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Theorem htpycc 21606
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 21509 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
65a1i 11 . . . 4  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
7 eqid 2457 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
8 eqid 2457 . . . . 5  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )
9 eqid 2457 . . . . 5  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )
10 dfii2 21512 . . . . 5  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
11 0red 9614 . . . . 5  |-  ( ph  ->  0  e.  RR )
12 1red 9628 . . . . 5  |-  ( ph  ->  1  e.  RR )
13 halfre 10775 . . . . . . 7  |-  ( 1  /  2 )  e.  RR
14 0re 9613 . . . . . . . 8  |-  0  e.  RR
15 halfgt0 10777 . . . . . . . 8  |-  0  <  ( 1  /  2
)
1614, 13, 15ltleii 9724 . . . . . . 7  |-  0  <_  ( 1  /  2
)
17 1re 9612 . . . . . . . 8  |-  1  e.  RR
18 halflt1 10778 . . . . . . . 8  |-  ( 1  /  2 )  <  1
1913, 17, 18ltleii 9724 . . . . . . 7  |-  ( 1  /  2 )  <_ 
1
2014, 17elicc2i 11615 . . . . . . 7  |-  ( ( 1  /  2 )  e.  ( 0 [,] 1 )  <->  ( (
1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_ 
1 ) )
2113, 16, 19, 20mpbir3an 1178 . . . . . 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 21600 . . . . . . . . . . 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 21600 . . . . . . . . . . 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 2501 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  X )  ->  (
s L 1 )  =  ( s M 0 ) )
3130ralrimiva 2871 . . . . . . . 8  |-  ( ph  ->  A. s  e.  X  ( s L 1 )  =  ( s M 0 ) )
32 oveq1 6303 . . . . . . . . . 10  |-  ( s  =  x  ->  (
s L 1 )  =  ( x L 1 ) )
33 oveq1 6303 . . . . . . . . . 10  |-  ( s  =  x  ->  (
s M 0 )  =  ( x M 0 ) )
3432, 33eqeq12d 2479 . . . . . . . . 9  |-  ( s  =  x  ->  (
( s L 1 )  =  ( s M 0 )  <->  ( x L 1 )  =  ( x M 0 ) ) )
3534rspccva 3209 . . . . . . . 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 756 . . . . . . . . 9  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
y  =  ( 1  /  2 ) )
3938oveq2d 6312 . . . . . . . 8  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( 2  x.  y
)  =  ( 2  x.  ( 1  / 
2 ) ) )
40 2cn 10627 . . . . . . . . 9  |-  2  e.  CC
41 2ne0 10649 . . . . . . . . 9  |-  2  =/=  0
4240, 41recidi 10296 . . . . . . . 8  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
4339, 42syl6eq 2514 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( 2  x.  y
)  =  1 )
4443oveq2d 6312 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x L ( 2  x.  y ) )  =  ( x L 1 ) )
4543oveq1d 6311 . . . . . . . 8  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( ( 2  x.  y )  -  1 )  =  ( 1  -  1 ) )
46 1m1e0 10625 . . . . . . . 8  |-  ( 1  -  1 )  =  0
4745, 46syl6eq 2514 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( ( 2  x.  y )  -  1 )  =  0 )
4847oveq2d 6312 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x M ( ( 2  x.  y
)  -  1 ) )  =  ( x M 0 ) )
4937, 44, 483eqtr4d 2508 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x L ( 2  x.  y ) )  =  ( x M ( ( 2  x.  y )  - 
1 ) ) )
50 retopon 21396 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
51 iccssre 11631 . . . . . . . . 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 19789 . . . . . . . 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 20296 . . . . . 6  |-  ( ph  ->  ( y  e.  ( 0 [,] ( 1  /  2 ) ) ,  x  e.  X  |->  x )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  J )  Cn  J
) )
5755, 1cnmpt1st 20295 . . . . . . 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 21558 . . . . . . . 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 6304 . . . . . . 7  |-  ( z  =  y  ->  (
2  x.  z )  =  ( 2  x.  y ) )
6155, 1, 57, 55, 59, 60cnmpt21 20298 . . . . . 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 21599 . . . . . . 7  |-  ( ph  ->  ( F ( J Htpy 
K ) G ) 
C_  ( ( J 
tX  II )  Cn  K ) )
6362, 24sseldd 3500 . . . . . 6  |-  ( ph  ->  L  e.  ( ( J  tX  II )  Cn  K ) )
6455, 1, 56, 61, 63cnmpt22f 20302 . . . . 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 11631 . . . . . . . . 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 19789 . . . . . . . 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 20296 . . . . . 6  |-  ( ph  ->  ( y  e.  ( ( 1  /  2
) [,] 1 ) ,  x  e.  X  |->  x )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( ( 1  /  2 ) [,] 1 ) )  tX  J )  Cn  J
) )
7169, 1cnmpt1st 20295 . . . . . . 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 21560 . . . . . . . 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 6311 . . . . . . 7  |-  ( z  =  y  ->  (
( 2  x.  z
)  -  1 )  =  ( ( 2  x.  y )  - 
1 ) )
7569, 1, 71, 69, 73, 74cnmpt21 20298 . . . . . 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 21599 . . . . . . 7  |-  ( ph  ->  ( G ( J Htpy 
K ) H ) 
C_  ( ( J 
tX  II )  Cn  K ) )
7776, 27sseldd 3500 . . . . . 6  |-  ( ph  ->  M  e.  ( ( J  tX  II )  Cn  K ) )
7869, 1, 70, 75, 77cnmpt22f 20302 . . . . 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 21554 . . . 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 20305 . . 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 2549 . 2  |-  ( ph  ->  N  e.  ( ( J  tX  II )  Cn  K ) )
82 simpr 461 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  s  e.  X )
83 0elunit 11663 . . . 4  |-  0  e.  ( 0 [,] 1
)
84 simpr 461 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  y  =  0 )
8584, 16syl6eqbr 4493 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  y  <_  (
1  /  2 ) )
8685iftrued 3952 . . . . . 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 ) ) )
87 simpl 457 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  x  =  s )
8884oveq2d 6312 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 2  x.  y )  =  ( 2  x.  0 ) )
89 2t0e0 10712 . . . . . . . 8  |-  ( 2  x.  0 )  =  0
9088, 89syl6eq 2514 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 2  x.  y )  =  0 )
9187, 90oveq12d 6314 . . . . . 6  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( x L ( 2  x.  y
) )  =  ( s L 0 ) )
9286, 91eqtrd 2498 . . . . 5  |-  ( ( x  =  s  /\  y  =  0 )  ->  if ( y  <_  ( 1  / 
2 ) ,  ( x L ( 2  x.  y ) ) ,  ( x M ( ( 2  x.  y )  -  1 ) ) )  =  ( s L 0 ) )
93 ovex 6324 . . . . 5  |-  ( s L 0 )  e. 
_V
9492, 4, 93ovmpt2a 6432 . . . 4  |-  ( ( s  e.  X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s N 0 )  =  ( s L 0 ) )
9582, 83, 94sylancl 662 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( s L 0 ) )
9625simpld 459 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s L 0 )  =  ( F `  s ) )
9795, 96eqtrd 2498 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( F `  s ) )
98 1elunit 11664 . . . 4  |-  1  e.  ( 0 [,] 1
)
9913, 17ltnlei 9722 . . . . . . . . 9  |-  ( ( 1  /  2 )  <  1  <->  -.  1  <_  ( 1  /  2
) )
10018, 99mpbi 208 . . . . . . . 8  |-  -.  1  <_  ( 1  /  2
)
101 simpr 461 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  y  =  1 )
102101breq1d 4466 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( y  <_ 
( 1  /  2
)  <->  1  <_  (
1  /  2 ) ) )
103100, 102mtbiri 303 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  -.  y  <_  ( 1  /  2 ) )
104103iffalsed 3955 . . . . . 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 ) ) )
105 simpl 457 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  x  =  s )
106101oveq2d 6312 . . . . . . . . . 10  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 2  x.  y )  =  ( 2  x.  1 ) )
107 2t1e2 10705 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
108106, 107syl6eq 2514 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 2  x.  y )  =  2 )
109108oveq1d 6311 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 2  x.  y )  - 
1 )  =  ( 2  -  1 ) )
110 2m1e1 10671 . . . . . . . 8  |-  ( 2  -  1 )  =  1
111109, 110syl6eq 2514 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 2  x.  y )  - 
1 )  =  1 )
112105, 111oveq12d 6314 . . . . . 6  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( x M ( ( 2  x.  y )  -  1 ) )  =  ( s M 1 ) )
113104, 112eqtrd 2498 . . . . 5  |-  ( ( x  =  s  /\  y  =  1 )  ->  if ( y  <_  ( 1  / 
2 ) ,  ( x L ( 2  x.  y ) ) ,  ( x M ( ( 2  x.  y )  -  1 ) ) )  =  ( s M 1 ) )
114 ovex 6324 . . . . 5  |-  ( s M 1 )  e. 
_V
115113, 4, 114ovmpt2a 6432 . . . 4  |-  ( ( s  e.  X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s N 1 )  =  ( s M 1 ) )
11682, 98, 115sylancl 662 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( s M 1 ) )
11728simprd 463 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s M 1 )  =  ( H `  s ) )
118116, 117eqtrd 2498 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( H `  s ) )
1191, 2, 3, 81, 97, 118ishtpyd 21601 1  |-  ( ph  ->  N  e.  ( F ( J Htpy  K ) H ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807    C_ wss 3471   ifcif 3944   class class class wbr 4456    |-> cmpt 4515   ran crn 5009   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   RRcr 9508   0cc0 9509   1c1 9510    x. cmul 9514    < clt 9645    <_ cle 9646    - cmin 9824    / cdiv 10227   2c2 10606   (,)cioo 11554   [,]cicc 11557   ↾t crest 14838   topGenctg 14855  TopOnctopon 19522    Cn ccn 19852    tX ctx 20187   IIcii 21505   Htpy chtpy 21593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-icc 11561  df-fz 11698  df-fzo 11822  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-cn 19855  df-cnp 19856  df-tx 20189  df-hmeo 20382  df-xms 20949  df-ms 20950  df-tms 20951  df-ii 21507  df-htpy 21596
This theorem is referenced by:  phtpycc  21617
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