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Theorem reparphti 21260
Description: Lemma for reparpht 21261. (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 19561 . . 3  |-  ( ( G  e.  ( II 
Cn  II )  /\  F  e.  ( II  Cn  J ) )  -> 
( F  o.  G
)  e.  ( II 
Cn  J ) )
41, 2, 3syl2anc 661 . 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 21146 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
76a1i 11 . . . 4  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
8 eqid 2467 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
98cnfldtop 21054 . . . . . . . . . 10  |-  ( TopOpen ` fld )  e.  Top
10 cnrest2r 19582 . . . . . . . . . 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 19933 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( II  tX  II )  Cn  II ) )
13 iirevcn 21193 . . . . . . . . . . . 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 6292 . . . . . . . . . . 11  |-  ( z  =  y  ->  (
1  -  z )  =  ( 1  -  y ) )
167, 7, 12, 7, 14, 15cnmpt21 19935 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( 1  -  y
) )  e.  ( ( II  tX  II )  Cn  II ) )
178dfii3 21150 . . . . . . . . . . 11  |-  II  =  ( ( TopOpen ` fld )t  ( 0 [,] 1 ) )
1817oveq2i 6295 . . . . . . . . . 10  |-  ( ( II  tX  II )  Cn  II )  =  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) )
1916, 18syl6eleq 2565 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( 1  -  y
) )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
2011, 19sseldd 3505 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( 1  -  y
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
217, 7cnmpt1st 19932 . . . . . . . . . . 11  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( ( II  tX  II )  Cn  II ) )
227, 7, 21, 1cnmpt21f 19936 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( G `  x
) )  e.  ( ( II  tX  II )  Cn  II ) )
2322, 18syl6eleq 2565 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( G `  x
) )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
2411, 23sseldd 3505 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( G `  x
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
258mulcn 21134 . . . . . . . . 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 19939 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( 1  -  y )  x.  ( G `  x )
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
2812, 18syl6eleq 2565 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
2911, 28sseldd 3505 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
3021, 18syl6eleq 2565 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( ( II  tX  II )  Cn  ( ( TopOpen ` fld )t  (
0 [,] 1 ) ) ) )
3111, 30sseldd 3505 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
327, 7, 29, 31, 26cnmpt22f 19939 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( y  x.  x
) )  e.  ( ( II  tX  II )  Cn  ( TopOpen ` fld ) ) )
338addcn 21132 . . . . . . . 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 19939 . . . . . 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 21053 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
3736a1i 11 . . . . . . 7  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
38 iiuni 21148 . . . . . . . . . . . . . . 15  |-  ( 0 [,] 1 )  = 
U. II
3938, 38cnf 19541 . . . . . . . . . . . . . 14  |-  ( G  e.  ( II  Cn  II )  ->  G :
( 0 [,] 1
) --> ( 0 [,] 1 ) )
401, 39syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  G : ( 0 [,] 1 ) --> ( 0 [,] 1 ) )
4140ffvelrnda 6021 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( 0 [,] 1
) )  ->  ( G `  x )  e.  ( 0 [,] 1
) )
4241adantrr 716 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( G `  x
)  e.  ( 0 [,] 1 ) )
43 simprl 755 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  ->  x  e.  ( 0 [,] 1 ) )
44 simprr 756 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
y  e.  ( 0 [,] 1 ) )
45 0re 9596 . . . . . . . . . . . 12  |-  0  e.  RR
46 1re 9595 . . . . . . . . . . . 12  |-  1  e.  RR
47 icccvx 21213 . . . . . . . . . . . 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 672 . . . . . . . . . . 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 1228 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) )  e.  ( 0 [,] 1 ) )
5049ralrimivva 2885 . . . . . . . . 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 2467 . . . . . . . . . 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 6851 . . . . . . . . 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 5737 . . . . . . . 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 11667 . . . . . . . . 9  |-  ( 0 [,] 1 )  C_  RR
57 ax-resscn 9549 . . . . . . . . 9  |-  RR  C_  CC
5856, 57sstri 3513 . . . . . . . 8  |-  ( 0 [,] 1 )  C_  CC
5958a1i 11 . . . . . . 7  |-  ( ph  ->  ( 0 [,] 1
)  C_  CC )
60 cnrest2 19581 . . . . . . 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 1228 . . . . . 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 2566 . . . 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 19936 . . 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 2559 . 2  |-  ( ph  ->  H  e.  ( ( II  tX  II )  Cn  J ) )
6640ffvelrnda 6021 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  s )  e.  ( 0 [,] 1
) )
6758, 66sseldi 3502 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  s )  e.  CC )
6867mulid2d 9614 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  x.  ( G `
 s ) )  =  ( G `  s ) )
6958sseli 3500 . . . . . . . 8  |-  ( s  e.  ( 0 [,] 1 )  ->  s  e.  CC )
7069adantl 466 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  CC )
7170mul02d 9777 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  x.  s )  =  0 )
7268, 71oveq12d 6302 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  x.  ( G `  s )
)  +  ( 0  x.  s ) )  =  ( ( G `
 s )  +  0 ) )
7367addid1d 9779 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( G `  s
)  +  0 )  =  ( G `  s ) )
7472, 73eqtrd 2508 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  x.  ( G `  s )
)  +  ( 0  x.  s ) )  =  ( G `  s ) )
7574fveq2d 5870 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
1  x.  ( G `
 s ) )  +  ( 0  x.  s ) ) )  =  ( F `  ( G `  s ) ) )
76 simpr 461 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
77 0elunit 11638 . . . 4  |-  0  e.  ( 0 [,] 1
)
78 simpr 461 . . . . . . . . . 10  |-  ( ( x  =  s  /\  y  =  0 )  ->  y  =  0 )
7978oveq2d 6300 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 1  -  y )  =  ( 1  -  0 ) )
80 1m0e1 10646 . . . . . . . . 9  |-  ( 1  -  0 )  =  1
8179, 80syl6eq 2524 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 1  -  y )  =  1 )
82 simpl 457 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  0 )  ->  x  =  s )
8382fveq2d 5870 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( G `  x )  =  ( G `  s ) )
8481, 83oveq12d 6302 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( 1  x.  ( G `
 s ) ) )
8578, 82oveq12d 6302 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( y  x.  x )  =  ( 0  x.  s ) )
8684, 85oveq12d 6302 . . . . . 6  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( 1  x.  ( G `  s )
)  +  ( 0  x.  s ) ) )
8786fveq2d 5870 . . . . 5  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( F `  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  =  ( F `  ( ( 1  x.  ( G `
 s ) )  +  ( 0  x.  s ) ) ) )
88 fvex 5876 . . . . 5  |-  ( F `
 ( ( 1  x.  ( G `  s ) )  +  ( 0  x.  s
) ) )  e. 
_V
8987, 5, 88ovmpt2a 6417 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s H 0 )  =  ( F `  ( ( 1  x.  ( G `
 s ) )  +  ( 0  x.  s ) ) ) )
9076, 77, 89sylancl 662 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 0 )  =  ( F `  ( ( 1  x.  ( G `  s
) )  +  ( 0  x.  s ) ) ) )
91 fvco3 5944 . . . 4  |-  ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  G ) `  s )  =  ( F `  ( G `
 s ) ) )
9240, 91sylan 471 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  G
) `  s )  =  ( F `  ( G `  s ) ) )
9375, 90, 923eqtr4d 2518 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 0 )  =  ( ( F  o.  G ) `  s ) )
94 1elunit 11639 . . . 4  |-  1  e.  ( 0 [,] 1
)
95 simpr 461 . . . . . . . . . 10  |-  ( ( x  =  s  /\  y  =  1 )  ->  y  =  1 )
9695oveq2d 6300 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 1  -  y )  =  ( 1  -  1 ) )
97 1m1e0 10604 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
9896, 97syl6eq 2524 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 1  -  y )  =  0 )
99 simpl 457 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  x  =  s )
10099fveq2d 5870 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( G `  x )  =  ( G `  s ) )
10198, 100oveq12d 6302 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( 0  x.  ( G `
 s ) ) )
10295, 99oveq12d 6302 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( y  x.  x )  =  ( 1  x.  s ) )
103101, 102oveq12d 6302 . . . . . 6  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( 0  x.  ( G `  s )
)  +  ( 1  x.  s ) ) )
104103fveq2d 5870 . . . . 5  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( F `  ( ( ( 1  -  y )  x.  ( G `  x
) )  +  ( y  x.  x ) ) )  =  ( F `  ( ( 0  x.  ( G `
 s ) )  +  ( 1  x.  s ) ) ) )
105 fvex 5876 . . . . 5  |-  ( F `
 ( ( 0  x.  ( G `  s ) )  +  ( 1  x.  s
) ) )  e. 
_V
106104, 5, 105ovmpt2a 6417 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s H 1 )  =  ( F `  ( ( 0  x.  ( G `
 s ) )  +  ( 1  x.  s ) ) ) )
10776, 94, 106sylancl 662 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 1 )  =  ( F `  ( ( 0  x.  ( G `  s
) )  +  ( 1  x.  s ) ) ) )
10867mul02d 9777 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  x.  ( G `
 s ) )  =  0 )
10970mulid2d 9614 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  x.  s )  =  s )
110108, 109oveq12d 6302 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0  x.  ( G `  s )
)  +  ( 1  x.  s ) )  =  ( 0  +  s ) )
11170addid2d 9780 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  +  s )  =  s )
112110, 111eqtrd 2508 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0  x.  ( G `  s )
)  +  ( 1  x.  s ) )  =  s )
113112fveq2d 5870 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
0  x.  ( G `
 s ) )  +  ( 1  x.  s ) ) )  =  ( F `  s ) )
114107, 113eqtrd 2508 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s H 1 )  =  ( F `  s ) )
115 reparpht.4 . . . . . . . . 9  |-  ( ph  ->  ( G `  0
)  =  0 )
116115adantr 465 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  0 )  =  0 )
117116oveq2d 6300 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 0 ) )  =  ( ( 1  -  s )  x.  0 ) )
118 ax-1cn 9550 . . . . . . . . 9  |-  1  e.  CC
119 subcl 9819 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  s  e.  CC )  ->  ( 1  -  s
)  e.  CC )
120118, 70, 119sylancr 663 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  -  s )  e.  CC )
121120mul01d 9778 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  0 )  =  0 )
122117, 121eqtrd 2508 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 0 ) )  =  0 )
12370mul01d 9778 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s  x.  0 )  =  0 )
124122, 123oveq12d 6302 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  0 )
)  +  ( s  x.  0 ) )  =  ( 0  +  0 ) )
125 00id 9754 . . . . 5  |-  ( 0  +  0 )  =  0
126124, 125syl6eq 2524 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  0 )
)  +  ( s  x.  0 ) )  =  0 )
127126fveq2d 5870 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
( 1  -  s
)  x.  ( G `
 0 ) )  +  ( s  x.  0 ) ) )  =  ( F ` 
0 ) )
128 simpr 461 . . . . . . . . 9  |-  ( ( x  =  0  /\  y  =  s )  ->  y  =  s )
129128oveq2d 6300 . . . . . . . 8  |-  ( ( x  =  0  /\  y  =  s )  ->  ( 1  -  y )  =  ( 1  -  s ) )
130 simpl 457 . . . . . . . . 9  |-  ( ( x  =  0  /\  y  =  s )  ->  x  =  0 )
131130fveq2d 5870 . . . . . . . 8  |-  ( ( x  =  0  /\  y  =  s )  ->  ( G `  x )  =  ( G `  0 ) )
132129, 131oveq12d 6302 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( ( 1  -  s
)  x.  ( G `
 0 ) ) )
133128, 130oveq12d 6302 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( y  x.  x )  =  ( s  x.  0 ) )
134132, 133oveq12d 6302 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( ( 1  -  s )  x.  ( G `  0 )
)  +  ( s  x.  0 ) ) )
135134fveq2d 5870 . . . . 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 5876 . . . . 5  |-  ( F `
 ( ( ( 1  -  s )  x.  ( G ` 
0 ) )  +  ( s  x.  0 ) ) )  e. 
_V
137135, 5, 136ovmpt2a 6417 . . . 4  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 0 H s )  =  ( F `  ( ( ( 1  -  s
)  x.  ( G `
 0 ) )  +  ( s  x.  0 ) ) ) )
13877, 76, 137sylancr 663 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 H s )  =  ( F `  ( ( ( 1  -  s )  x.  ( G `  0
) )  +  ( s  x.  0 ) ) ) )
139 fvco3 5944 . . . . . 6  |-  ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  G ) `  0 )  =  ( F `  ( G `  0 )
) )
14040, 77, 139sylancl 662 . . . . 5  |-  ( ph  ->  ( ( F  o.  G ) `  0
)  =  ( F `
 ( G ` 
0 ) ) )
141115fveq2d 5870 . . . . 5  |-  ( ph  ->  ( F `  ( G `  0 )
)  =  ( F `
 0 ) )
142140, 141eqtrd 2508 . . . 4  |-  ( ph  ->  ( ( F  o.  G ) `  0
)  =  ( F `
 0 ) )
143142adantr 465 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  G
) `  0 )  =  ( F ` 
0 ) )
144127, 138, 1433eqtr4d 2518 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 H s )  =  ( ( F  o.  G ) ` 
0 ) )
145 reparpht.5 . . . . . . . . 9  |-  ( ph  ->  ( G `  1
)  =  1 )
146145adantr 465 . . . . . . . 8  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( G `  1 )  =  1 )
147146oveq2d 6300 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 1 ) )  =  ( ( 1  -  s )  x.  1 ) )
148120mulid1d 9613 . . . . . . 7  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  1 )  =  ( 1  -  s ) )
149147, 148eqtrd 2508 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( G `
 1 ) )  =  ( 1  -  s ) )
15070mulid1d 9613 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s  x.  1 )  =  s )
151149, 150oveq12d 6302 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  1 )
)  +  ( s  x.  1 ) )  =  ( ( 1  -  s )  +  s ) )
152 npcan 9829 . . . . . 6  |-  ( ( 1  e.  CC  /\  s  e.  CC )  ->  ( ( 1  -  s )  +  s )  =  1 )
153118, 70, 152sylancr 663 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  +  s )  =  1 )
154151, 153eqtrd 2508 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  s )  x.  ( G `  1 )
)  +  ( s  x.  1 ) )  =  1 )
155154fveq2d 5870 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( F `  ( (
( 1  -  s
)  x.  ( G `
 1 ) )  +  ( s  x.  1 ) ) )  =  ( F ` 
1 ) )
156 simpr 461 . . . . . . . . 9  |-  ( ( x  =  1  /\  y  =  s )  ->  y  =  s )
157156oveq2d 6300 . . . . . . . 8  |-  ( ( x  =  1  /\  y  =  s )  ->  ( 1  -  y )  =  ( 1  -  s ) )
158 simpl 457 . . . . . . . . 9  |-  ( ( x  =  1  /\  y  =  s )  ->  x  =  1 )
159158fveq2d 5870 . . . . . . . 8  |-  ( ( x  =  1  /\  y  =  s )  ->  ( G `  x )  =  ( G `  1 ) )
160157, 159oveq12d 6302 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( 1  -  y )  x.  ( G `  x
) )  =  ( ( 1  -  s
)  x.  ( G `
 1 ) ) )
161156, 158oveq12d 6302 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( y  x.  x )  =  ( s  x.  1 ) )
162160, 161oveq12d 6302 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  ( y  x.  x
) )  =  ( ( ( 1  -  s )  x.  ( G `  1 )
)  +  ( s  x.  1 ) ) )
163162fveq2d 5870 . . . . 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 5876 . . . . 5  |-  ( F `
 ( ( ( 1  -  s )  x.  ( G ` 
1 ) )  +  ( s  x.  1 ) ) )  e. 
_V
165163, 5, 164ovmpt2a 6417 . . . 4  |-  ( ( 1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 1 H s )  =  ( F `  ( ( ( 1  -  s
)  x.  ( G `
 1 ) )  +  ( s  x.  1 ) ) ) )
16694, 76, 165sylancr 663 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 H s )  =  ( F `  ( ( ( 1  -  s )  x.  ( G `  1
) )  +  ( s  x.  1 ) ) ) )
167 fvco3 5944 . . . . . 6  |-  ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  ( ( F  o.  G ) `  1 )  =  ( F `  ( G `  1 )
) )
16840, 94, 167sylancl 662 . . . . 5  |-  ( ph  ->  ( ( F  o.  G ) `  1
)  =  ( F `
 ( G ` 
1 ) ) )
169145fveq2d 5870 . . . . 5  |-  ( ph  ->  ( F `  ( G `  1 )
)  =  ( F `
 1 ) )
170168, 169eqtrd 2508 . . . 4  |-  ( ph  ->  ( ( F  o.  G ) `  1
)  =  ( F `
 1 ) )
171170adantr 465 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F  o.  G
) `  1 )  =  ( F ` 
1 ) )
172155, 166, 1713eqtr4d 2518 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 H s )  =  ( ( F  o.  G ) ` 
1 ) )
1734, 2, 65, 93, 114, 144, 172isphtpy2d 21250 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 973    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476    |-> cmpt 4505    X. cxp 4997   ran crn 5000    o. ccom 5003   -->wf 5584   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    - cmin 9805   [,]cicc 11532   ↾t crest 14676   TopOpenctopn 14677  ℂfldccnfld 18219   Topctop 19189  TopOnctopon 19190    Cn ccn 19519    tX ctx 19824   IIcii 21142   PHtpycphtpy 21231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-icc 11536  df-fz 11673  df-fzo 11793  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cn 19522  df-cnp 19523  df-tx 19826  df-hmeo 20019  df-xms 20586  df-ms 20587  df-tms 20588  df-ii 21144  df-htpy 21233  df-phtpy 21234
This theorem is referenced by:  reparpht  21261
  Copyright terms: Public domain W3C validator