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Theorem htpycc 18958
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 18862 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
65a1i 11 . . . 4  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
7 eqid 2404 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
8 eqid 2404 . . . . 5  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )
9 eqid 2404 . . . . 5  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )
10 dfii2 18865 . . . . 5  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
11 0re 9047 . . . . . 6  |-  0  e.  RR
1211a1i 11 . . . . 5  |-  ( ph  ->  0  e.  RR )
13 1re 9046 . . . . . 6  |-  1  e.  RR
1413a1i 11 . . . . 5  |-  ( ph  ->  1  e.  RR )
1513rehalfcli 10172 . . . . . . 7  |-  ( 1  /  2 )  e.  RR
16 halfgt0 10144 . . . . . . . 8  |-  0  <  ( 1  /  2
)
1711, 15, 16ltleii 9152 . . . . . . 7  |-  0  <_  ( 1  /  2
)
18 halflt1 10145 . . . . . . . 8  |-  ( 1  /  2 )  <  1
1915, 13, 18ltleii 9152 . . . . . . 7  |-  ( 1  /  2 )  <_ 
1
2011, 13elicc2i 10932 . . . . . . 7  |-  ( ( 1  /  2 )  e.  ( 0 [,] 1 )  <->  ( (
1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_ 
1 ) )
2115, 17, 19, 20mpbir3an 1136 . . . . . 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 18952 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  X )  ->  (
( s L 0 )  =  ( F `
 s )  /\  ( s L 1 )  =  ( G `
 s ) ) )
2625simprd 450 . . . . . . . . . 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 18952 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  X )  ->  (
( s M 0 )  =  ( G `
 s )  /\  ( s M 1 )  =  ( H `
 s ) ) )
2928simpld 446 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  X )  ->  (
s M 0 )  =  ( G `  s ) )
3026, 29eqtr4d 2439 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  X )  ->  (
s L 1 )  =  ( s M 0 ) )
3130ralrimiva 2749 . . . . . . . 8  |-  ( ph  ->  A. s  e.  X  ( s L 1 )  =  ( s M 0 ) )
32 oveq1 6047 . . . . . . . . . 10  |-  ( s  =  x  ->  (
s L 1 )  =  ( x L 1 ) )
33 oveq1 6047 . . . . . . . . . 10  |-  ( s  =  x  ->  (
s M 0 )  =  ( x M 0 ) )
3432, 33eqeq12d 2418 . . . . . . . . 9  |-  ( s  =  x  ->  (
( s L 1 )  =  ( s M 0 )  <->  ( x L 1 )  =  ( x M 0 ) ) )
3534rspccva 3011 . . . . . . . 8  |-  ( ( A. s  e.  X  ( s L 1 )  =  ( s M 0 )  /\  x  e.  X )  ->  ( x L 1 )  =  ( x M 0 ) )
3631, 35sylan 458 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
x L 1 )  =  ( x M 0 ) )
3736adantrl 697 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x L 1 )  =  ( x M 0 ) )
38 simprl 733 . . . . . . . . 9  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
y  =  ( 1  /  2 ) )
3938oveq2d 6056 . . . . . . . 8  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( 2  x.  y
)  =  ( 2  x.  ( 1  / 
2 ) ) )
40 2cn 10026 . . . . . . . . 9  |-  2  e.  CC
41 2ne0 10039 . . . . . . . . 9  |-  2  =/=  0
4240, 41recidi 9701 . . . . . . . 8  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
4339, 42syl6eq 2452 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( 2  x.  y
)  =  1 )
4443oveq2d 6056 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x L ( 2  x.  y ) )  =  ( x L 1 ) )
4543oveq1d 6055 . . . . . . . 8  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( ( 2  x.  y )  -  1 )  =  ( 1  -  1 ) )
46 1m1e0 10024 . . . . . . . 8  |-  ( 1  -  1 )  =  0
4745, 46syl6eq 2452 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( ( 2  x.  y )  -  1 )  =  0 )
4847oveq2d 6056 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x M ( ( 2  x.  y
)  -  1 ) )  =  ( x M 0 ) )
4937, 44, 483eqtr4d 2446 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x L ( 2  x.  y ) )  =  ( x M ( ( 2  x.  y )  - 
1 ) ) )
50 retopon 18750 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
51 iccssre 10948 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( 0 [,] ( 1  /  2
) )  C_  RR )
5211, 15, 51mp2an 654 . . . . . . . 8  |-  ( 0 [,] ( 1  / 
2 ) )  C_  RR
53 resttopon 17179 . . . . . . . 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 654 . . . . . . 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 17654 . . . . . 6  |-  ( ph  ->  ( y  e.  ( 0 [,] ( 1  /  2 ) ) ,  x  e.  X  |->  x )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  J )  Cn  J
) )
5755, 1cnmpt1st 17653 . . . . . . 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 18910 . . . . . . . 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 6048 . . . . . . 7  |-  ( z  =  y  ->  (
2  x.  z )  =  ( 2  x.  y ) )
6155, 1, 57, 55, 59, 60cnmpt21 17656 . . . . . 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 18951 . . . . . . 7  |-  ( ph  ->  ( F ( J Htpy 
K ) G ) 
C_  ( ( J 
tX  II )  Cn  K ) )
6362, 24sseldd 3309 . . . . . 6  |-  ( ph  ->  L  e.  ( ( J  tX  II )  Cn  K ) )
6455, 1, 56, 61, 63cnmpt22f 17660 . . . . 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 10948 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1  / 
2 ) [,] 1
)  C_  RR )
6615, 13, 65mp2an 654 . . . . . . . 8  |-  ( ( 1  /  2 ) [,] 1 )  C_  RR
67 resttopon 17179 . . . . . . . 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 654 . . . . . . 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 17654 . . . . . 6  |-  ( ph  ->  ( y  e.  ( ( 1  /  2
) [,] 1 ) ,  x  e.  X  |->  x )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( ( 1  /  2 ) [,] 1 ) )  tX  J )  Cn  J
) )
7169, 1cnmpt1st 17653 . . . . . . 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 18912 . . . . . . . 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 6055 . . . . . . 7  |-  ( z  =  y  ->  (
( 2  x.  z
)  -  1 )  =  ( ( 2  x.  y )  - 
1 ) )
7569, 1, 71, 69, 73, 74cnmpt21 17656 . . . . . 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 18951 . . . . . . 7  |-  ( ph  ->  ( G ( J Htpy 
K ) H ) 
C_  ( ( J 
tX  II )  Cn  K ) )
7776, 27sseldd 3309 . . . . . 6  |-  ( ph  ->  M  e.  ( ( J  tX  II )  Cn  K ) )
7869, 1, 70, 75, 77cnmpt22f 17660 . . . . 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, 12, 14, 22, 1, 49, 64, 78cnmpt2pc 18906 . . . 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 17663 . . 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 2488 . 2  |-  ( ph  ->  N  e.  ( ( J  tX  II )  Cn  K ) )
82 simpr 448 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  s  e.  X )
83 0elunit 10971 . . . 4  |-  0  e.  ( 0 [,] 1
)
84 simpr 448 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  y  =  0 )
8584, 17syl6eqbr 4209 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  y  <_  (
1  /  2 ) )
86 iftrue 3705 . . . . . . 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 444 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  x  =  s )
8984oveq2d 6056 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 2  x.  y )  =  ( 2  x.  0 ) )
9040mul01i 9212 . . . . . . . 8  |-  ( 2  x.  0 )  =  0
9189, 90syl6eq 2452 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 2  x.  y )  =  0 )
9288, 91oveq12d 6058 . . . . . 6  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( x L ( 2  x.  y
) )  =  ( s L 0 ) )
9387, 92eqtrd 2436 . . . . 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 6065 . . . . 5  |-  ( s L 0 )  e. 
_V
9593, 4, 94ovmpt2a 6163 . . . 4  |-  ( ( s  e.  X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s N 0 )  =  ( s L 0 ) )
9682, 83, 95sylancl 644 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( s L 0 ) )
9725simpld 446 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s L 0 )  =  ( F `  s ) )
9896, 97eqtrd 2436 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( F `  s ) )
99 1elunit 10972 . . . 4  |-  1  e.  ( 0 [,] 1
)
10015, 13ltnlei 9150 . . . . . . . . 9  |-  ( ( 1  /  2 )  <  1  <->  -.  1  <_  ( 1  /  2
) )
10118, 100mpbi 200 . . . . . . . 8  |-  -.  1  <_  ( 1  /  2
)
102 simpr 448 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  y  =  1 )
103102breq1d 4182 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( y  <_ 
( 1  /  2
)  <->  1  <_  (
1  /  2 ) ) )
104101, 103mtbiri 295 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  -.  y  <_  ( 1  /  2 ) )
105 iffalse 3706 . . . . . . 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 444 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  x  =  s )
108102oveq2d 6056 . . . . . . . . . 10  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 2  x.  y )  =  ( 2  x.  1 ) )
10940mulid1i 9048 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
110108, 109syl6eq 2452 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 2  x.  y )  =  2 )
111110oveq1d 6055 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 2  x.  y )  - 
1 )  =  ( 2  -  1 ) )
112 2m1e1 10051 . . . . . . . 8  |-  ( 2  -  1 )  =  1
113111, 112syl6eq 2452 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 2  x.  y )  - 
1 )  =  1 )
114107, 113oveq12d 6058 . . . . . 6  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( x M ( ( 2  x.  y )  -  1 ) )  =  ( s M 1 ) )
115106, 114eqtrd 2436 . . . . 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 6065 . . . . 5  |-  ( s M 1 )  e. 
_V
117115, 4, 116ovmpt2a 6163 . . . 4  |-  ( ( s  e.  X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s N 1 )  =  ( s M 1 ) )
11882, 99, 117sylancl 644 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( s M 1 ) )
11928simprd 450 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s M 1 )  =  ( H `  s ) )
120118, 119eqtrd 2436 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( H `  s ) )
1211, 2, 3, 81, 98, 120ishtpyd 18953 1  |-  ( ph  ->  N  e.  ( F ( J Htpy  K ) H ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666    C_ wss 3280   ifcif 3699   class class class wbr 4172    e. cmpt 4226   ran crn 4838   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   2c2 10005   (,)cioo 10872   [,]cicc 10875   ↾t crest 13603   topGenctg 13620  TopOnctopon 16914    Cn ccn 17242    tX ctx 17545   IIcii 18858   Htpy chtpy 18945
This theorem is referenced by:  phtpycc  18969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-cn 17245  df-cnp 17246  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305  df-ii 18860  df-htpy 18948
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