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Theorem reparphti 20413
Description: Lemma for reparpht 20414. (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 18714 . . 3  |-  ( ( G  e.  ( II 
Cn  II )  /\  F  e.  ( II  Cn  J ) )  -> 
( F  o.  G
)  e.  ( II 
Cn  J ) )
41, 2, 3syl2anc 656 . 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 20299 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
76a1i 11 . . . 4  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
8 eqid 2435 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
98cnfldtop 20207 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  Top
10 cnrest2r 18735 . . . . . . . . . 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 19086 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( II  tX  II )  Cn  II ) )
13 iirevcn 20346 . . . . . . . . . . . 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 6090 . . . . . . . . . . 11  |-  ( z  =  y  ->  (
1  -  z )  =  ( 1  -  y ) )
167, 7, 12, 7, 14, 15cnmpt21 19088 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( 1  -  y
) )  e.  ( ( II  tX  II )  Cn  II ) )
178dfii3 20303 . . . . . . . . . . 11  |-  II  =  ( ( TopOpen ` fld )t  ( 0 [,] 1 ) )
1817oveq2i 6093 . . . . . . . . . 10  |-  ( ( II  tX  II )  Cn  II )  =  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) )
1916, 18syl6eleq 2525 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( 1  -  y
) )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
2011, 19sseldd 3347 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( 1  -  y
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
217, 7cnmpt1st 19085 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( ( II  tX  II )  Cn  II ) )
227, 7, 21, 1cnmpt21f 19089 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( G `  x
) )  e.  ( ( II  tX  II )  Cn  II ) )
2322, 18syl6eleq 2525 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( G `  x
) )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
2411, 23sseldd 3347 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( G `  x
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
258mulcn 20287 . . . . . . . . 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 19092 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( 1  -  y )  x.  ( G `  x )
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
2812, 18syl6eleq 2525 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
2911, 28sseldd 3347 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
3021, 18syl6eleq 2525 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
3111, 30sseldd 3347 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
327, 7, 29, 31, 26cnmpt22f 19092 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( y  x.  x
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
338addcn 20285 . . . . . . . 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 19092 . . . . . 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 20206 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
3736a1i 11 . . . . . . 7  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
38 iiuni 20301 . . . . . . . . . . . . . . 15  |-  ( 0 [,] 1 )  = 
U. II
3938, 38cnf 18694 . . . . . . . . . . . . . 14  |-  ( G  e.  ( II  Cn  II )  ->  G :
( 0 [,] 1
) --> ( 0 [,] 1 ) )
401, 39syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  G : ( 0 [,] 1 ) --> ( 0 [,] 1 ) )
4140ffvelrnda 5833 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 0 [,] 1
) )  ->  ( G `  x )  e.  ( 0 [,] 1
) )
4241adantrr 711 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( G `  x
)  e.  ( 0 [,] 1 ) )
43 simprl 750 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  ->  x  e.  ( 0 [,] 1 ) )
44 simprr 751 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
y  e.  ( 0 [,] 1 ) )
45 0re 9376 . . . . . . . . . . . 12  |-  0  e.  RR
46 1re 9375 . . . . . . . . . . . 12  |-  1  e.  RR
47 icccvx 20366 . . . . . . . . . . . 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 667 . . . . . . . . . . 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 1213 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) )  e.  ( 0 [,] 1 ) )
5049ralrimivva 2800 . . . . . . . . 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 2435 . . . . . . . . . 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 6632 . . . . . . . . 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 196 . . . . . . . 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 5555 . . . . . . . 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 11421 . . . . . . . . 9  |-  ( 0 [,] 1 )  C_  RR
57 ax-resscn 9329 . . . . . . . . 9  |-  RR  C_  CC
5856, 57sstri 3355 . . . . . . . 8  |-  ( 0 [,] 1 )  C_  CC
5958a1i 11 . . . . . . 7  |-  ( ph  ->  ( 0 [,] 1
)  C_  CC )
60 cnrest2 18734 . . . . . . 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 1213 . . . . . 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 210 . . . . 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 2526 . . . 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 19089 . . 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 2519 . 2  |-  ( ph  ->  H  e.  ( ( II  tX  II )  Cn  J ) )
6640ffvelrnda 5833 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  s )  e.  ( 0 [,] 1
) )
6758, 66sseldi 3344 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  s )  e.  CC )
6867mulid2d 9394 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  x.  ( G `
 s ) )  =  ( G `  s ) )
6958sseli 3342 . . . . . . . 8  |-  ( s  e.  ( 0 [,] 1 )  ->  s  e.  CC )
7069adantl 463 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  CC )
7170mul02d 9557 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  x.  s )  =  0 )
7268, 71oveq12d 6100 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  x.  ( G `  s )
)  +  ( 0  x.  s ) )  =  ( ( G `
 s )  +  0 ) )
7367addid1d 9559 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( G `  s
)  +  0 )  =  ( G `  s ) )
7472, 73eqtrd 2467 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  x.  ( G `  s )
)  +  ( 0  x.  s ) )  =  ( G `  s ) )
7574fveq2d 5685 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
1  x.  ( G `
 s ) )  +  ( 0  x.  s ) ) )  =  ( F `  ( G `  s ) ) )
76 simpr 458 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
77 0elunit 11392 . . . 4  |-  0  e.  ( 0 [,] 1
)
78 simpr 458 . . . . . . . . . 10  |-  ( ( x  =  s  /\  y  =  0 )  ->  y  =  0 )
7978oveq2d 6098 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 1  -  y )  =  ( 1  -  0 ) )
80 1m0e1 10422 . . . . . . . . 9  |-  ( 1  -  0 )  =  1
8179, 80syl6eq 2483 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 1  -  y )  =  1 )
82 simpl 454 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  0 )  ->  x  =  s )
8382fveq2d 5685 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( G `  x )  =  ( G `  s ) )
8481, 83oveq12d 6100 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( 1  x.  ( G `
 s ) ) )
8578, 82oveq12d 6100 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( y  x.  x )  =  ( 0  x.  s ) )
8684, 85oveq12d 6100 . . . . . 6  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( 1  x.  ( G `  s )
)  +  ( 0  x.  s ) ) )
8786fveq2d 5685 . . . . 5  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( F `  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  =  ( F `  ( ( 1  x.  ( G `
 s ) )  +  ( 0  x.  s ) ) ) )
88 fvex 5691 . . . . 5  |-  ( F `
 ( ( 1  x.  ( G `  s ) )  +  ( 0  x.  s
) ) )  e. 
_V
8987, 5, 88ovmpt2a 6212 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s H 0 )  =  ( F `  ( ( 1  x.  ( G `
 s ) )  +  ( 0  x.  s ) ) ) )
9076, 77, 89sylancl 657 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 0 )  =  ( F `  ( ( 1  x.  ( G `  s
) )  +  ( 0  x.  s ) ) ) )
91 fvco3 5758 . . . 4  |-  ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  G ) `  s )  =  ( F `  ( G `
 s ) ) )
9240, 91sylan 468 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  G
) `  s )  =  ( F `  ( G `  s ) ) )
9375, 90, 923eqtr4d 2477 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 0 )  =  ( ( F  o.  G ) `  s ) )
94 1elunit 11393 . . . 4  |-  1  e.  ( 0 [,] 1
)
95 simpr 458 . . . . . . . . . 10  |-  ( ( x  =  s  /\  y  =  1 )  ->  y  =  1 )
9695oveq2d 6098 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 1  -  y )  =  ( 1  -  1 ) )
97 1m1e0 10380 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
9896, 97syl6eq 2483 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 1  -  y )  =  0 )
99 simpl 454 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  x  =  s )
10099fveq2d 5685 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( G `  x )  =  ( G `  s ) )
10198, 100oveq12d 6100 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( 0  x.  ( G `
 s ) ) )
10295, 99oveq12d 6100 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( y  x.  x )  =  ( 1  x.  s ) )
103101, 102oveq12d 6100 . . . . . 6  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( 0  x.  ( G `  s )
)  +  ( 1  x.  s ) ) )
104103fveq2d 5685 . . . . 5  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( F `  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  =  ( F `  ( ( 0  x.  ( G `
 s ) )  +  ( 1  x.  s ) ) ) )
105 fvex 5691 . . . . 5  |-  ( F `
 ( ( 0  x.  ( G `  s ) )  +  ( 1  x.  s
) ) )  e. 
_V
106104, 5, 105ovmpt2a 6212 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s H 1 )  =  ( F `  ( ( 0  x.  ( G `
 s ) )  +  ( 1  x.  s ) ) ) )
10776, 94, 106sylancl 657 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 1 )  =  ( F `  ( ( 0  x.  ( G `  s
) )  +  ( 1  x.  s ) ) ) )
10867mul02d 9557 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  x.  ( G `
 s ) )  =  0 )
10970mulid2d 9394 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  x.  s )  =  s )
110108, 109oveq12d 6100 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0  x.  ( G `  s )
)  +  ( 1  x.  s ) )  =  ( 0  +  s ) )
11170addid2d 9560 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  +  s )  =  s )
112110, 111eqtrd 2467 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0  x.  ( G `  s )
)  +  ( 1  x.  s ) )  =  s )
113112fveq2d 5685 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
0  x.  ( G `
 s ) )  +  ( 1  x.  s ) ) )  =  ( F `  s ) )
114107, 113eqtrd 2467 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 1 )  =  ( F `  s ) )
115 reparpht.4 . . . . . . . . 9  |-  ( ph  ->  ( G `  0
)  =  0 )
116115adantr 462 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  0 )  =  0 )
117116oveq2d 6098 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 0 ) )  =  ( ( 1  -  s )  x.  0 ) )
118 ax-1cn 9330 . . . . . . . . 9  |-  1  e.  CC
119 subcl 9599 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  s  e.  CC )  ->  ( 1  -  s
)  e.  CC )
120118, 70, 119sylancr 658 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  -  s )  e.  CC )
121120mul01d 9558 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  0 )  =  0 )
122117, 121eqtrd 2467 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 0 ) )  =  0 )
12370mul01d 9558 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s  x.  0 )  =  0 )
124122, 123oveq12d 6100 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  0 )
)  +  ( s  x.  0 ) )  =  ( 0  +  0 ) )
125 00id 9534 . . . . 5  |-  ( 0  +  0 )  =  0
126124, 125syl6eq 2483 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  0 )
)  +  ( s  x.  0 ) )  =  0 )
127126fveq2d 5685 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
( 1  -  s
)  x.  ( G `
 0 ) )  +  ( s  x.  0 ) ) )  =  ( F ` 
0 ) )
128 simpr 458 . . . . . . . . 9  |-  ( ( x  =  0  /\  y  =  s )  ->  y  =  s )
129128oveq2d 6098 . . . . . . . 8  |-  ( ( x  =  0  /\  y  =  s )  ->  ( 1  -  y )  =  ( 1  -  s ) )
130 simpl 454 . . . . . . . . 9  |-  ( ( x  =  0  /\  y  =  s )  ->  x  =  0 )
131130fveq2d 5685 . . . . . . . 8  |-  ( ( x  =  0  /\  y  =  s )  ->  ( G `  x )  =  ( G `  0 ) )
132129, 131oveq12d 6100 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( ( 1  -  s
)  x.  ( G `
 0 ) ) )
133128, 130oveq12d 6100 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( y  x.  x )  =  ( s  x.  0 ) )
134132, 133oveq12d 6100 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( ( 1  -  s )  x.  ( G `  0 )
)  +  ( s  x.  0 ) ) )
135134fveq2d 5685 . . . . 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 5691 . . . . 5  |-  ( F `
 ( ( ( 1  -  s )  x.  ( G ` 
0 ) )  +  ( s  x.  0 ) ) )  e. 
_V
137135, 5, 136ovmpt2a 6212 . . . 4  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 0 H s )  =  ( F `  ( ( ( 1  -  s
)  x.  ( G `
 0 ) )  +  ( s  x.  0 ) ) ) )
13877, 76, 137sylancr 658 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 H s )  =  ( F `  ( ( ( 1  -  s )  x.  ( G `  0
) )  +  ( s  x.  0 ) ) ) )
139 fvco3 5758 . . . . . 6  |-  ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  G ) `  0 )  =  ( F `  ( G `  0 )
) )
14040, 77, 139sylancl 657 . . . . 5  |-  ( ph  ->  ( ( F  o.  G ) `  0
)  =  ( F `
 ( G ` 
0 ) ) )
141115fveq2d 5685 . . . . 5  |-  ( ph  ->  ( F `  ( G `  0 )
)  =  ( F `
 0 ) )
142140, 141eqtrd 2467 . . . 4  |-  ( ph  ->  ( ( F  o.  G ) `  0
)  =  ( F `
 0 ) )
143142adantr 462 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  G
) `  0 )  =  ( F ` 
0 ) )
144127, 138, 1433eqtr4d 2477 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 H s )  =  ( ( F  o.  G ) ` 
0 ) )
145 reparpht.5 . . . . . . . . 9  |-  ( ph  ->  ( G `  1
)  =  1 )
146145adantr 462 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  1 )  =  1 )
147146oveq2d 6098 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 1 ) )  =  ( ( 1  -  s )  x.  1 ) )
148120mulid1d 9393 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  1 )  =  ( 1  -  s ) )
149147, 148eqtrd 2467 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 1 ) )  =  ( 1  -  s ) )
15070mulid1d 9393 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s  x.  1 )  =  s )
151149, 150oveq12d 6100 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  1 )
)  +  ( s  x.  1 ) )  =  ( ( 1  -  s )  +  s ) )
152 npcan 9609 . . . . . 6  |-  ( ( 1  e.  CC  /\  s  e.  CC )  ->  ( ( 1  -  s )  +  s )  =  1 )
153118, 70, 152sylancr 658 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  +  s )  =  1 )
154151, 153eqtrd 2467 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  1 )
)  +  ( s  x.  1 ) )  =  1 )
155154fveq2d 5685 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
( 1  -  s
)  x.  ( G `
 1 ) )  +  ( s  x.  1 ) ) )  =  ( F ` 
1 ) )
156 simpr 458 . . . . . . . . 9  |-  ( ( x  =  1  /\  y  =  s )  ->  y  =  s )
157156oveq2d 6098 . . . . . . . 8  |-  ( ( x  =  1  /\  y  =  s )  ->  ( 1  -  y )  =  ( 1  -  s ) )
158 simpl 454 . . . . . . . . 9  |-  ( ( x  =  1  /\  y  =  s )  ->  x  =  1 )
159158fveq2d 5685 . . . . . . . 8  |-  ( ( x  =  1  /\  y  =  s )  ->  ( G `  x )  =  ( G `  1 ) )
160157, 159oveq12d 6100 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( ( 1  -  s
)  x.  ( G `
 1 ) ) )
161156, 158oveq12d 6100 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( y  x.  x )  =  ( s  x.  1 ) )
162160, 161oveq12d 6100 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( ( 1  -  s )  x.  ( G `  1 )
)  +  ( s  x.  1 ) ) )
163162fveq2d 5685 . . . . 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 5691 . . . . 5  |-  ( F `
 ( ( ( 1  -  s )  x.  ( G ` 
1 ) )  +  ( s  x.  1 ) ) )  e. 
_V
165163, 5, 164ovmpt2a 6212 . . . 4  |-  ( ( 1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 1 H s )  =  ( F `  ( ( ( 1  -  s
)  x.  ( G `
 1 ) )  +  ( s  x.  1 ) ) ) )
16694, 76, 165sylancr 658 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 H s )  =  ( F `  ( ( ( 1  -  s )  x.  ( G `  1
) )  +  ( s  x.  1 ) ) ) )
167 fvco3 5758 . . . . . 6  |-  ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  G ) `  1 )  =  ( F `  ( G `  1 )
) )
16840, 94, 167sylancl 657 . . . . 5  |-  ( ph  ->  ( ( F  o.  G ) `  1
)  =  ( F `
 ( G ` 
1 ) ) )
169145fveq2d 5685 . . . . 5  |-  ( ph  ->  ( F `  ( G `  1 )
)  =  ( F `
 1 ) )
170168, 169eqtrd 2467 . . . 4  |-  ( ph  ->  ( ( F  o.  G ) `  1
)  =  ( F `
 1 ) )
171170adantr 462 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  G
) `  1 )  =  ( F ` 
1 ) )
172155, 166, 1713eqtr4d 2477 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 H s )  =  ( ( F  o.  G ) ` 
1 ) )
1734, 2, 65, 93, 114, 144, 172isphtpy2d 20403 1  |-  ( ph  ->  H  e.  ( ( F  o.  G ) ( PHtpy `  J ) F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1757   A.wral 2707    C_ wss 3318    e. cmpt 4340    X. cxp 4827   ran crn 4830    o. ccom 4833   -->wf 5404   ` cfv 5408  (class class class)co 6082    e. cmpt2 6084   CCcc 9270   RRcr 9271   0cc0 9272   1c1 9273    + caddc 9275    x. cmul 9277    - cmin 9585   [,]cicc 11293   ↾t crest 14344   TopOpenctopn 14345  ℂfldccnfld 17664   Topctop 18342  TopOnctopon 18343    Cn ccn 18672    tX ctx 18977   IIcii 20295   PHtpycphtpy 20384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-rep 4393  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-inf2 7837  ax-cnex 9328  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348  ax-pre-mulgt0 9349  ax-pre-sup 9350  ax-addf 9351  ax-mulf 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-pss 3334  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-tp 3872  df-op 3874  df-uni 4082  df-int 4119  df-iun 4163  df-iin 4164  df-br 4283  df-opab 4341  df-mpt 4342  df-tr 4376  df-eprel 4621  df-id 4625  df-po 4630  df-so 4631  df-fr 4668  df-se 4669  df-we 4670  df-ord 4711  df-on 4712  df-lim 4713  df-suc 4714  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-isom 5417  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6311  df-om 6468  df-1st 6568  df-2nd 6569  df-supp 6682  df-recs 6820  df-rdg 6854  df-1o 6910  df-2o 6911  df-oadd 6914  df-er 7091  df-map 7206  df-ixp 7254  df-en 7301  df-dom 7302  df-sdom 7303  df-fin 7304  df-fsupp 7611  df-fi 7651  df-sup 7681  df-oi 7714  df-card 8099  df-cda 8327  df-pnf 9410  df-mnf 9411  df-xr 9412  df-ltxr 9413  df-le 9414  df-sub 9587  df-neg 9588  df-div 9984  df-nn 10313  df-2 10370  df-3 10371  df-4 10372  df-5 10373  df-6 10374  df-7 10375  df-8 10376  df-9 10377  df-10 10378  df-n0 10570  df-z 10637  df-dec 10746  df-uz 10852  df-q 10944  df-rp 10982  df-xneg 11079  df-xadd 11080  df-xmul 11081  df-ioo 11294  df-icc 11297  df-fz 11427  df-fzo 11535  df-seq 11793  df-exp 11852  df-hash 12090  df-cj 12574  df-re 12575  df-im 12576  df-sqr 12710  df-abs 12711  df-struct 14161  df-ndx 14162  df-slot 14163  df-base 14164  df-sets 14165  df-ress 14166  df-plusg 14236  df-mulr 14237  df-starv 14238  df-sca 14239  df-vsca 14240  df-ip 14241  df-tset 14242  df-ple 14243  df-ds 14245  df-unif 14246  df-hom 14247  df-cco 14248  df-rest 14346  df-topn 14347  df-0g 14365  df-gsum 14366  df-topgen 14367  df-pt 14368  df-prds 14371  df-xrs 14425  df-qtop 14430  df-imas 14431  df-xps 14433  df-mre 14509  df-mrc 14510  df-acs 14512  df-mnd 15400  df-submnd 15450  df-mulg 15530  df-cntz 15817  df-cmn 16261  df-psmet 17655  df-xmet 17656  df-met 17657  df-bl 17658  df-mopn 17659  df-cnfld 17665  df-top 18347  df-bases 18349  df-topon 18350  df-topsp 18351  df-cn 18675  df-cnp 18676  df-tx 18979  df-hmeo 19172  df-xms 19739  df-ms 19740  df-tms 19741  df-ii 20297  df-htpy 20386  df-phtpy 20387
This theorem is referenced by:  reparpht  20414
  Copyright terms: Public domain W3C validator