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Theorem isphtpy 18479
Description: Membership in the class of path homotopies between two continuous functions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
isphtpy.2  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
isphtpy.3  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
Assertion
Ref Expression
isphtpy  |-  ( ph  ->  ( H  e.  ( F ( PHtpy `  J
) G )  <->  ( H  e.  ( F ( II Htpy  J ) G )  /\  A. s  e.  ( 0 [,] 1
) ( ( 0 H s )  =  ( F `  0
)  /\  ( 1 H s )  =  ( F `  1
) ) ) ) )
Distinct variable groups:    F, s    G, s    H, s    J, s    ph, s

Proof of Theorem isphtpy
Dummy variables  f 
g  h  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isphtpy.2 . . . . 5  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
2 cntop2 16971 . . . . 5  |-  ( F  e.  ( II  Cn  J )  ->  J  e.  Top )
3 oveq2 5866 . . . . . . 7  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
4 oveq2 5866 . . . . . . . . 9  |-  ( j  =  J  ->  (
II Htpy  j )  =  ( II Htpy  J ) )
54oveqd 5875 . . . . . . . 8  |-  ( j  =  J  ->  (
f ( II Htpy  j
) g )  =  ( f ( II Htpy  J ) g ) )
6 rabeq 2782 . . . . . . . 8  |-  ( ( f ( II Htpy  j
) g )  =  ( f ( II Htpy  J ) g )  ->  { h  e.  ( f ( II Htpy 
j ) g )  |  A. s  e.  ( 0 [,] 1
) ( ( 0 h s )  =  ( f `  0
)  /\  ( 1 h s )  =  ( f `  1
) ) }  =  { h  e.  (
f ( II Htpy  J
) g )  | 
A. s  e.  ( 0 [,] 1 ) ( ( 0 h s )  =  ( f `  0 )  /\  ( 1 h s )  =  ( f `  1 ) ) } )
75, 6syl 15 . . . . . . 7  |-  ( j  =  J  ->  { h  e.  ( f ( II Htpy 
j ) g )  |  A. s  e.  ( 0 [,] 1
) ( ( 0 h s )  =  ( f `  0
)  /\  ( 1 h s )  =  ( f `  1
) ) }  =  { h  e.  (
f ( II Htpy  J
) g )  | 
A. s  e.  ( 0 [,] 1 ) ( ( 0 h s )  =  ( f `  0 )  /\  ( 1 h s )  =  ( f `  1 ) ) } )
83, 3, 7mpt2eq123dv 5910 . . . . . 6  |-  ( j  =  J  ->  (
f  e.  ( II 
Cn  j ) ,  g  e.  ( II 
Cn  j )  |->  { h  e.  ( f ( II Htpy  j ) g )  |  A. s  e.  ( 0 [,] 1 ) ( ( 0 h s )  =  ( f `
 0 )  /\  ( 1 h s )  =  ( f `
 1 ) ) } )  =  ( f  e.  ( II 
Cn  J ) ,  g  e.  ( II 
Cn  J )  |->  { h  e.  ( f ( II Htpy  J ) g )  |  A. s  e.  ( 0 [,] 1 ) ( ( 0 h s )  =  ( f `
 0 )  /\  ( 1 h s )  =  ( f `
 1 ) ) } ) )
9 df-phtpy 18469 . . . . . 6  |-  PHtpy  =  ( j  e.  Top  |->  ( f  e.  ( II 
Cn  j ) ,  g  e.  ( II 
Cn  j )  |->  { h  e.  ( f ( II Htpy  j ) g )  |  A. s  e.  ( 0 [,] 1 ) ( ( 0 h s )  =  ( f `
 0 )  /\  ( 1 h s )  =  ( f `
 1 ) ) } ) )
10 ovex 5883 . . . . . . 7  |-  ( II 
Cn  J )  e. 
_V
1110, 10mpt2ex 6198 . . . . . 6  |-  ( f  e.  ( II  Cn  J ) ,  g  e.  ( II  Cn  J )  |->  { h  e.  ( f ( II Htpy  J ) g )  |  A. s  e.  ( 0 [,] 1
) ( ( 0 h s )  =  ( f `  0
)  /\  ( 1 h s )  =  ( f `  1
) ) } )  e.  _V
128, 9, 11fvmpt 5602 . . . . 5  |-  ( J  e.  Top  ->  ( PHtpy `  J )  =  ( f  e.  ( II  Cn  J ) ,  g  e.  ( II  Cn  J ) 
|->  { h  e.  ( f ( II Htpy  J
) g )  | 
A. s  e.  ( 0 [,] 1 ) ( ( 0 h s )  =  ( f `  0 )  /\  ( 1 h s )  =  ( f `  1 ) ) } ) )
131, 2, 123syl 18 . . . 4  |-  ( ph  ->  ( PHtpy `  J )  =  ( f  e.  ( II  Cn  J
) ,  g  e.  ( II  Cn  J
)  |->  { h  e.  ( f ( II Htpy  J ) g )  |  A. s  e.  ( 0 [,] 1
) ( ( 0 h s )  =  ( f `  0
)  /\  ( 1 h s )  =  ( f `  1
) ) } ) )
14 oveq12 5867 . . . . . 6  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f ( II Htpy  J ) g )  =  ( F ( II Htpy  J ) G ) )
15 simpl 443 . . . . . . . . . 10  |-  ( ( f  =  F  /\  g  =  G )  ->  f  =  F )
1615fveq1d 5527 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f `  0
)  =  ( F `
 0 ) )
1716eqeq2d 2294 . . . . . . . 8  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( 0 h s )  =  ( f `  0 )  <-> 
( 0 h s )  =  ( F `
 0 ) ) )
1815fveq1d 5527 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f `  1
)  =  ( F `
 1 ) )
1918eqeq2d 2294 . . . . . . . 8  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( 1 h s )  =  ( f `  1 )  <-> 
( 1 h s )  =  ( F `
 1 ) ) )
2017, 19anbi12d 691 . . . . . . 7  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( ( 0 h s )  =  ( f `  0
)  /\  ( 1 h s )  =  ( f `  1
) )  <->  ( (
0 h s )  =  ( F ` 
0 )  /\  (
1 h s )  =  ( F ` 
1 ) ) ) )
2120ralbidv 2563 . . . . . 6  |-  ( ( f  =  F  /\  g  =  G )  ->  ( A. s  e.  ( 0 [,] 1
) ( ( 0 h s )  =  ( f `  0
)  /\  ( 1 h s )  =  ( f `  1
) )  <->  A. s  e.  ( 0 [,] 1
) ( ( 0 h s )  =  ( F `  0
)  /\  ( 1 h s )  =  ( F `  1
) ) ) )
2214, 21rabeqbidv 2783 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  { h  e.  ( f ( II Htpy  J
) g )  | 
A. s  e.  ( 0 [,] 1 ) ( ( 0 h s )  =  ( f `  0 )  /\  ( 1 h s )  =  ( f `  1 ) ) }  =  {
h  e.  ( F ( II Htpy  J ) G )  |  A. s  e.  ( 0 [,] 1 ) ( ( 0 h s )  =  ( F `
 0 )  /\  ( 1 h s )  =  ( F `
 1 ) ) } )
2322adantl 452 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  ->  { h  e.  (
f ( II Htpy  J
) g )  | 
A. s  e.  ( 0 [,] 1 ) ( ( 0 h s )  =  ( f `  0 )  /\  ( 1 h s )  =  ( f `  1 ) ) }  =  {
h  e.  ( F ( II Htpy  J ) G )  |  A. s  e.  ( 0 [,] 1 ) ( ( 0 h s )  =  ( F `
 0 )  /\  ( 1 h s )  =  ( F `
 1 ) ) } )
24 isphtpy.3 . . . 4  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
25 ovex 5883 . . . . . 6  |-  ( F ( II Htpy  J ) G )  e.  _V
2625rabex 4165 . . . . 5  |-  { h  e.  ( F ( II Htpy  J ) G )  |  A. s  e.  ( 0 [,] 1
) ( ( 0 h s )  =  ( F `  0
)  /\  ( 1 h s )  =  ( F `  1
) ) }  e.  _V
2726a1i 10 . . . 4  |-  ( ph  ->  { h  e.  ( F ( II Htpy  J
) G )  | 
A. s  e.  ( 0 [,] 1 ) ( ( 0 h s )  =  ( F `  0 )  /\  ( 1 h s )  =  ( F `  1 ) ) }  e.  _V )
2813, 23, 1, 24, 27ovmpt2d 5975 . . 3  |-  ( ph  ->  ( F ( PHtpy `  J ) G )  =  { h  e.  ( F ( II Htpy  J ) G )  |  A. s  e.  ( 0 [,] 1
) ( ( 0 h s )  =  ( F `  0
)  /\  ( 1 h s )  =  ( F `  1
) ) } )
2928eleq2d 2350 . 2  |-  ( ph  ->  ( H  e.  ( F ( PHtpy `  J
) G )  <->  H  e.  { h  e.  ( F ( II Htpy  J ) G )  |  A. s  e.  ( 0 [,] 1 ) ( ( 0 h s )  =  ( F `
 0 )  /\  ( 1 h s )  =  ( F `
 1 ) ) } ) )
30 oveq 5864 . . . . . 6  |-  ( h  =  H  ->  (
0 h s )  =  ( 0 H s ) )
3130eqeq1d 2291 . . . . 5  |-  ( h  =  H  ->  (
( 0 h s )  =  ( F `
 0 )  <->  ( 0 H s )  =  ( F `  0
) ) )
32 oveq 5864 . . . . . 6  |-  ( h  =  H  ->  (
1 h s )  =  ( 1 H s ) )
3332eqeq1d 2291 . . . . 5  |-  ( h  =  H  ->  (
( 1 h s )  =  ( F `
 1 )  <->  ( 1 H s )  =  ( F `  1
) ) )
3431, 33anbi12d 691 . . . 4  |-  ( h  =  H  ->  (
( ( 0 h s )  =  ( F `  0 )  /\  ( 1 h s )  =  ( F `  1 ) )  <->  ( ( 0 H s )  =  ( F `  0
)  /\  ( 1 H s )  =  ( F `  1
) ) ) )
3534ralbidv 2563 . . 3  |-  ( h  =  H  ->  ( A. s  e.  (
0 [,] 1 ) ( ( 0 h s )  =  ( F `  0 )  /\  ( 1 h s )  =  ( F `  1 ) )  <->  A. s  e.  ( 0 [,] 1 ) ( ( 0 H s )  =  ( F `  0 )  /\  ( 1 H s )  =  ( F `  1 ) ) ) )
3635elrab 2923 . 2  |-  ( H  e.  { h  e.  ( F ( II Htpy  J ) G )  |  A. s  e.  ( 0 [,] 1
) ( ( 0 h s )  =  ( F `  0
)  /\  ( 1 h s )  =  ( F `  1
) ) }  <->  ( H  e.  ( F ( II Htpy  J ) G )  /\  A. s  e.  ( 0 [,] 1
) ( ( 0 H s )  =  ( F `  0
)  /\  ( 1 H s )  =  ( F `  1
) ) ) )
3729, 36syl6bb 252 1  |-  ( ph  ->  ( H  e.  ( F ( PHtpy `  J
) G )  <->  ( H  e.  ( F ( II Htpy  J ) G )  /\  A. s  e.  ( 0 [,] 1
) ( ( 0 H s )  =  ( F `  0
)  /\  ( 1 H s )  =  ( F `  1
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   0cc0 8737   1c1 8738   [,]cicc 10659   Topctop 16631    Cn ccn 16954   IIcii 18379   Htpy chtpy 18465   PHtpycphtpy 18466
This theorem is referenced by:  phtpyhtpy  18480  phtpyi  18482  isphtpyd  18484
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957  df-phtpy 18469
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