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Theorem htpycc 18494
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 18399 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
65a1i 10 . . . 4  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
7 eqid 2296 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
8 eqid 2296 . . . . 5  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )
9 eqid 2296 . . . . 5  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )
10 dfii2 18402 . . . . 5  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
11 0re 8854 . . . . . 6  |-  0  e.  RR
1211a1i 10 . . . . 5  |-  ( ph  ->  0  e.  RR )
13 1re 8853 . . . . . 6  |-  1  e.  RR
1413a1i 10 . . . . 5  |-  ( ph  ->  1  e.  RR )
15 rehalfcl 9954 . . . . . . . 8  |-  ( 1  e.  RR  ->  (
1  /  2 )  e.  RR )
1613, 15ax-mp 8 . . . . . . 7  |-  ( 1  /  2 )  e.  RR
17 halfgt0 9948 . . . . . . . 8  |-  0  <  ( 1  /  2
)
1811, 16, 17ltleii 8957 . . . . . . 7  |-  0  <_  ( 1  /  2
)
19 halflt1 9949 . . . . . . . 8  |-  ( 1  /  2 )  <  1
2016, 13, 19ltleii 8957 . . . . . . 7  |-  ( 1  /  2 )  <_ 
1
2111, 13elicc2i 10732 . . . . . . 7  |-  ( ( 1  /  2 )  e.  ( 0 [,] 1 )  <->  ( (
1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_ 
1 ) )
2216, 18, 20, 21mpbir3an 1134 . . . . . 6  |-  ( 1  /  2 )  e.  ( 0 [,] 1
)
2322a1i 10 . . . . 5  |-  ( ph  ->  ( 1  /  2
)  e.  ( 0 [,] 1 ) )
24 htpycc.5 . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
25 htpycc.7 . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  ( F ( J Htpy  K ) G ) )
261, 2, 24, 25htpyi 18488 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  X )  ->  (
( s L 0 )  =  ( F `
 s )  /\  ( s L 1 )  =  ( G `
 s ) ) )
2726simprd 449 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  X )  ->  (
s L 1 )  =  ( G `  s ) )
28 htpycc.8 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  ( G ( J Htpy  K ) H ) )
291, 24, 3, 28htpyi 18488 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  X )  ->  (
( s M 0 )  =  ( G `
 s )  /\  ( s M 1 )  =  ( H `
 s ) ) )
3029simpld 445 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  X )  ->  (
s M 0 )  =  ( G `  s ) )
3127, 30eqtr4d 2331 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  X )  ->  (
s L 1 )  =  ( s M 0 ) )
3231ralrimiva 2639 . . . . . . . 8  |-  ( ph  ->  A. s  e.  X  ( s L 1 )  =  ( s M 0 ) )
33 oveq1 5881 . . . . . . . . . 10  |-  ( s  =  x  ->  (
s L 1 )  =  ( x L 1 ) )
34 oveq1 5881 . . . . . . . . . 10  |-  ( s  =  x  ->  (
s M 0 )  =  ( x M 0 ) )
3533, 34eqeq12d 2310 . . . . . . . . 9  |-  ( s  =  x  ->  (
( s L 1 )  =  ( s M 0 )  <->  ( x L 1 )  =  ( x M 0 ) ) )
3635rspccva 2896 . . . . . . . 8  |-  ( ( A. s  e.  X  ( s L 1 )  =  ( s M 0 )  /\  x  e.  X )  ->  ( x L 1 )  =  ( x M 0 ) )
3732, 36sylan 457 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
x L 1 )  =  ( x M 0 ) )
3837adantrl 696 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x L 1 )  =  ( x M 0 ) )
39 simprl 732 . . . . . . . . 9  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
y  =  ( 1  /  2 ) )
4039oveq2d 5890 . . . . . . . 8  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( 2  x.  y
)  =  ( 2  x.  ( 1  / 
2 ) ) )
41 2cn 9832 . . . . . . . . 9  |-  2  e.  CC
42 2ne0 9845 . . . . . . . . 9  |-  2  =/=  0
4341, 42recidi 9507 . . . . . . . 8  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
4440, 43syl6eq 2344 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( 2  x.  y
)  =  1 )
4544oveq2d 5890 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x L ( 2  x.  y ) )  =  ( x L 1 ) )
4644oveq1d 5889 . . . . . . . 8  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( ( 2  x.  y )  -  1 )  =  ( 1  -  1 ) )
47 1m1e0 9830 . . . . . . . 8  |-  ( 1  -  1 )  =  0
4846, 47syl6eq 2344 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( ( 2  x.  y )  -  1 )  =  0 )
4948oveq2d 5890 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x M ( ( 2  x.  y
)  -  1 ) )  =  ( x M 0 ) )
5038, 45, 493eqtr4d 2338 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  x  e.  X ) )  -> 
( x L ( 2  x.  y ) )  =  ( x M ( ( 2  x.  y )  - 
1 ) ) )
51 retopon 18288 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
52 iccssre 10747 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( 0 [,] ( 1  /  2
) )  C_  RR )
5311, 16, 52mp2an 653 . . . . . . . 8  |-  ( 0 [,] ( 1  / 
2 ) )  C_  RR
54 resttopon 16908 . . . . . . . 8  |-  ( ( ( topGen `  ran  (,) )  e.  (TopOn `  RR )  /\  ( 0 [,] (
1  /  2 ) )  C_  RR )  ->  ( ( topGen `  ran  (,) )t  ( 0 [,] (
1  /  2 ) ) )  e.  (TopOn `  ( 0 [,] (
1  /  2 ) ) ) )
5551, 53, 54mp2an 653 . . . . . . 7  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  e.  (TopOn `  ( 0 [,] (
1  /  2 ) ) )
5655a1i 10 . . . . . 6  |-  ( ph  ->  ( ( topGen `  ran  (,) )t  ( 0 [,] (
1  /  2 ) ) )  e.  (TopOn `  ( 0 [,] (
1  /  2 ) ) ) )
5756, 1cnmpt2nd 17379 . . . . . 6  |-  ( ph  ->  ( y  e.  ( 0 [,] ( 1  /  2 ) ) ,  x  e.  X  |->  x )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  J )  Cn  J
) )
5856, 1cnmpt1st 17378 . . . . . . 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 ) ) ) ) )
598iihalf1cn 18446 . . . . . . . 8  |-  ( z  e.  ( 0 [,] ( 1  /  2
) )  |->  ( 2  x.  z ) )  e.  ( ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  Cn  II )
6059a1i 10 . . . . . . 7  |-  ( ph  ->  ( z  e.  ( 0 [,] ( 1  /  2 ) ) 
|->  ( 2  x.  z
) )  e.  ( ( ( topGen `  ran  (,) )t  ( 0 [,] (
1  /  2 ) ) )  Cn  II ) )
61 oveq2 5882 . . . . . . 7  |-  ( z  =  y  ->  (
2  x.  z )  =  ( 2  x.  y ) )
6256, 1, 58, 56, 60, 61cnmpt21 17381 . . . . . 6  |-  ( ph  ->  ( y  e.  ( 0 [,] ( 1  /  2 ) ) ,  x  e.  X  |->  ( 2  x.  y
) )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  J )  Cn  II ) )
631, 2, 24htpycn 18487 . . . . . . 7  |-  ( ph  ->  ( F ( J Htpy 
K ) G ) 
C_  ( ( J 
tX  II )  Cn  K ) )
6463, 25sseldd 3194 . . . . . 6  |-  ( ph  ->  L  e.  ( ( J  tX  II )  Cn  K ) )
6556, 1, 57, 62, 64cnmpt22f 17385 . . . . 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
) )
66 iccssre 10747 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1  / 
2 ) [,] 1
)  C_  RR )
6716, 13, 66mp2an 653 . . . . . . . 8  |-  ( ( 1  /  2 ) [,] 1 )  C_  RR
68 resttopon 16908 . . . . . . . 8  |-  ( ( ( topGen `  ran  (,) )  e.  (TopOn `  RR )  /\  ( ( 1  / 
2 ) [,] 1
)  C_  RR )  ->  ( ( topGen `  ran  (,) )t  ( ( 1  / 
2 ) [,] 1
) )  e.  (TopOn `  ( ( 1  / 
2 ) [,] 1
) ) )
6951, 67, 68mp2an 653 . . . . . . 7  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  e.  (TopOn `  ( ( 1  / 
2 ) [,] 1
) )
7069a1i 10 . . . . . 6  |-  ( ph  ->  ( ( topGen `  ran  (,) )t  ( ( 1  / 
2 ) [,] 1
) )  e.  (TopOn `  ( ( 1  / 
2 ) [,] 1
) ) )
7170, 1cnmpt2nd 17379 . . . . . 6  |-  ( ph  ->  ( y  e.  ( ( 1  /  2
) [,] 1 ) ,  x  e.  X  |->  x )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( ( 1  /  2 ) [,] 1 ) )  tX  J )  Cn  J
) )
7270, 1cnmpt1st 17378 . . . . . . 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 ) ) ) )
739iihalf2cn 18448 . . . . . . . 8  |-  ( z  e.  ( ( 1  /  2 ) [,] 1 )  |->  ( ( 2  x.  z )  -  1 ) )  e.  ( ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  Cn  II )
7473a1i 10 . . . . . . 7  |-  ( ph  ->  ( z  e.  ( ( 1  /  2
) [,] 1 ) 
|->  ( ( 2  x.  z )  -  1 ) )  e.  ( ( ( topGen `  ran  (,) )t  ( ( 1  / 
2 ) [,] 1
) )  Cn  II ) )
7561oveq1d 5889 . . . . . . 7  |-  ( z  =  y  ->  (
( 2  x.  z
)  -  1 )  =  ( ( 2  x.  y )  - 
1 ) )
7670, 1, 72, 70, 74, 75cnmpt21 17381 . . . . . 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 ) )
771, 24, 3htpycn 18487 . . . . . . 7  |-  ( ph  ->  ( G ( J Htpy 
K ) H ) 
C_  ( ( J 
tX  II )  Cn  K ) )
7877, 28sseldd 3194 . . . . . 6  |-  ( ph  ->  M  e.  ( ( J  tX  II )  Cn  K ) )
7970, 1, 71, 76, 78cnmpt22f 17385 . . . . 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
) )
807, 8, 9, 10, 12, 14, 23, 1, 50, 65, 79cnmpt2pc 18442 . . . 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
) )
816, 1, 80cnmptcom 17388 . . 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
) )
824, 81syl5eqel 2380 . 2  |-  ( ph  ->  N  e.  ( ( J  tX  II )  Cn  K ) )
83 simpr 447 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  s  e.  X )
84 0elunit 10770 . . . 4  |-  0  e.  ( 0 [,] 1
)
85 simpr 447 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  y  =  0 )
8685, 18syl6eqbr 4076 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  y  <_  (
1  /  2 ) )
87 iftrue 3584 . . . . . . 7  |-  ( y  <_  ( 1  / 
2 )  ->  if ( y  <_  (
1  /  2 ) ,  ( x L ( 2  x.  y
) ) ,  ( x M ( ( 2  x.  y )  -  1 ) ) )  =  ( x L ( 2  x.  y ) ) )
8886, 87syl 15 . . . . . 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 ) ) )
89 simpl 443 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  x  =  s )
9085oveq2d 5890 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 2  x.  y )  =  ( 2  x.  0 ) )
9141mul01i 9018 . . . . . . . 8  |-  ( 2  x.  0 )  =  0
9290, 91syl6eq 2344 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( 2  x.  y )  =  0 )
9389, 92oveq12d 5892 . . . . . 6  |-  ( ( x  =  s  /\  y  =  0 )  ->  ( x L ( 2  x.  y
) )  =  ( s L 0 ) )
9488, 93eqtrd 2328 . . . . 5  |-  ( ( x  =  s  /\  y  =  0 )  ->  if ( y  <_  ( 1  / 
2 ) ,  ( x L ( 2  x.  y ) ) ,  ( x M ( ( 2  x.  y )  -  1 ) ) )  =  ( s L 0 ) )
95 ovex 5899 . . . . 5  |-  ( s L 0 )  e. 
_V
9694, 4, 95ovmpt2a 5994 . . . 4  |-  ( ( s  e.  X  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s N 0 )  =  ( s L 0 ) )
9783, 84, 96sylancl 643 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( s L 0 ) )
9826simpld 445 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s L 0 )  =  ( F `  s ) )
9997, 98eqtrd 2328 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 0 )  =  ( F `  s ) )
100 1elunit 10771 . . . 4  |-  1  e.  ( 0 [,] 1
)
10116, 13ltnlei 8955 . . . . . . . . 9  |-  ( ( 1  /  2 )  <  1  <->  -.  1  <_  ( 1  /  2
) )
10219, 101mpbi 199 . . . . . . . 8  |-  -.  1  <_  ( 1  /  2
)
103 simpr 447 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  y  =  1 )
104103breq1d 4049 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( y  <_ 
( 1  /  2
)  <->  1  <_  (
1  /  2 ) ) )
105102, 104mtbiri 294 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  -.  y  <_  ( 1  /  2 ) )
106 iffalse 3585 . . . . . . 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 ) ) )
107105, 106syl 15 . . . . . 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 ) ) )
108 simpl 443 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  x  =  s )
109103oveq2d 5890 . . . . . . . . . 10  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 2  x.  y )  =  ( 2  x.  1 ) )
11041mulid1i 8855 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
111109, 110syl6eq 2344 . . . . . . . . 9  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( 2  x.  y )  =  2 )
112111oveq1d 5889 . . . . . . . 8  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 2  x.  y )  - 
1 )  =  ( 2  -  1 ) )
113 ax-1cn 8811 . . . . . . . . 9  |-  1  e.  CC
114 1p1e2 9856 . . . . . . . . 9  |-  ( 1  +  1 )  =  2
11541, 113, 113, 114subaddrii 9151 . . . . . . . 8  |-  ( 2  -  1 )  =  1
116112, 115syl6eq 2344 . . . . . . 7  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( ( 2  x.  y )  - 
1 )  =  1 )
117108, 116oveq12d 5892 . . . . . 6  |-  ( ( x  =  s  /\  y  =  1 )  ->  ( x M ( ( 2  x.  y )  -  1 ) )  =  ( s M 1 ) )
118107, 117eqtrd 2328 . . . . 5  |-  ( ( x  =  s  /\  y  =  1 )  ->  if ( y  <_  ( 1  / 
2 ) ,  ( x L ( 2  x.  y ) ) ,  ( x M ( ( 2  x.  y )  -  1 ) ) )  =  ( s M 1 ) )
119 ovex 5899 . . . . 5  |-  ( s M 1 )  e. 
_V
120118, 4, 119ovmpt2a 5994 . . . 4  |-  ( ( s  e.  X  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s N 1 )  =  ( s M 1 ) )
12183, 100, 120sylancl 643 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( s M 1 ) )
12229simprd 449 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
s M 1 )  =  ( H `  s ) )
123121, 122eqtrd 2328 . 2  |-  ( (
ph  /\  s  e.  X )  ->  (
s N 1 )  =  ( H `  s ) )
1241, 2, 3, 82, 99, 123ishtpyd 18489 1  |-  ( ph  ->  N  e.  ( F ( J Htpy  K ) H ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ifcif 3578   class class class wbr 4039    e. cmpt 4093   ran crn 4706   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   2c2 9811   (,)cioo 10672   [,]cicc 10675   ↾t crest 13341   topGenctg 13358  TopOnctopon 16648    Cn ccn 16970    tX ctx 17271   IIcii 18395   Htpy chtpy 18481
This theorem is referenced by:  phtpycc  18505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-cn 16973  df-cnp 16974  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903  df-ii 18397  df-htpy 18484
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