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Theorem htpycc 20687
Description: Concatenate two homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
htpycc.1  |-  N  =  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  if ( y  <_ 
( 1  /  2
) ,  ( x L ( 2  x.  y ) ) ,  ( x M ( ( 2  x.  y
)  -  1 ) ) ) )
htpycc.2  |-  ( ph  ->  J  e.  (TopOn `  X ) )
htpycc.4  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
htpycc.5  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
htpycc.6  |-  ( ph  ->  H  e.  ( J  Cn  K ) )
htpycc.7  |-  ( ph  ->  L  e.  ( F ( J Htpy  K ) G ) )
htpycc.8  |-  ( ph  ->  M  e.  ( G ( J Htpy  K ) H ) )
Assertion
Ref Expression
htpycc  |-  ( ph  ->  N  e.  ( F ( J Htpy  K ) H ) )
Distinct variable groups:    x, y, J    x, K, y    x, L, y    x, M, y   
x, X, y    ph, x, y
Allowed substitution hints:    F( x, y)    G( x, y)    H( x, y)    N( x, y)

Proof of Theorem htpycc
Dummy variables  s 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 htpycc.2 . 2  |-  ( ph  ->  J  e.  (TopOn `  X ) )
2 htpycc.4 . 2  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
3 htpycc.6 . 2  |-  ( ph  ->  H  e.  ( J  Cn  K ) )
4 htpycc.1 . . 3  |-  N  =  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  if ( y  <_ 
( 1  /  2
) ,  ( x L ( 2  x.  y ) ) ,  ( x M ( ( 2  x.  y
)  -  1 ) ) ) )
5 iitopon 20590 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
65a1i 11 . . . 4  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
7 eqid 2454 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
8 eqid 2454 . . . . 5  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )
9 eqid 2454 . . . . 5  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )
10 dfii2 20593 . . . . 5  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
11 0red 9501 . . . . 5  |-  ( ph  ->  0  e.  RR )
12 1red 9515 . . . . 5  |-  ( ph  ->  1  e.  RR )
13 halfre 10654 . . . . . . 7  |-  ( 1  /  2 )  e.  RR
14 0re 9500 . . . . . . . 8  |-  0  e.  RR
15 halfgt0 10656 . . . . . . . 8  |-  0  <  ( 1  /  2
)
1614, 13, 15ltleii 9611 . . . . . . 7  |-  0  <_  ( 1  /  2
)
17 1re 9499 . . . . . . . 8  |-  1  e.  RR
18 halflt1 10657 . . . . . . . 8  |-  ( 1  /  2 )  <  1
1913, 17, 18ltleii 9611 . . . . . . 7  |-  ( 1  /  2 )  <_ 
1
2014, 17elicc2i 11475 . . . . . . 7  |-  ( ( 1  /  2 )  e.  ( 0 [,] 1 )  <->  ( (
1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_ 
1 ) )
2113, 16, 19, 20mpbir3an 1170 . . . . . 6  |-  ( 1  /  2 )  e.  ( 0 [,] 1
)
2221a1i 11 . . . . 5  |-  ( ph  ->  ( 1  /  2
)  e.  ( 0 [,] 1 ) )
23 htpycc.5 . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
24 htpycc.7 . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  ( F ( J Htpy  K ) G ) )
251, 2, 23, 24htpyi 20681 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  X )  ->  (
( s L 0 )  =  ( F `
 s )  /\  ( s L 1 )  =  ( G `
 s ) ) )
2625simprd 463 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  X )  ->  (
s L 1 )  =  ( G `  s ) )
27 htpycc.8 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  ( G ( J Htpy  K ) H ) )
281, 23, 3, 27htpyi 20681 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  X )  ->  (
( s M 0 )  =  ( G `
 s )  /\  ( s M 1 )  =  ( H `
 s ) ) )
2928simpld 459 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  X )  ->  (
s M 0 )  =  ( G `  s ) )
3026, 29eqtr4d 2498 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  X )  ->  (
s L 1 )  =  ( s M 0 ) )
3130ralrimiva 2830 . . . . . . . 8  |-  ( ph  ->  A. s  e.  X  ( s L 1 )  =  ( s M 0 ) )
32 oveq1 6210 . . . . . . . . . 10  |-  ( s  =  x  ->  (
s L 1 )  =  ( x L 1 ) )
33 oveq1 6210 . . . . . . . . . 10  |-  ( s  =  x  ->  (
s M 0 )  =  ( x M 0 ) )
3432, 33eqeq12d 2476 . . . . . . . . 9  |-  ( s  =  x  ->  (
( s L 1 )  =  ( s M 0 )  <->  ( x L 1 )  =  ( x M 0 ) ) )
3534rspccva 3178 . . . . . . . 8  |-  ( ( A. s  e.  X  ( s L 1 )  =  ( s M 0 )  /\  x  e.  X )  ->  ( x L 1 )  =  ( x M 0 ) )
3631, 35sylan 471 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
x L 1 )  =  ( x M 0 ) )
3736adantrl 715 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x L 1 )  =  ( x M 0 ) )
38 simprl 755 . . . . . . . . 9  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
y  =  ( 1  /  2 ) )
3938oveq2d 6219 . . . . . . . 8  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( 2  x.  y
)  =  ( 2  x.  ( 1  / 
2 ) ) )
40 2cn 10506 . . . . . . . . 9  |-  2  e.  CC
41 2ne0 10528 . . . . . . . . 9  |-  2  =/=  0
4240, 41recidi 10176 . . . . . . . 8  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
4339, 42syl6eq 2511 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( 2  x.  y
)  =  1 )
4443oveq2d 6219 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x L ( 2  x.  y ) )  =  ( x L 1 ) )
4543oveq1d 6218 . . . . . . . 8  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( ( 2  x.  y )  -  1 )  =  ( 1  -  1 ) )
46 1m1e0 10504 . . . . . . . 8  |-  ( 1  -  1 )  =  0
4745, 46syl6eq 2511 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( ( 2  x.  y )  -  1 )  =  0 )
4847oveq2d 6219 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x M ( ( 2  x.  y
)  -  1 ) )  =  ( x M 0 ) )
4937, 44, 483eqtr4d 2505 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x L ( 2  x.  y ) )  =  ( x M ( ( 2  x.  y )  - 
1 ) ) )
50 retopon 20477 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
51 iccssre 11491 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( 0 [,] ( 1  /  2
) )  C_  RR )
5214, 13, 51mp2an 672 . . . . . . . 8  |-  ( 0 [,] ( 1  / 
2 ) )  C_  RR
53 resttopon 18900 . . . . . . . 8  |-  ( ( ( topGen `  ran  (,) )  e.  (TopOn `  RR )  /\  ( 0 [,] (
1  /  2 ) )  C_  RR )  ->  ( ( topGen `  ran  (,) )t  ( 0 [,] (
1  /  2 ) ) )  e.  (TopOn `  ( 0 [,] (
1  /  2 ) ) ) )
5450, 52, 53mp2an 672 . . . . . . 7  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  e.  (TopOn `  ( 0 [,] (
1  /  2 ) ) )
5554a1i 11 . . . . . 6  |-  ( ph  ->  ( ( topGen `  ran  (,) )t  ( 0 [,] (
1  /  2 ) ) )  e.  (TopOn `  ( 0 [,] (
1  /  2 ) ) ) )
5655, 1cnmpt2nd 19377 . . . . . 6  |-  ( ph  ->  ( y  e.  ( 0 [,] ( 1  /  2 ) ) ,  x  e.  X  |->  x )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  J )  Cn  J
) )
5755, 1cnmpt1st 19376 . . . . . . 7  |-  ( ph  ->  ( y  e.  ( 0 [,] ( 1  /  2 ) ) ,  x  e.  X  |->  y )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  J )  Cn  (
( topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) ) ) )
588iihalf1cn 20639 . . . . . . . 8  |-  ( z  e.  ( 0 [,] ( 1  /  2
) )  |->  ( 2  x.  z ) )  e.  ( ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  Cn  II )
5958a1i 11 . . . . . . 7  |-  ( ph  ->  ( z  e.  ( 0 [,] ( 1  /  2 ) ) 
|->  ( 2  x.  z
) )  e.  ( ( ( topGen `  ran  (,) )t  ( 0 [,] (
1  /  2 ) ) )  Cn  II ) )
60 oveq2 6211 . . . . . . 7  |-  ( z  =  y  ->  (
2  x.  z )  =  ( 2  x.  y ) )
6155, 1, 57, 55, 59, 60cnmpt21 19379 . . . . . 6  |-  ( ph  ->  ( y  e.  ( 0 [,] ( 1  /  2 ) ) ,  x  e.  X  |->  ( 2  x.  y
) )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  J )  Cn  II ) )
621, 2, 23htpycn 20680 . . . . . . 7  |-  ( ph  ->  ( F ( J Htpy 
K ) G ) 
C_  ( ( J 
tX  II )  Cn  K ) )
6362, 24sseldd 3468 . . . . . 6  |-  ( ph  ->  L  e.  ( ( J  tX  II )  Cn  K ) )
6455, 1, 56, 61, 63cnmpt22f 19383 . . . . 5  |-  ( ph  ->  ( y  e.  ( 0 [,] ( 1  /  2 ) ) ,  x  e.  X  |->  ( x L ( 2  x.  y ) ) )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  J )  Cn  K
) )
65 iccssre 11491 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1  / 
2 ) [,] 1
)  C_  RR )
6613, 17, 65mp2an 672 . . . . . . . 8  |-  ( ( 1  /  2 ) [,] 1 )  C_  RR
67 resttopon 18900 . . . . . . . 8  |-  ( ( ( topGen `  ran  (,) )  e.  (TopOn `  RR )  /\  ( ( 1  / 
2 ) [,] 1
)  C_  RR )  ->  ( ( topGen `  ran  (,) )t  ( ( 1  / 
2 ) [,] 1
) )  e.  (TopOn `  ( ( 1  / 
2 ) [,] 1
) ) )
6850, 66, 67mp2an 672 . . . . . . 7  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  e.  (TopOn `  ( ( 1  / 
2 ) [,] 1
) )
6968a1i 11 . . . . . 6  |-  ( ph  ->  ( ( topGen `  ran  (,) )t  ( ( 1  / 
2 ) [,] 1
) )  e.  (TopOn `  ( ( 1  / 
2 ) [,] 1
) ) )
7069, 1cnmpt2nd 19377 . . . . . 6  |-  ( ph  ->  ( y  e.  ( ( 1  /  2
) [,] 1 ) ,  x  e.  X  |->  x )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( ( 1  /  2 ) [,] 1 ) )  tX  J )  Cn  J
) )
7169, 1cnmpt1st 19376 . . . . . . 7  |-  ( ph  ->  ( y  e.  ( ( 1  /  2
) [,] 1 ) ,  x  e.  X  |->  y )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( ( 1  /  2 ) [,] 1 ) )  tX  J )  Cn  (
( topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) ) ) )
729iihalf2cn 20641 . . . . . . . 8  |-  ( z  e.  ( ( 1  /  2 ) [,] 1 )  |->  ( ( 2  x.  z )  -  1 ) )  e.  ( ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  Cn  II )
7372a1i 11 . . . . . . 7  |-  ( ph  ->  ( z  e.  ( ( 1  /  2
) [,] 1 ) 
|->  ( ( 2  x.  z )  -  1 ) )  e.  ( ( ( topGen `  ran  (,) )t  ( ( 1  / 
2 ) [,] 1
) )  Cn  II ) )
7460oveq1d 6218 . . . . . . 7  |-  ( z  =  y  ->  (
( 2  x.  z
)  -  1 )  =  ( ( 2  x.  y )  - 
1 ) )
7569, 1, 71, 69, 73, 74cnmpt21 19379 . . . . . 6  |-  ( ph  ->  ( y  e.  ( ( 1  /  2
) [,] 1 ) ,  x  e.  X  |->  ( ( 2  x.  y )  -  1 ) )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( ( 1  /  2 ) [,] 1 ) )  tX  J )  Cn  II ) )
761, 23, 3htpycn 20680 . . . . . . 7  |-  ( ph  ->  ( G ( J Htpy 
K ) H ) 
C_  ( ( J 
tX  II )  Cn  K ) )
7776, 27sseldd 3468 . . . . . 6  |-  ( ph  ->  M  e.  ( ( J  tX  II )  Cn  K ) )
7869, 1, 70, 75, 77cnmpt22f 19383 . . . . 5  |-  ( ph  ->  ( y  e.  ( ( 1  /  2
) [,] 1 ) ,  x  e.  X  |->  ( x M ( ( 2  x.  y
)  -  1 ) ) )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( ( 1  /  2 ) [,] 1 ) )  tX  J )  Cn  K
) )
797, 8, 9, 10, 11, 12, 22, 1, 49, 64, 78cnmpt2pc 20635 . . . 4  |-  ( ph  ->  ( y  e.  ( 0 [,] 1 ) ,  x  e.  X  |->  if ( y  <_ 
( 1  /  2
) ,  ( x L ( 2  x.  y ) ) ,  ( x M ( ( 2  x.  y
)  -  1 ) ) ) )  e.  ( ( II  tX  J )  Cn  K
) )
806, 1, 79cnmptcom 19386 . . 3  |-  ( ph  ->  ( x  e.  X ,  y  e.  (
0 [,] 1 ) 
|->  if ( y  <_ 
( 1  /  2
) ,  ( x L ( 2  x.  y ) ) ,  ( x M ( ( 2  x.  y
)  -  1 ) ) ) )  e.  ( ( J  tX  II )  Cn  K
) )
814, 80syl5eqel 2546 . 2  |-  ( ph  ->  N  e.  ( ( J  tX  II )  Cn  K ) )
82 simpr 461 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  s  e.  X )
83 0elunit 11523 . . . 4  |-  0  e.  ( 0 [,] 1
)
84 simpr 461 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  y  =  0 )
8584, 16syl6eqbr 4440 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  y  <_  (
1  /  2 ) )
86 iftrue 3908 . . . . . . 7  |-  ( y  <_  ( 1  / 
2 )  ->  if ( y  <_  (
1  /  2 ) ,  ( x L ( 2  x.  y
) ) ,  ( x M ( ( 2  x.  y )  -  1 ) ) )  =  ( x L ( 2  x.  y ) ) )
8785, 86syl 16 . . . . . 6  |-  ( ( x  =  s  /\  y  =  0 )  ->  if ( y  <_  ( 1  / 
2 ) ,  ( x L ( 2  x.  y ) ) ,  ( x M ( ( 2  x.  y )  -  1 ) ) )  =  ( x L ( 2  x.  y ) ) )
88 simpl 457 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  x  =  s )
8984oveq2d 6219 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 2  x.  y )  =  ( 2  x.  0 ) )
90 2t0e0 10591 . . . . . . . 8  |-  ( 2  x.  0 )  =  0
9189, 90syl6eq 2511 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 2  x.  y )  =  0 )
9288, 91oveq12d 6221 . . . . . 6  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( x L ( 2  x.  y
) )  =  ( s L 0 ) )
9387, 92eqtrd 2495 . . . . 5  |-  ( ( x  =  s  /\  y  =  0 )  ->  if ( y  <_  ( 1  / 
2 ) ,  ( x L ( 2  x.  y ) ) ,  ( x M ( ( 2  x.  y )  -  1 ) ) )  =  ( s L 0 ) )
94 ovex 6228 . . . . 5  |-  ( s L 0 )  e. 
_V
9593, 4, 94ovmpt2a 6334 . . . 4  |-  ( ( s  e.  X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s N 0 )  =  ( s L 0 ) )
9682, 83, 95sylancl 662 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( s L 0 ) )
9725simpld 459 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s L 0 )  =  ( F `  s ) )
9896, 97eqtrd 2495 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( F `  s ) )
99 1elunit 11524 . . . 4  |-  1  e.  ( 0 [,] 1
)
10013, 17ltnlei 9609 . . . . . . . . 9  |-  ( ( 1  /  2 )  <  1  <->  -.  1  <_  ( 1  /  2
) )
10118, 100mpbi 208 . . . . . . . 8  |-  -.  1  <_  ( 1  /  2
)
102 simpr 461 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  y  =  1 )
103102breq1d 4413 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( y  <_ 
( 1  /  2
)  <->  1  <_  (
1  /  2 ) ) )
104101, 103mtbiri 303 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  -.  y  <_  ( 1  /  2 ) )
105 iffalse 3910 . . . . . . 7  |-  ( -.  y  <_  ( 1  /  2 )  ->  if ( y  <_  (
1  /  2 ) ,  ( x L ( 2  x.  y
) ) ,  ( x M ( ( 2  x.  y )  -  1 ) ) )  =  ( x M ( ( 2  x.  y )  - 
1 ) ) )
106104, 105syl 16 . . . . . 6  |-  ( ( x  =  s  /\  y  =  1 )  ->  if ( y  <_  ( 1  / 
2 ) ,  ( x L ( 2  x.  y ) ) ,  ( x M ( ( 2  x.  y )  -  1 ) ) )  =  ( x M ( ( 2  x.  y
)  -  1 ) ) )
107 simpl 457 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  x  =  s )
108102oveq2d 6219 . . . . . . . . . 10  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 2  x.  y )  =  ( 2  x.  1 ) )
109 2t1e2 10584 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
110108, 109syl6eq 2511 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 2  x.  y )  =  2 )
111110oveq1d 6218 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 2  x.  y )  - 
1 )  =  ( 2  -  1 ) )
112 2m1e1 10550 . . . . . . . 8  |-  ( 2  -  1 )  =  1
113111, 112syl6eq 2511 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 2  x.  y )  - 
1 )  =  1 )
114107, 113oveq12d 6221 . . . . . 6  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( x M ( ( 2  x.  y )  -  1 ) )  =  ( s M 1 ) )
115106, 114eqtrd 2495 . . . . 5  |-  ( ( x  =  s  /\  y  =  1 )  ->  if ( y  <_  ( 1  / 
2 ) ,  ( x L ( 2  x.  y ) ) ,  ( x M ( ( 2  x.  y )  -  1 ) ) )  =  ( s M 1 ) )
116 ovex 6228 . . . . 5  |-  ( s M 1 )  e. 
_V
117115, 4, 116ovmpt2a 6334 . . . 4  |-  ( ( s  e.  X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s N 1 )  =  ( s M 1 ) )
11882, 99, 117sylancl 662 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( s M 1 ) )
11928simprd 463 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s M 1 )  =  ( H `  s ) )
120118, 119eqtrd 2495 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( H `  s ) )
1211, 2, 3, 81, 98, 120ishtpyd 20682 1  |-  ( ph  ->  N  e.  ( F ( J Htpy  K ) H ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799    C_ wss 3439   ifcif 3902   class class class wbr 4403    |-> cmpt 4461   ran crn 4952   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   RRcr 9395   0cc0 9396   1c1 9397    x. cmul 9401    < clt 9532    <_ cle 9533    - cmin 9709    / cdiv 10107   2c2 10485   (,)cioo 11414   [,]cicc 11417   ↾t crest 14481   topGenctg 14498  TopOnctopon 18634    Cn ccn 18963    tX ctx 19268   IIcii 20586   Htpy chtpy 20674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474  ax-mulf 9476
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-map 7329  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-fi 7775  df-sup 7805  df-oi 7838  df-card 8223  df-cda 8451  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-q 11068  df-rp 11106  df-xneg 11203  df-xadd 11204  df-xmul 11205  df-ioo 11418  df-icc 11421  df-fz 11558  df-fzo 11669  df-seq 11927  df-exp 11986  df-hash 12224  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-mulr 14374  df-starv 14375  df-sca 14376  df-vsca 14377  df-ip 14378  df-tset 14379  df-ple 14380  df-ds 14382  df-unif 14383  df-hom 14384  df-cco 14385  df-rest 14483  df-topn 14484  df-0g 14502  df-gsum 14503  df-topgen 14504  df-pt 14505  df-prds 14508  df-xrs 14562  df-qtop 14567  df-imas 14568  df-xps 14570  df-mre 14646  df-mrc 14647  df-acs 14649  df-mnd 15537  df-submnd 15587  df-mulg 15670  df-cntz 15957  df-cmn 16403  df-psmet 17937  df-xmet 17938  df-met 17939  df-bl 17940  df-mopn 17941  df-cnfld 17947  df-top 18638  df-bases 18640  df-topon 18641  df-topsp 18642  df-cld 18758  df-cn 18966  df-cnp 18967  df-tx 19270  df-hmeo 19463  df-xms 20030  df-ms 20031  df-tms 20032  df-ii 20588  df-htpy 20677
This theorem is referenced by:  phtpycc  20698
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