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Theorem pcohtpylem 22050
Description: Lemma for pcohtpy 22051. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)
Hypotheses
Ref Expression
pcohtpy.4  |-  ( ph  ->  ( F `  1
)  =  ( G `
 0 ) )
pcohtpy.5  |-  ( ph  ->  F (  ~=ph  `  J
) H )
pcohtpy.6  |-  ( ph  ->  G (  ~=ph  `  J
) K )
pcohtpylem.7  |-  P  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( ( 2  x.  x ) M y ) ,  ( ( ( 2  x.  x )  - 
1 ) N y ) ) )
pcohtpylem.8  |-  ( ph  ->  M  e.  ( F ( PHtpy `  J ) H ) )
pcohtpylem.9  |-  ( ph  ->  N  e.  ( G ( PHtpy `  J ) K ) )
Assertion
Ref Expression
pcohtpylem  |-  ( ph  ->  P  e.  ( ( F ( *p `  J ) G ) ( PHtpy `  J )
( H ( *p
`  J ) K ) ) )
Distinct variable groups:    x, y, F    x, M, y    x, N, y    ph, x, y   
x, G, y    x, H, y    x, J, y   
x, K, y
Allowed substitution hints:    P( x, y)

Proof of Theorem pcohtpylem
Dummy variables  s 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pcohtpy.5 . . . . 5  |-  ( ph  ->  F (  ~=ph  `  J
) H )
2 isphtpc 22025 . . . . 5  |-  ( F (  ~=ph  `  J ) H  <->  ( F  e.  ( II  Cn  J
)  /\  H  e.  ( II  Cn  J
)  /\  ( F
( PHtpy `  J ) H )  =/=  (/) ) )
31, 2sylib 200 . . . 4  |-  ( ph  ->  ( F  e.  ( II  Cn  J )  /\  H  e.  ( II  Cn  J )  /\  ( F (
PHtpy `  J ) H )  =/=  (/) ) )
43simp1d 1020 . . 3  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
5 pcohtpy.6 . . . . 5  |-  ( ph  ->  G (  ~=ph  `  J
) K )
6 isphtpc 22025 . . . . 5  |-  ( G (  ~=ph  `  J ) K  <->  ( G  e.  ( II  Cn  J
)  /\  K  e.  ( II  Cn  J
)  /\  ( G
( PHtpy `  J ) K )  =/=  (/) ) )
75, 6sylib 200 . . . 4  |-  ( ph  ->  ( G  e.  ( II  Cn  J )  /\  K  e.  ( II  Cn  J )  /\  ( G (
PHtpy `  J ) K )  =/=  (/) ) )
87simp1d 1020 . . 3  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
9 pcohtpy.4 . . 3  |-  ( ph  ->  ( F `  1
)  =  ( G `
 0 ) )
104, 8, 9pcocn 22048 . 2  |-  ( ph  ->  ( F ( *p
`  J ) G )  e.  ( II 
Cn  J ) )
113simp2d 1021 . . 3  |-  ( ph  ->  H  e.  ( II 
Cn  J ) )
127simp2d 1021 . . 3  |-  ( ph  ->  K  e.  ( II 
Cn  J ) )
13 pcohtpylem.8 . . . . . 6  |-  ( ph  ->  M  e.  ( F ( PHtpy `  J ) H ) )
144, 11, 13phtpy01 22016 . . . . 5  |-  ( ph  ->  ( ( F ` 
0 )  =  ( H `  0 )  /\  ( F ` 
1 )  =  ( H `  1 ) ) )
1514simprd 465 . . . 4  |-  ( ph  ->  ( F `  1
)  =  ( H `
 1 ) )
16 pcohtpylem.9 . . . . . 6  |-  ( ph  ->  N  e.  ( G ( PHtpy `  J ) K ) )
178, 12, 16phtpy01 22016 . . . . 5  |-  ( ph  ->  ( ( G ` 
0 )  =  ( K `  0 )  /\  ( G ` 
1 )  =  ( K `  1 ) ) )
1817simpld 461 . . . 4  |-  ( ph  ->  ( G `  0
)  =  ( K `
 0 ) )
199, 15, 183eqtr3d 2493 . . 3  |-  ( ph  ->  ( H `  1
)  =  ( K `
 0 ) )
2011, 12, 19pcocn 22048 . 2  |-  ( ph  ->  ( H ( *p
`  J ) K )  e.  ( II 
Cn  J ) )
21 pcohtpylem.7 . . 3  |-  P  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( ( 2  x.  x ) M y ) ,  ( ( ( 2  x.  x )  - 
1 ) N y ) ) )
22 eqid 2451 . . . 4  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
23 eqid 2451 . . . 4  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )
24 eqid 2451 . . . 4  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )
25 dfii2 21914 . . . 4  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
26 0red 9644 . . . 4  |-  ( ph  ->  0  e.  RR )
27 1red 9658 . . . 4  |-  ( ph  ->  1  e.  RR )
28 halfre 10828 . . . . . 6  |-  ( 1  /  2 )  e.  RR
29 0re 9643 . . . . . . 7  |-  0  e.  RR
30 halfgt0 10830 . . . . . . 7  |-  0  <  ( 1  /  2
)
3129, 28, 30ltleii 9757 . . . . . 6  |-  0  <_  ( 1  /  2
)
32 1re 9642 . . . . . . 7  |-  1  e.  RR
33 halflt1 10831 . . . . . . 7  |-  ( 1  /  2 )  <  1
3428, 32, 33ltleii 9757 . . . . . 6  |-  ( 1  /  2 )  <_ 
1
3529, 32elicc2i 11700 . . . . . 6  |-  ( ( 1  /  2 )  e.  ( 0 [,] 1 )  <->  ( (
1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_ 
1 ) )
3628, 31, 34, 35mpbir3an 1190 . . . . 5  |-  ( 1  /  2 )  e.  ( 0 [,] 1
)
3736a1i 11 . . . 4  |-  ( ph  ->  ( 1  /  2
)  e.  ( 0 [,] 1 ) )
38 iitopon 21911 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3938a1i 11 . . . 4  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
409adantr 467 . . . . . 6  |-  ( (
ph  /\  ( x  =  ( 1  / 
2 )  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( F `  1
)  =  ( G `
 0 ) )
414, 11, 13phtpyi 22015 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( 0 [,] 1
) )  ->  (
( 0 M y )  =  ( F `
 0 )  /\  ( 1 M y )  =  ( F `
 1 ) ) )
4241simprd 465 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( 0 [,] 1
) )  ->  (
1 M y )  =  ( F ` 
1 ) )
4342adantrl 722 . . . . . 6  |-  ( (
ph  /\  ( x  =  ( 1  / 
2 )  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( 1 M y )  =  ( F `
 1 ) )
448, 12, 16phtpyi 22015 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( 0 [,] 1
) )  ->  (
( 0 N y )  =  ( G `
 0 )  /\  ( 1 N y )  =  ( G `
 1 ) ) )
4544simpld 461 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( 0 [,] 1
) )  ->  (
0 N y )  =  ( G ` 
0 ) )
4645adantrl 722 . . . . . 6  |-  ( (
ph  /\  ( x  =  ( 1  / 
2 )  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( 0 N y )  =  ( G `
 0 ) )
4740, 43, 463eqtr4d 2495 . . . . 5  |-  ( (
ph  /\  ( x  =  ( 1  / 
2 )  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( 1 M y )  =  ( 0 N y ) )
48 simprl 764 . . . . . . . 8  |-  ( (
ph  /\  ( x  =  ( 1  / 
2 )  /\  y  e.  ( 0 [,] 1
) ) )  ->  x  =  ( 1  /  2 ) )
4948oveq2d 6306 . . . . . . 7  |-  ( (
ph  /\  ( x  =  ( 1  / 
2 )  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( 2  x.  x
)  =  ( 2  x.  ( 1  / 
2 ) ) )
50 2cn 10680 . . . . . . . 8  |-  2  e.  CC
51 2ne0 10702 . . . . . . . 8  |-  2  =/=  0
5250, 51recidi 10338 . . . . . . 7  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
5349, 52syl6eq 2501 . . . . . 6  |-  ( (
ph  /\  ( x  =  ( 1  / 
2 )  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( 2  x.  x
)  =  1 )
5453oveq1d 6305 . . . . 5  |-  ( (
ph  /\  ( x  =  ( 1  / 
2 )  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( ( 2  x.  x ) M y )  =  ( 1 M y ) )
5553oveq1d 6305 . . . . . . 7  |-  ( (
ph  /\  ( x  =  ( 1  / 
2 )  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( ( 2  x.  x )  -  1 )  =  ( 1  -  1 ) )
56 1m1e0 10678 . . . . . . 7  |-  ( 1  -  1 )  =  0
5755, 56syl6eq 2501 . . . . . 6  |-  ( (
ph  /\  ( x  =  ( 1  / 
2 )  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( ( 2  x.  x )  -  1 )  =  0 )
5857oveq1d 6305 . . . . 5  |-  ( (
ph  /\  ( x  =  ( 1  / 
2 )  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( ( ( 2  x.  x )  - 
1 ) N y )  =  ( 0 N y ) )
5947, 54, 583eqtr4d 2495 . . . 4  |-  ( (
ph  /\  ( x  =  ( 1  / 
2 )  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( ( 2  x.  x ) M y )  =  ( ( ( 2  x.  x
)  -  1 ) N y ) )
60 retopon 21784 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
61 iccssre 11716 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( 0 [,] ( 1  /  2
) )  C_  RR )
6229, 28, 61mp2an 678 . . . . . . 7  |-  ( 0 [,] ( 1  / 
2 ) )  C_  RR
63 resttopon 20177 . . . . . . 7  |-  ( ( ( topGen `  ran  (,) )  e.  (TopOn `  RR )  /\  ( 0 [,] (
1  /  2 ) )  C_  RR )  ->  ( ( topGen `  ran  (,) )t  ( 0 [,] (
1  /  2 ) ) )  e.  (TopOn `  ( 0 [,] (
1  /  2 ) ) ) )
6460, 62, 63mp2an 678 . . . . . 6  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  e.  (TopOn `  ( 0 [,] (
1  /  2 ) ) )
6564a1i 11 . . . . 5  |-  ( ph  ->  ( ( topGen `  ran  (,) )t  ( 0 [,] (
1  /  2 ) ) )  e.  (TopOn `  ( 0 [,] (
1  /  2 ) ) ) )
6665, 39cnmpt1st 20683 . . . . . 6  |-  ( ph  ->  ( x  e.  ( 0 [,] ( 1  /  2 ) ) ,  y  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  II )  Cn  (
( topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) ) ) )
6723iihalf1cn 21960 . . . . . . 7  |-  ( z  e.  ( 0 [,] ( 1  /  2
) )  |->  ( 2  x.  z ) )  e.  ( ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  Cn  II )
6867a1i 11 . . . . . 6  |-  ( ph  ->  ( z  e.  ( 0 [,] ( 1  /  2 ) ) 
|->  ( 2  x.  z
) )  e.  ( ( ( topGen `  ran  (,) )t  ( 0 [,] (
1  /  2 ) ) )  Cn  II ) )
69 oveq2 6298 . . . . . 6  |-  ( z  =  x  ->  (
2  x.  z )  =  ( 2  x.  x ) )
7065, 39, 66, 65, 68, 69cnmpt21 20686 . . . . 5  |-  ( ph  ->  ( x  e.  ( 0 [,] ( 1  /  2 ) ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( 2  x.  x
) )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  II )  Cn  II ) )
7165, 39cnmpt2nd 20684 . . . . 5  |-  ( ph  ->  ( x  e.  ( 0 [,] ( 1  /  2 ) ) ,  y  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  II )  Cn  II ) )
724, 11phtpycn 22014 . . . . . 6  |-  ( ph  ->  ( F ( PHtpy `  J ) H ) 
C_  ( ( II 
tX  II )  Cn  J ) )
7372, 13sseldd 3433 . . . . 5  |-  ( ph  ->  M  e.  ( ( II  tX  II )  Cn  J ) )
7465, 39, 70, 71, 73cnmpt22f 20690 . . . 4  |-  ( ph  ->  ( x  e.  ( 0 [,] ( 1  /  2 ) ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( 2  x.  x ) M y ) )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  II )  Cn  J
) )
75 iccssre 11716 . . . . . . . 8  |-  ( ( ( 1  /  2
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1  / 
2 ) [,] 1
)  C_  RR )
7628, 32, 75mp2an 678 . . . . . . 7  |-  ( ( 1  /  2 ) [,] 1 )  C_  RR
77 resttopon 20177 . . . . . . 7  |-  ( ( ( topGen `  ran  (,) )  e.  (TopOn `  RR )  /\  ( ( 1  / 
2 ) [,] 1
)  C_  RR )  ->  ( ( topGen `  ran  (,) )t  ( ( 1  / 
2 ) [,] 1
) )  e.  (TopOn `  ( ( 1  / 
2 ) [,] 1
) ) )
7860, 76, 77mp2an 678 . . . . . 6  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  e.  (TopOn `  ( ( 1  / 
2 ) [,] 1
) )
7978a1i 11 . . . . 5  |-  ( ph  ->  ( ( topGen `  ran  (,) )t  ( ( 1  / 
2 ) [,] 1
) )  e.  (TopOn `  ( ( 1  / 
2 ) [,] 1
) ) )
8079, 39cnmpt1st 20683 . . . . . 6  |-  ( ph  ->  ( x  e.  ( ( 1  /  2
) [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( ( 1  /  2 ) [,] 1 ) )  tX  II )  Cn  (
( topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) ) ) )
8124iihalf2cn 21962 . . . . . . 7  |-  ( z  e.  ( ( 1  /  2 ) [,] 1 )  |->  ( ( 2  x.  z )  -  1 ) )  e.  ( ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  Cn  II )
8281a1i 11 . . . . . 6  |-  ( ph  ->  ( z  e.  ( ( 1  /  2
) [,] 1 ) 
|->  ( ( 2  x.  z )  -  1 ) )  e.  ( ( ( topGen `  ran  (,) )t  ( ( 1  / 
2 ) [,] 1
) )  Cn  II ) )
8369oveq1d 6305 . . . . . 6  |-  ( z  =  x  ->  (
( 2  x.  z
)  -  1 )  =  ( ( 2  x.  x )  - 
1 ) )
8479, 39, 80, 79, 82, 83cnmpt21 20686 . . . . 5  |-  ( ph  ->  ( x  e.  ( ( 1  /  2
) [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( 2  x.  x )  -  1 ) )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( ( 1  /  2 ) [,] 1 ) )  tX  II )  Cn  II ) )
8579, 39cnmpt2nd 20684 . . . . 5  |-  ( ph  ->  ( x  e.  ( ( 1  /  2
) [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( ( 1  /  2 ) [,] 1 ) )  tX  II )  Cn  II ) )
868, 12phtpycn 22014 . . . . . 6  |-  ( ph  ->  ( G ( PHtpy `  J ) K ) 
C_  ( ( II 
tX  II )  Cn  J ) )
8786, 16sseldd 3433 . . . . 5  |-  ( ph  ->  N  e.  ( ( II  tX  II )  Cn  J ) )
8879, 39, 84, 85, 87cnmpt22f 20690 . . . 4  |-  ( ph  ->  ( x  e.  ( ( 1  /  2
) [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( ( 2  x.  x )  - 
1 ) N y ) )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( ( 1  /  2 ) [,] 1 ) )  tX  II )  Cn  J
) )
8922, 23, 24, 25, 26, 27, 37, 39, 59, 74, 88cnmpt2pc 21956 . . 3  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( ( 2  x.  x ) M y ) ,  ( ( ( 2  x.  x )  - 
1 ) N y ) ) )  e.  ( ( II  tX  II )  Cn  J
) )
9021, 89syl5eqel 2533 . 2  |-  ( ph  ->  P  e.  ( ( II  tX  II )  Cn  J ) )
91 simpll 760 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  ph )
92 elii1 21963 . . . . . . . . 9  |-  ( s  e.  ( 0 [,] ( 1  /  2
) )  <->  ( s  e.  ( 0 [,] 1
)  /\  s  <_  ( 1  /  2 ) ) )
93 iihalf1 21959 . . . . . . . . 9  |-  ( s  e.  ( 0 [,] ( 1  /  2
) )  ->  (
2  x.  s )  e.  ( 0 [,] 1 ) )
9492, 93sylbir 217 . . . . . . . 8  |-  ( ( s  e.  ( 0 [,] 1 )  /\  s  <_  ( 1  / 
2 ) )  -> 
( 2  x.  s
)  e.  ( 0 [,] 1 ) )
9594adantll 720 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
2  x.  s )  e.  ( 0 [,] 1 ) )
964, 11phtpyhtpy 22013 . . . . . . . . 9  |-  ( ph  ->  ( F ( PHtpy `  J ) H ) 
C_  ( F ( II Htpy  J ) H ) )
9796, 13sseldd 3433 . . . . . . . 8  |-  ( ph  ->  M  e.  ( F ( II Htpy  J ) H ) )
9839, 4, 11, 97htpyi 22005 . . . . . . 7  |-  ( (
ph  /\  ( 2  x.  s )  e.  ( 0 [,] 1
) )  ->  (
( ( 2  x.  s ) M 0 )  =  ( F `
 ( 2  x.  s ) )  /\  ( ( 2  x.  s ) M 1 )  =  ( H `
 ( 2  x.  s ) ) ) )
9991, 95, 98syl2anc 667 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
( ( 2  x.  s ) M 0 )  =  ( F `
 ( 2  x.  s ) )  /\  ( ( 2  x.  s ) M 1 )  =  ( H `
 ( 2  x.  s ) ) ) )
10099simpld 461 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
( 2  x.  s
) M 0 )  =  ( F `  ( 2  x.  s
) ) )
101 iftrue 3887 . . . . . 6  |-  ( s  <_  ( 1  / 
2 )  ->  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 0 ) ,  ( ( ( 2  x.  s )  -  1 ) N 0 ) )  =  ( ( 2  x.  s ) M 0 ) )
102101adantl 468 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 0 ) ,  ( ( ( 2  x.  s )  -  1 ) N 0 ) )  =  ( ( 2  x.  s ) M 0 ) )
103 iftrue 3887 . . . . . 6  |-  ( s  <_  ( 1  / 
2 )  ->  if ( s  <_  (
1  /  2 ) ,  ( F `  ( 2  x.  s
) ) ,  ( G `  ( ( 2  x.  s )  -  1 ) ) )  =  ( F `
 ( 2  x.  s ) ) )
104103adantl 468 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  if ( s  <_  (
1  /  2 ) ,  ( F `  ( 2  x.  s
) ) ,  ( G `  ( ( 2  x.  s )  -  1 ) ) )  =  ( F `
 ( 2  x.  s ) ) )
105100, 102, 1043eqtr4d 2495 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 0 ) ,  ( ( ( 2  x.  s )  -  1 ) N 0 ) )  =  if ( s  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  s ) ) ,  ( G `
 ( ( 2  x.  s )  - 
1 ) ) ) )
106 simpll 760 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  ->  ph )
107 elii2 21964 . . . . . . . . 9  |-  ( ( s  e.  ( 0 [,] 1 )  /\  -.  s  <_  ( 1  /  2 ) )  ->  s  e.  ( ( 1  /  2
) [,] 1 ) )
108107adantll 720 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
s  e.  ( ( 1  /  2 ) [,] 1 ) )
109 iihalf2 21961 . . . . . . . 8  |-  ( s  e.  ( ( 1  /  2 ) [,] 1 )  ->  (
( 2  x.  s
)  -  1 )  e.  ( 0 [,] 1 ) )
110108, 109syl 17 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( ( 2  x.  s )  -  1 )  e.  ( 0 [,] 1 ) )
1118, 12phtpyhtpy 22013 . . . . . . . . 9  |-  ( ph  ->  ( G ( PHtpy `  J ) K ) 
C_  ( G ( II Htpy  J ) K ) )
112111, 16sseldd 3433 . . . . . . . 8  |-  ( ph  ->  N  e.  ( G ( II Htpy  J ) K ) )
11339, 8, 12, 112htpyi 22005 . . . . . . 7  |-  ( (
ph  /\  ( (
2  x.  s )  -  1 )  e.  ( 0 [,] 1
) )  ->  (
( ( ( 2  x.  s )  - 
1 ) N 0 )  =  ( G `
 ( ( 2  x.  s )  - 
1 ) )  /\  ( ( ( 2  x.  s )  - 
1 ) N 1 )  =  ( K `
 ( ( 2  x.  s )  - 
1 ) ) ) )
114106, 110, 113syl2anc 667 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( ( ( ( 2  x.  s )  -  1 ) N 0 )  =  ( G `  ( ( 2  x.  s )  -  1 ) )  /\  ( ( ( 2  x.  s )  -  1 ) N 1 )  =  ( K `  ( ( 2  x.  s )  -  1 ) ) ) )
115114simpld 461 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( ( ( 2  x.  s )  - 
1 ) N 0 )  =  ( G `
 ( ( 2  x.  s )  - 
1 ) ) )
116 iffalse 3890 . . . . . 6  |-  ( -.  s  <_  ( 1  /  2 )  ->  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 0 ) ,  ( ( ( 2  x.  s )  -  1 ) N 0 ) )  =  ( ( ( 2  x.  s
)  -  1 ) N 0 ) )
117116adantl 468 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  ->  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 0 ) ,  ( ( ( 2  x.  s )  -  1 ) N 0 ) )  =  ( ( ( 2  x.  s
)  -  1 ) N 0 ) )
118 iffalse 3890 . . . . . 6  |-  ( -.  s  <_  ( 1  /  2 )  ->  if ( s  <_  (
1  /  2 ) ,  ( F `  ( 2  x.  s
) ) ,  ( G `  ( ( 2  x.  s )  -  1 ) ) )  =  ( G `
 ( ( 2  x.  s )  - 
1 ) ) )
119118adantl 468 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  ->  if ( s  <_  (
1  /  2 ) ,  ( F `  ( 2  x.  s
) ) ,  ( G `  ( ( 2  x.  s )  -  1 ) ) )  =  ( G `
 ( ( 2  x.  s )  - 
1 ) ) )
120115, 117, 1193eqtr4d 2495 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  ->  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 0 ) ,  ( ( ( 2  x.  s )  -  1 ) N 0 ) )  =  if ( s  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  s ) ) ,  ( G `
 ( ( 2  x.  s )  - 
1 ) ) ) )
121105, 120pm2.61dan 800 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 0 ) ,  ( ( ( 2  x.  s )  -  1 ) N 0 ) )  =  if ( s  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  s ) ) ,  ( G `
 ( ( 2  x.  s )  - 
1 ) ) ) )
122 simpr 463 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
123 0elunit 11750 . . . 4  |-  0  e.  ( 0 [,] 1
)
124 simpl 459 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  x  =  s )
125124breq1d 4412 . . . . . 6  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( x  <_ 
( 1  /  2
)  <->  s  <_  (
1  /  2 ) ) )
126124oveq2d 6306 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 2  x.  x )  =  ( 2  x.  s ) )
127 simpr 463 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  y  =  0 )
128126, 127oveq12d 6308 . . . . . 6  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( 2  x.  x ) M y )  =  ( ( 2  x.  s
) M 0 ) )
129126oveq1d 6305 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( 2  x.  x )  - 
1 )  =  ( ( 2  x.  s
)  -  1 ) )
130129, 127oveq12d 6308 . . . . . 6  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( ( ( 2  x.  x )  -  1 ) N y )  =  ( ( ( 2  x.  s )  -  1 ) N 0 ) )
131125, 128, 130ifbieq12d 3908 . . . . 5  |-  ( ( x  =  s  /\  y  =  0 )  ->  if ( x  <_  ( 1  / 
2 ) ,  ( ( 2  x.  x
) M y ) ,  ( ( ( 2  x.  x )  -  1 ) N y ) )  =  if ( s  <_ 
( 1  /  2
) ,  ( ( 2  x.  s ) M 0 ) ,  ( ( ( 2  x.  s )  - 
1 ) N 0 ) ) )
132 ovex 6318 . . . . . 6  |-  ( ( 2  x.  s ) M 0 )  e. 
_V
133 ovex 6318 . . . . . 6  |-  ( ( ( 2  x.  s
)  -  1 ) N 0 )  e. 
_V
134132, 133ifex 3949 . . . . 5  |-  if ( s  <_  ( 1  /  2 ) ,  ( ( 2  x.  s ) M 0 ) ,  ( ( ( 2  x.  s
)  -  1 ) N 0 ) )  e.  _V
135131, 21, 134ovmpt2a 6427 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s P 0 )  =  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 0 ) ,  ( ( ( 2  x.  s )  -  1 ) N 0 ) ) )
136122, 123, 135sylancl 668 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s P 0 )  =  if ( s  <_  ( 1  / 
2 ) ,  ( ( 2  x.  s
) M 0 ) ,  ( ( ( 2  x.  s )  -  1 ) N 0 ) ) )
1374, 8pcovalg 22043 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F ( *p
`  J ) G ) `  s )  =  if ( s  <_  ( 1  / 
2 ) ,  ( F `  ( 2  x.  s ) ) ,  ( G `  ( ( 2  x.  s )  -  1 ) ) ) )
138121, 136, 1373eqtr4d 2495 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s P 0 )  =  ( ( F ( *p `  J
) G ) `  s ) )
13999simprd 465 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
( 2  x.  s
) M 1 )  =  ( H `  ( 2  x.  s
) ) )
140 iftrue 3887 . . . . . 6  |-  ( s  <_  ( 1  / 
2 )  ->  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 1 ) ,  ( ( ( 2  x.  s )  -  1 ) N 1 ) )  =  ( ( 2  x.  s ) M 1 ) )
141140adantl 468 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 1 ) ,  ( ( ( 2  x.  s )  -  1 ) N 1 ) )  =  ( ( 2  x.  s ) M 1 ) )
142 iftrue 3887 . . . . . 6  |-  ( s  <_  ( 1  / 
2 )  ->  if ( s  <_  (
1  /  2 ) ,  ( H `  ( 2  x.  s
) ) ,  ( K `  ( ( 2  x.  s )  -  1 ) ) )  =  ( H `
 ( 2  x.  s ) ) )
143142adantl 468 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  if ( s  <_  (
1  /  2 ) ,  ( H `  ( 2  x.  s
) ) ,  ( K `  ( ( 2  x.  s )  -  1 ) ) )  =  ( H `
 ( 2  x.  s ) ) )
144139, 141, 1433eqtr4d 2495 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 1 ) ,  ( ( ( 2  x.  s )  -  1 ) N 1 ) )  =  if ( s  <_  ( 1  /  2 ) ,  ( H `  (
2  x.  s ) ) ,  ( K `
 ( ( 2  x.  s )  - 
1 ) ) ) )
145114simprd 465 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( ( ( 2  x.  s )  - 
1 ) N 1 )  =  ( K `
 ( ( 2  x.  s )  - 
1 ) ) )
146 iffalse 3890 . . . . . 6  |-  ( -.  s  <_  ( 1  /  2 )  ->  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 1 ) ,  ( ( ( 2  x.  s )  -  1 ) N 1 ) )  =  ( ( ( 2  x.  s
)  -  1 ) N 1 ) )
147146adantl 468 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  ->  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 1 ) ,  ( ( ( 2  x.  s )  -  1 ) N 1 ) )  =  ( ( ( 2  x.  s
)  -  1 ) N 1 ) )
148 iffalse 3890 . . . . . 6  |-  ( -.  s  <_  ( 1  /  2 )  ->  if ( s  <_  (
1  /  2 ) ,  ( H `  ( 2  x.  s
) ) ,  ( K `  ( ( 2  x.  s )  -  1 ) ) )  =  ( K `
 ( ( 2  x.  s )  - 
1 ) ) )
149148adantl 468 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  ->  if ( s  <_  (
1  /  2 ) ,  ( H `  ( 2  x.  s
) ) ,  ( K `  ( ( 2  x.  s )  -  1 ) ) )  =  ( K `
 ( ( 2  x.  s )  - 
1 ) ) )
150145, 147, 1493eqtr4d 2495 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  ->  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 1 ) ,  ( ( ( 2  x.  s )  -  1 ) N 1 ) )  =  if ( s  <_  ( 1  /  2 ) ,  ( H `  (
2  x.  s ) ) ,  ( K `
 ( ( 2  x.  s )  - 
1 ) ) ) )
151144, 150pm2.61dan 800 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 1 ) ,  ( ( ( 2  x.  s )  -  1 ) N 1 ) )  =  if ( s  <_  ( 1  /  2 ) ,  ( H `  (
2  x.  s ) ) ,  ( K `
 ( ( 2  x.  s )  - 
1 ) ) ) )
152 1elunit 11751 . . . 4  |-  1  e.  ( 0 [,] 1
)
153 simpl 459 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  x  =  s )
154153breq1d 4412 . . . . . 6  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( x  <_ 
( 1  /  2
)  <->  s  <_  (
1  /  2 ) ) )
155153oveq2d 6306 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 2  x.  x )  =  ( 2  x.  s ) )
156 simpr 463 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  y  =  1 )
157155, 156oveq12d 6308 . . . . . 6  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 2  x.  x ) M y )  =  ( ( 2  x.  s
) M 1 ) )
158155oveq1d 6305 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 2  x.  x )  - 
1 )  =  ( ( 2  x.  s
)  -  1 ) )
159158, 156oveq12d 6308 . . . . . 6  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( ( 2  x.  x )  -  1 ) N y )  =  ( ( ( 2  x.  s )  -  1 ) N 1 ) )
160154, 157, 159ifbieq12d 3908 . . . . 5  |-  ( ( x  =  s  /\  y  =  1 )  ->  if ( x  <_  ( 1  / 
2 ) ,  ( ( 2  x.  x
) M y ) ,  ( ( ( 2  x.  x )  -  1 ) N y ) )  =  if ( s  <_ 
( 1  /  2
) ,  ( ( 2  x.  s ) M 1 ) ,  ( ( ( 2  x.  s )  - 
1 ) N 1 ) ) )
161 ovex 6318 . . . . . 6  |-  ( ( 2  x.  s ) M 1 )  e. 
_V
162 ovex 6318 . . . . . 6  |-  ( ( ( 2  x.  s
)  -  1 ) N 1 )  e. 
_V
163161, 162ifex 3949 . . . . 5  |-  if ( s  <_  ( 1  /  2 ) ,  ( ( 2  x.  s ) M 1 ) ,  ( ( ( 2  x.  s
)  -  1 ) N 1 ) )  e.  _V
164160, 21, 163ovmpt2a 6427 . . . 4  |-  ( ( s  e.  ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s P 1 )  =  if ( s  <_  (
1  /  2 ) ,  ( ( 2  x.  s ) M 1 ) ,  ( ( ( 2  x.  s )  -  1 ) N 1 ) ) )
165122, 152, 164sylancl 668 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s P 1 )  =  if ( s  <_  ( 1  / 
2 ) ,  ( ( 2  x.  s
) M 1 ) ,  ( ( ( 2  x.  s )  -  1 ) N 1 ) ) )
16611, 12pcovalg 22043 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( H ( *p
`  J ) K ) `  s )  =  if ( s  <_  ( 1  / 
2 ) ,  ( H `  ( 2  x.  s ) ) ,  ( K `  ( ( 2  x.  s )  -  1 ) ) ) )
167151, 165, 1663eqtr4d 2495 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
s P 1 )  =  ( ( H ( *p `  J
) K ) `  s ) )
1684, 11, 13phtpyi 22015 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0 M s )  =  ( F `
 0 )  /\  ( 1 M s )  =  ( F `
 1 ) ) )
169168simpld 461 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 M s )  =  ( F ` 
0 ) )
170 simpl 459 . . . . . . . 8  |-  ( ( x  =  0  /\  y  =  s )  ->  x  =  0 )
171170, 31syl6eqbr 4440 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  x  <_  (
1  /  2 ) )
172171iftrued 3889 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  if ( x  <_  ( 1  / 
2 ) ,  ( ( 2  x.  x
) M y ) ,  ( ( ( 2  x.  x )  -  1 ) N y ) )  =  ( ( 2  x.  x ) M y ) )
173170oveq2d 6306 . . . . . . . 8  |-  ( ( x  =  0  /\  y  =  s )  ->  ( 2  x.  x )  =  ( 2  x.  0 ) )
174 2t0e0 10765 . . . . . . . 8  |-  ( 2  x.  0 )  =  0
175173, 174syl6eq 2501 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( 2  x.  x )  =  0 )
176 simpr 463 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  y  =  s )
177175, 176oveq12d 6308 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( ( 2  x.  x ) M y )  =  ( 0 M s ) )
178172, 177eqtrd 2485 . . . . 5  |-  ( ( x  =  0  /\  y  =  s )  ->  if ( x  <_  ( 1  / 
2 ) ,  ( ( 2  x.  x
) M y ) ,  ( ( ( 2  x.  x )  -  1 ) N y ) )  =  ( 0 M s ) )
179 ovex 6318 . . . . 5  |-  ( 0 M s )  e. 
_V
180178, 21, 179ovmpt2a 6427 . . . 4  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 0 P s )  =  ( 0 M s ) )
181123, 122, 180sylancr 669 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 P s )  =  ( 0 M s ) )
1824, 8pco0 22045 . . . 4  |-  ( ph  ->  ( ( F ( *p `  J ) G ) `  0
)  =  ( F `
 0 ) )
183182adantr 467 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F ( *p
`  J ) G ) `  0 )  =  ( F ` 
0 ) )
184169, 181, 1833eqtr4d 2495 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 P s )  =  ( ( F ( *p `  J
) G ) ` 
0 ) )
1858, 12, 16phtpyi 22015 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0 N s )  =  ( G `
 0 )  /\  ( 1 N s )  =  ( G `
 1 ) ) )
186185simprd 465 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 N s )  =  ( G ` 
1 ) )
18728, 32ltnlei 9755 . . . . . . . . 9  |-  ( ( 1  /  2 )  <  1  <->  -.  1  <_  ( 1  /  2
) )
18833, 187mpbi 212 . . . . . . . 8  |-  -.  1  <_  ( 1  /  2
)
189 simpl 459 . . . . . . . . 9  |-  ( ( x  =  1  /\  y  =  s )  ->  x  =  1 )
190189breq1d 4412 . . . . . . . 8  |-  ( ( x  =  1  /\  y  =  s )  ->  ( x  <_ 
( 1  /  2
)  <->  1  <_  (
1  /  2 ) ) )
191188, 190mtbiri 305 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  -.  x  <_  ( 1  /  2 ) )
192191iffalsed 3892 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  if ( x  <_  ( 1  / 
2 ) ,  ( ( 2  x.  x
) M y ) ,  ( ( ( 2  x.  x )  -  1 ) N y ) )  =  ( ( ( 2  x.  x )  - 
1 ) N y ) )
193189oveq2d 6306 . . . . . . . . . 10  |-  ( ( x  =  1  /\  y  =  s )  ->  ( 2  x.  x )  =  ( 2  x.  1 ) )
194 2t1e2 10758 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
195193, 194syl6eq 2501 . . . . . . . . 9  |-  ( ( x  =  1  /\  y  =  s )  ->  ( 2  x.  x )  =  2 )
196195oveq1d 6305 . . . . . . . 8  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( 2  x.  x )  - 
1 )  =  ( 2  -  1 ) )
197 2m1e1 10724 . . . . . . . 8  |-  ( 2  -  1 )  =  1
198196, 197syl6eq 2501 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( 2  x.  x )  - 
1 )  =  1 )
199 simpr 463 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  y  =  s )
200198, 199oveq12d 6308 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( ( 2  x.  x )  -  1 ) N y )  =  ( 1 N s ) )
201192, 200eqtrd 2485 . . . . 5  |-  ( ( x  =  1  /\  y  =  s )  ->  if ( x  <_  ( 1  / 
2 ) ,  ( ( 2  x.  x
) M y ) ,  ( ( ( 2  x.  x )  -  1 ) N y ) )  =  ( 1 N s ) )
202 ovex 6318 . . . . 5  |-  ( 1 N s )  e. 
_V
203201, 21, 202ovmpt2a 6427 . . . 4  |-  ( ( 1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 1 P s )  =  ( 1 N s ) )
204152, 122, 203sylancr 669 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 P s )  =  ( 1 N s ) )
2054, 8pco1 22046 . . . 4  |-  ( ph  ->  ( ( F ( *p `  J ) G ) `  1
)  =  ( G `
 1 ) )
206205adantr 467 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( F ( *p
`  J ) G ) `  1 )  =  ( G ` 
1 ) )
207186, 204, 2063eqtr4d 2495 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 P s )  =  ( ( F ( *p `  J
) G ) ` 
1 ) )
20810, 20, 90, 138, 167, 184, 207isphtpy2d 22018 1  |-  ( ph  ->  P  e.  ( ( F ( *p `  J ) G ) ( PHtpy `  J )
( H ( *p
`  J ) K ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622    C_ wss 3404   (/)c0 3731   ifcif 3881   class class class wbr 4402    |-> cmpt 4461   ran crn 4835   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   RRcr 9538   0cc0 9539   1c1 9540    x. cmul 9544    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   2c2 10659   (,)cioo 11635   [,]cicc 11638   ↾t crest 15319   topGenctg 15336  TopOnctopon 19918    Cn ccn 20240    tX ctx 20575   IIcii 21907   Htpy chtpy 21998   PHtpycphtpy 21999    ~=ph cphtpc 22000   *pcpco 22031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-cn 20243  df-cnp 20244  df-tx 20577  df-hmeo 20770  df-xms 21335  df-ms 21336  df-tms 21337  df-ii 21909  df-htpy 22001  df-phtpy 22002  df-phtpc 22023  df-pco 22036
This theorem is referenced by:  pcohtpy  22051
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