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Theorem htpycc 21904
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 21807 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
65a1i 11 . . . 4  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
7 eqid 2429 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
8 eqid 2429 . . . . 5  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )
9 eqid 2429 . . . . 5  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )
10 dfii2 21810 . . . . 5  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
11 0red 9643 . . . . 5  |-  ( ph  ->  0  e.  RR )
12 1red 9657 . . . . 5  |-  ( ph  ->  1  e.  RR )
13 halfre 10828 . . . . . . 7  |-  ( 1  /  2 )  e.  RR
14 0re 9642 . . . . . . . 8  |-  0  e.  RR
15 halfgt0 10830 . . . . . . . 8  |-  0  <  ( 1  /  2
)
1614, 13, 15ltleii 9756 . . . . . . 7  |-  0  <_  ( 1  /  2
)
17 1re 9641 . . . . . . . 8  |-  1  e.  RR
18 halflt1 10831 . . . . . . . 8  |-  ( 1  /  2 )  <  1
1913, 17, 18ltleii 9756 . . . . . . 7  |-  ( 1  /  2 )  <_ 
1
2014, 17elicc2i 11700 . . . . . . 7  |-  ( ( 1  /  2 )  e.  ( 0 [,] 1 )  <->  ( (
1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_ 
1 ) )
2113, 16, 19, 20mpbir3an 1187 . . . . . 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 21898 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  X )  ->  (
( s L 0 )  =  ( F `
 s )  /\  ( s L 1 )  =  ( G `
 s ) ) )
2625simprd 464 . . . . . . . . . 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 21898 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  X )  ->  (
( s M 0 )  =  ( G `
 s )  /\  ( s M 1 )  =  ( H `
 s ) ) )
2928simpld 460 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  X )  ->  (
s M 0 )  =  ( G `  s ) )
3026, 29eqtr4d 2473 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  X )  ->  (
s L 1 )  =  ( s M 0 ) )
3130ralrimiva 2846 . . . . . . . 8  |-  ( ph  ->  A. s  e.  X  ( s L 1 )  =  ( s M 0 ) )
32 oveq1 6312 . . . . . . . . . 10  |-  ( s  =  x  ->  (
s L 1 )  =  ( x L 1 ) )
33 oveq1 6312 . . . . . . . . . 10  |-  ( s  =  x  ->  (
s M 0 )  =  ( x M 0 ) )
3432, 33eqeq12d 2451 . . . . . . . . 9  |-  ( s  =  x  ->  (
( s L 1 )  =  ( s M 0 )  <->  ( x L 1 )  =  ( x M 0 ) ) )
3534rspccva 3187 . . . . . . . 8  |-  ( ( A. s  e.  X  ( s L 1 )  =  ( s M 0 )  /\  x  e.  X )  ->  ( x L 1 )  =  ( x M 0 ) )
3631, 35sylan 473 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
x L 1 )  =  ( x M 0 ) )
3736adantrl 720 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x L 1 )  =  ( x M 0 ) )
38 simprl 762 . . . . . . . . 9  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
y  =  ( 1  /  2 ) )
3938oveq2d 6321 . . . . . . . 8  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( 2  x.  y
)  =  ( 2  x.  ( 1  / 
2 ) ) )
40 2cn 10680 . . . . . . . . 9  |-  2  e.  CC
41 2ne0 10702 . . . . . . . . 9  |-  2  =/=  0
4240, 41recidi 10337 . . . . . . . 8  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
4339, 42syl6eq 2486 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( 2  x.  y
)  =  1 )
4443oveq2d 6321 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x L ( 2  x.  y ) )  =  ( x L 1 ) )
4543oveq1d 6320 . . . . . . . 8  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( ( 2  x.  y )  -  1 )  =  ( 1  -  1 ) )
46 1m1e0 10678 . . . . . . . 8  |-  ( 1  -  1 )  =  0
4745, 46syl6eq 2486 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( ( 2  x.  y )  -  1 )  =  0 )
4847oveq2d 6321 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x M ( ( 2  x.  y
)  -  1 ) )  =  ( x M 0 ) )
4937, 44, 483eqtr4d 2480 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x L ( 2  x.  y ) )  =  ( x M ( ( 2  x.  y )  - 
1 ) ) )
50 retopon 21695 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
51 iccssre 11716 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( 0 [,] ( 1  /  2
) )  C_  RR )
5214, 13, 51mp2an 676 . . . . . . . 8  |-  ( 0 [,] ( 1  / 
2 ) )  C_  RR
53 resttopon 20108 . . . . . . . 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 676 . . . . . . 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 20615 . . . . . 6  |-  ( ph  ->  ( y  e.  ( 0 [,] ( 1  /  2 ) ) ,  x  e.  X  |->  x )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  J )  Cn  J
) )
5755, 1cnmpt1st 20614 . . . . . . 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 21856 . . . . . . . 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 6313 . . . . . . 7  |-  ( z  =  y  ->  (
2  x.  z )  =  ( 2  x.  y ) )
6155, 1, 57, 55, 59, 60cnmpt21 20617 . . . . . 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 21897 . . . . . . 7  |-  ( ph  ->  ( F ( J Htpy 
K ) G ) 
C_  ( ( J 
tX  II )  Cn  K ) )
6362, 24sseldd 3471 . . . . . 6  |-  ( ph  ->  L  e.  ( ( J  tX  II )  Cn  K ) )
6455, 1, 56, 61, 63cnmpt22f 20621 . . . . 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 11716 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1  / 
2 ) [,] 1
)  C_  RR )
6613, 17, 65mp2an 676 . . . . . . . 8  |-  ( ( 1  /  2 ) [,] 1 )  C_  RR
67 resttopon 20108 . . . . . . . 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 676 . . . . . . 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 20615 . . . . . 6  |-  ( ph  ->  ( y  e.  ( ( 1  /  2
) [,] 1 ) ,  x  e.  X  |->  x )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( ( 1  /  2 ) [,] 1 ) )  tX  J )  Cn  J
) )
7169, 1cnmpt1st 20614 . . . . . . 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 21858 . . . . . . . 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 6320 . . . . . . 7  |-  ( z  =  y  ->  (
( 2  x.  z
)  -  1 )  =  ( ( 2  x.  y )  - 
1 ) )
7569, 1, 71, 69, 73, 74cnmpt21 20617 . . . . . 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 21897 . . . . . . 7  |-  ( ph  ->  ( G ( J Htpy 
K ) H ) 
C_  ( ( J 
tX  II )  Cn  K ) )
7776, 27sseldd 3471 . . . . . 6  |-  ( ph  ->  M  e.  ( ( J  tX  II )  Cn  K ) )
7869, 1, 70, 75, 77cnmpt22f 20621 . . . . 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 21852 . . . 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 20624 . . 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 2521 . 2  |-  ( ph  ->  N  e.  ( ( J  tX  II )  Cn  K ) )
82 simpr 462 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  s  e.  X )
83 0elunit 11748 . . . 4  |-  0  e.  ( 0 [,] 1
)
84 simpr 462 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  y  =  0 )
8584, 16syl6eqbr 4463 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  y  <_  (
1  /  2 ) )
8685iftrued 3923 . . . . . 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 458 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  x  =  s )
8884oveq2d 6321 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 2  x.  y )  =  ( 2  x.  0 ) )
89 2t0e0 10765 . . . . . . . 8  |-  ( 2  x.  0 )  =  0
9088, 89syl6eq 2486 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 2  x.  y )  =  0 )
9187, 90oveq12d 6323 . . . . . 6  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( x L ( 2  x.  y
) )  =  ( s L 0 ) )
9286, 91eqtrd 2470 . . . . 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 6333 . . . . 5  |-  ( s L 0 )  e. 
_V
9492, 4, 93ovmpt2a 6441 . . . 4  |-  ( ( s  e.  X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s N 0 )  =  ( s L 0 ) )
9582, 83, 94sylancl 666 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( s L 0 ) )
9625simpld 460 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s L 0 )  =  ( F `  s ) )
9795, 96eqtrd 2470 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( F `  s ) )
98 1elunit 11749 . . . 4  |-  1  e.  ( 0 [,] 1
)
9913, 17ltnlei 9754 . . . . . . . . 9  |-  ( ( 1  /  2 )  <  1  <->  -.  1  <_  ( 1  /  2
) )
10018, 99mpbi 211 . . . . . . . 8  |-  -.  1  <_  ( 1  /  2
)
101 simpr 462 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  y  =  1 )
102101breq1d 4436 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( y  <_ 
( 1  /  2
)  <->  1  <_  (
1  /  2 ) ) )
103100, 102mtbiri 304 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  -.  y  <_  ( 1  /  2 ) )
104103iffalsed 3926 . . . . . 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 458 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  x  =  s )
106101oveq2d 6321 . . . . . . . . . 10  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 2  x.  y )  =  ( 2  x.  1 ) )
107 2t1e2 10758 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
108106, 107syl6eq 2486 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 2  x.  y )  =  2 )
109108oveq1d 6320 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 2  x.  y )  - 
1 )  =  ( 2  -  1 ) )
110 2m1e1 10724 . . . . . . . 8  |-  ( 2  -  1 )  =  1
111109, 110syl6eq 2486 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 2  x.  y )  - 
1 )  =  1 )
112105, 111oveq12d 6323 . . . . . 6  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( x M ( ( 2  x.  y )  -  1 ) )  =  ( s M 1 ) )
113104, 112eqtrd 2470 . . . . 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 6333 . . . . 5  |-  ( s M 1 )  e. 
_V
115113, 4, 114ovmpt2a 6441 . . . 4  |-  ( ( s  e.  X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s N 1 )  =  ( s M 1 ) )
11682, 98, 115sylancl 666 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( s M 1 ) )
11728simprd 464 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s M 1 )  =  ( H `  s ) )
118116, 117eqtrd 2470 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( H `  s ) )
1191, 2, 3, 81, 97, 118ishtpyd 21899 1  |-  ( ph  ->  N  e.  ( F ( J Htpy  K ) H ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782    C_ wss 3442   ifcif 3915   class class class wbr 4426    |-> cmpt 4484   ran crn 4855   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   RRcr 9537   0cc0 9538   1c1 9539    x. cmul 9543    < clt 9674    <_ cle 9675    - cmin 9859    / cdiv 10268   2c2 10659   (,)cioo 11635   [,]cicc 11638   ↾t crest 15278   topGenctg 15295  TopOnctopon 19849    Cn ccn 20171    tX ctx 20506   IIcii 21803   Htpy chtpy 21891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  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 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-cn 20174  df-cnp 20175  df-tx 20508  df-hmeo 20701  df-xms 21266  df-ms 21267  df-tms 21268  df-ii 21805  df-htpy 21894
This theorem is referenced by:  phtpycc  21915
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