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Theorem isphtpy 19006
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 17305 . . . . 5  |-  ( F  e.  ( II  Cn  J )  ->  J  e.  Top )
3 oveq2 6089 . . . . . . 7  |-  ( j  =  J  ->  (
II  Cn  j )  =  ( II  Cn  J ) )
4 oveq2 6089 . . . . . . . . 9  |-  ( j  =  J  ->  (
II Htpy  j )  =  ( II Htpy  J ) )
54oveqd 6098 . . . . . . . 8  |-  ( j  =  J  ->  (
f ( II Htpy  j
) g )  =  ( f ( II Htpy  J ) g ) )
6 rabeq 2950 . . . . . . . 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 6136 . . . . . 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 18996 . . . . . 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 6106 . . . . . . 7  |-  ( II 
Cn  J )  e. 
_V
1110, 10mpt2ex 6425 . . . . . 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 5806 . . . . 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 19 . . . 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 6090 . . . . . 6  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f ( II Htpy  J ) g )  =  ( F ( II Htpy  J ) G ) )
15 simpl 444 . . . . . . . . . 10  |-  ( ( f  =  F  /\  g  =  G )  ->  f  =  F )
1615fveq1d 5730 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f `  0
)  =  ( F `
 0 ) )
1716eqeq2d 2447 . . . . . . . 8  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( 0 h s )  =  ( f `  0 )  <-> 
( 0 h s )  =  ( F `
 0 ) ) )
1815fveq1d 5730 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f `  1
)  =  ( F `
 1 ) )
1918eqeq2d 2447 . . . . . . . 8  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( 1 h s )  =  ( f `  1 )  <-> 
( 1 h s )  =  ( F `
 1 ) ) )
2017, 19anbi12d 692 . . . . . . 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 2725 . . . . . 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 2951 . . . . 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 453 . . . 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 6106 . . . . . 6  |-  ( F ( II Htpy  J ) G )  e.  _V
2625rabex 4354 . . . . 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 6201 . . 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 2503 . 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 6087 . . . . . 6  |-  ( h  =  H  ->  (
0 h s )  =  ( 0 H s ) )
3130eqeq1d 2444 . . . . 5  |-  ( h  =  H  ->  (
( 0 h s )  =  ( F `
 0 )  <->  ( 0 H s )  =  ( F `  0
) ) )
32 oveq 6087 . . . . . 6  |-  ( h  =  H  ->  (
1 h s )  =  ( 1 H s ) )
3332eqeq1d 2444 . . . . 5  |-  ( h  =  H  ->  (
( 1 h s )  =  ( F `
 1 )  <->  ( 1 H s )  =  ( F `  1
) ) )
3431, 33anbi12d 692 . . . 4  |-  ( h  =  H  ->  (
( ( 0 h s )  =  ( F `  0 )  /\  ( 1 h s )  =  ( F `  1 ) )  <->  ( ( 0 H s )  =  ( F `  0
)  /\  ( 1 H s )  =  ( F `  1
) ) ) )
3534ralbidv 2725 . . 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 3092 . 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 253 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   0cc0 8990   1c1 8991   [,]cicc 10919   Topctop 16958    Cn ccn 17288   IIcii 18905   Htpy chtpy 18992   PHtpycphtpy 18993
This theorem is referenced by:  phtpyhtpy  19007  phtpyi  19009  isphtpyd  19011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-map 7020  df-top 16963  df-topon 16966  df-cn 17291  df-phtpy 18996
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