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( ( 1  -  s
)  x.  ( G `
 0 ) ) )
133128, 130oveq12d 6058 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( y  x.  x )  =  ( s  x.  0 ) )
134132, 133oveq12d 6058 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( ( 1  -  s )  x.  ( G `  0 )
)  +  ( s  x.  0 ) ) )
135134fveq2d 5691 . . . . 5  |-  ( ( x  =  0  /\  y  =  s )  ->  ( F `  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  =  ( F `  ( ( ( 1  -  s
)  x.  ( G `
 0 ) )  +  ( s  x.  0 ) ) ) )
136 fvex 5701 . . . . 5  |-  ( F `
 ( ( ( 1  -  s )  x.  ( G ` 
0 ) )  +  ( s  x.  0 ) ) )  e. 
_V
137135, 5, 136ovmpt2a 6163 . . . 4  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 0 H s )  =  ( F `  ( ( ( 1  -  s
)  x.  ( G `
 0 ) )  +  ( s  x.  0 ) ) ) )
13877, 76, 137sylancr 645 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 H s )  =  ( F `  ( ( ( 1  -  s )  x.  ( G `  0
) )  +  ( s  x.  0 ) ) ) )
139 fvco3 5759 . . . . . 6  |-  ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  G ) `  0 )  =  ( F `  ( G `  0 )
) )
14040, 77, 139sylancl 644 . . . . 5  |-  ( ph  ->  ( ( F  o.  G ) `  0
)  =  ( F `
 ( G ` 
0 ) ) )
141116fveq2d 5691 . . . . 5  |-  ( ph  ->  ( F `  ( G `  0 )
)  =  ( F `
 0 ) )
142140, 141eqtrd 2436 . . . 4  |-  ( ph  ->  ( ( F  o.  G ) `  0
)  =  ( F `
 0 ) )
143142adantr 452 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  G
) `  0 )  =  ( F ` 
0 ) )
144127, 138, 1433eqtr4d 2446 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 H s )  =  ( ( F  o.  G ) ` 
0 ) )
145 reparpht.5 . . . . . . . . 9  |-  ( ph  ->  ( G `  1
)  =  1 )
146145adantr 452 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  1 )  =  1 )
147146oveq2d 6056 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 1 ) )  =  ( ( 1  -  s )  x.  1 ) )
148120mulid1d 9061 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  1 )  =  ( 1  -  s ) )
149147, 148eqtrd 2436 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 1 ) )  =  ( 1  -  s ) )
15070mulid1d 9061 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s  x.  1 )  =  s )
151149, 150oveq12d 6058 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  1 )
)  +  ( s  x.  1 ) )  =  ( ( 1  -  s )  +  s ) )
152 npcan 9270 . . . . . 6  |-  ( ( 1  e.  CC  /\  s  e.  CC )  ->  ( ( 1  -  s )  +  s )  =  1 )
15380, 70, 152sylancr 645 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  +  s )  =  1 )
154151, 153eqtrd 2436 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  1 )
)  +  ( s  x.  1 ) )  =  1 )
155154fveq2d 5691 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
( 1  -  s
)  x.  ( G `
 1 ) )  +  ( s  x.  1 ) ) )  =  ( F ` 
1 ) )
156 simpr 448 . . . . . . . . 9  |-  ( ( x  =  1  /\  y  =  s )  ->  y  =  s )
157156oveq2d 6056 . . . . . . . 8  |-  ( ( x  =  1  /\  y  =  s )  ->  ( 1  -  y )  =  ( 1  -  s ) )
158 simpl 444 . . . . . . . . 9  |-  ( ( x  =  1  /\  y  =  s )  ->  x  =  1 )
159158fveq2d 5691 . . . . . . . 8  |-  ( ( x  =  1  /\  y  =  s )  ->  ( G `  x )  =  ( G `  1 ) )
160157, 159oveq12d 6058 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( ( 1  -  s
)  x.  ( G `
 1 ) ) )
161156, 158oveq12d 6058 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( y  x.  x )  =  ( s  x.  1 ) )
162160, 161oveq12d 6058 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( ( 1  -  s )  x.  ( G `  1 )
)  +  ( s  x.  1 ) ) )
163162fveq2d 5691 . . . . 5  |-  ( ( x  =  1  /\  y  =  s )  ->  ( F `  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  =  ( F `  ( ( ( 1  -  s
)  x.  ( G `
 1 ) )  +  ( s  x.  1 ) ) ) )
164 fvex 5701 . . . . 5  |-  ( F `
 ( ( ( 1  -  s )  x.  ( G ` 
1 ) )  +  ( s  x.  1 ) ) )  e. 
_V
165163, 5, 164ovmpt2a 6163 . . . 4  |-  ( ( 1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 1 H s )  =  ( F `  ( ( ( 1  -  s
)  x.  ( G `
 1 ) )  +  ( s  x.  1 ) ) ) )
16695, 76, 165sylancr 645 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 H s )  =  ( F `  ( ( ( 1  -  s )  x.  ( G `  1
) )  +  ( s  x.  1 ) ) ) )
167 fvco3 5759 . . . . . 6  |-  ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  G ) `  1 )  =  ( F `  ( G `  1 )
) )
16840, 95, 167sylancl 644 . . . . 5  |-  ( ph  ->  ( ( F  o.  G ) `  1
)  =  ( F `
 ( G ` 
1 ) ) )
169145fveq2d 5691 . . . . 5  |-  ( ph  ->  ( F `  ( G `  1 )
)  =  ( F `
 1 ) )
170168, 169eqtrd 2436 . . . 4  |-  ( ph  ->  ( ( F  o.  G ) `  1
)  =  ( F `
 1 ) )
171170adantr 452 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  G
) `  1 )  =  ( F ` 
1 ) )
172155, 166, 1713eqtr4d 2446 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 H s )  =  ( ( F  o.  G ) ` 
1 ) )
1734, 2, 65, 94, 115, 144, 172isphtpy2d 18965 1  |-  ( ph  ->  H  e.  ( ( F  o.  G ) ( PHtpy `  J ) F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666    C_ wss 3280    e. cmpt 4226    X. cxp 4835   ran crn 4838    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247   [,]cicc 10875   ↾t crest 13603   TopOpenctopn 13604  ℂfldccnfld 16658   Topctop 16913  TopOnctopon 16914    Cn ccn 17242    tX ctx 17545   IIcii 18858   PHtpycphtpy 18946
This theorem is referenced by:  reparpht  18976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cn 17245  df-cnp 17246  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305  df-ii 18860  df-htpy 18948  df-phtpy 18949
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