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Theorem isphtpy 21209
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 19501 . . . . 5  |-  ( F  e.  ( II  Cn  J )  ->  J  e.  Top )
3 oveq2 6283 . . . . . . 7  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
4 oveq2 6283 . . . . . . . . 9  |-  ( j  =  J  ->  (
II Htpy  j )  =  ( II Htpy  J ) )
54oveqd 6292 . . . . . . . 8  |-  ( j  =  J  ->  (
f ( II Htpy  j
) g )  =  ( f ( II Htpy  J ) g ) )
6 rabeq 3100 . . . . . . . 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 16 . . . . . . 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 6334 . . . . . 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 21199 . . . . . 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 6300 . . . . . . 7  |-  ( II 
Cn  J )  e. 
_V
1110, 10mpt2ex 6850 . . . . . 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 5941 . . . . 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 20 . . . 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 6284 . . . . . 6  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f ( II Htpy  J ) g )  =  ( F ( II Htpy  J ) G ) )
15 simpl 457 . . . . . . . . . 10  |-  ( ( f  =  F  /\  g  =  G )  ->  f  =  F )
1615fveq1d 5859 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f `  0
)  =  ( F `
 0 ) )
1716eqeq2d 2474 . . . . . . . 8  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( 0 h s )  =  ( f `  0 )  <-> 
( 0 h s )  =  ( F `
 0 ) ) )
1815fveq1d 5859 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f `  1
)  =  ( F `
 1 ) )
1918eqeq2d 2474 . . . . . . . 8  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( 1 h s )  =  ( f `  1 )  <-> 
( 1 h s )  =  ( F `
 1 ) ) )
2017, 19anbi12d 710 . . . . . . 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 2896 . . . . . 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 3101 . . . . 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 466 . . . 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 6300 . . . . . 6  |-  ( F ( II Htpy  J ) G )  e.  _V
2625rabex 4591 . . . . 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 11 . . . 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 6405 . . 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 2530 . 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 6281 . . . . . 6  |-  ( h  =  H  ->  (
0 h s )  =  ( 0 H s ) )
3130eqeq1d 2462 . . . . 5  |-  ( h  =  H  ->  (
( 0 h s )  =  ( F `
 0 )  <->  ( 0 H s )  =  ( F `  0
) ) )
32 oveq 6281 . . . . . 6  |-  ( h  =  H  ->  (
1 h s )  =  ( 1 H s ) )
3332eqeq1d 2462 . . . . 5  |-  ( h  =  H  ->  (
( 1 h s )  =  ( F `
 1 )  <->  ( 1 H s )  =  ( F `  1
) ) )
3431, 33anbi12d 710 . . . 4  |-  ( h  =  H  ->  (
( ( 0 h s )  =  ( F `  0 )  /\  ( 1 h s )  =  ( F `  1 ) )  <->  ( ( 0 H s )  =  ( F `  0
)  /\  ( 1 H s )  =  ( F `  1
) ) ) )
3534ralbidv 2896 . . 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 3254 . 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 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   {crab 2811   _Vcvv 3106   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   0cc0 9481   1c1 9482   [,]cicc 11521   Topctop 19154    Cn ccn 19484   IIcii 21107   Htpy chtpy 21195   PHtpycphtpy 21196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-map 7412  df-top 19159  df-topon 19162  df-cn 19487  df-phtpy 21199
This theorem is referenced by:  phtpyhtpy  21210  phtpyi  21212  isphtpyd  21214
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