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Theorem phtpyi 19009
Description: Membership in the class of path homotopies between two continuous functions. (Contributed 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 ) )
phtpyi.1  |-  ( ph  ->  H  e.  ( F ( PHtpy `  J ) G ) )
Assertion
Ref Expression
phtpyi  |-  ( (
ph  /\  A  e.  ( 0 [,] 1
) )  ->  (
( 0 H A )  =  ( F `
 0 )  /\  ( 1 H A )  =  ( F `
 1 ) ) )

Proof of Theorem phtpyi
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 phtpyi.1 . . . 4  |-  ( ph  ->  H  e.  ( F ( PHtpy `  J ) G ) )
2 isphtpy.2 . . . . 5  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
3 isphtpy.3 . . . . 5  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
42, 3isphtpy 19006 . . . 4  |-  ( 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
) ) ) ) )
51, 4mpbid 202 . . 3  |-  ( ph  ->  ( H  e.  ( F ( II Htpy  J
) G )  /\  A. s  e.  ( 0 [,] 1 ) ( ( 0 H s )  =  ( F `
 0 )  /\  ( 1 H s )  =  ( F `
 1 ) ) ) )
65simprd 450 . 2  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( ( 0 H s )  =  ( F `  0 )  /\  ( 1 H s )  =  ( F `  1 ) ) )
7 oveq2 6089 . . . . 5  |-  ( s  =  A  ->  (
0 H s )  =  ( 0 H A ) )
87eqeq1d 2444 . . . 4  |-  ( s  =  A  ->  (
( 0 H s )  =  ( F `
 0 )  <->  ( 0 H A )  =  ( F `  0
) ) )
9 oveq2 6089 . . . . 5  |-  ( s  =  A  ->  (
1 H s )  =  ( 1 H A ) )
109eqeq1d 2444 . . . 4  |-  ( s  =  A  ->  (
( 1 H s )  =  ( F `
 1 )  <->  ( 1 H A )  =  ( F `  1
) ) )
118, 10anbi12d 692 . . 3  |-  ( s  =  A  ->  (
( ( 0 H s )  =  ( F `  0 )  /\  ( 1 H s )  =  ( F `  1 ) )  <->  ( ( 0 H A )  =  ( F `  0
)  /\  ( 1 H A )  =  ( F `  1
) ) ) )
1211rspccva 3051 . 2  |-  ( ( A. s  e.  ( 0 [,] 1 ) ( ( 0 H s )  =  ( F `  0 )  /\  ( 1 H s )  =  ( F `  1 ) )  /\  A  e.  ( 0 [,] 1
) )  ->  (
( 0 H A )  =  ( F `
 0 )  /\  ( 1 H A )  =  ( F `
 1 ) ) )
136, 12sylan 458 1  |-  ( (
ph  /\  A  e.  ( 0 [,] 1
) )  ->  (
( 0 H A )  =  ( F `
 0 )  /\  ( 1 H A )  =  ( F `
 1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   ` cfv 5454  (class class class)co 6081   0cc0 8990   1c1 8991   [,]cicc 10919    Cn ccn 17288   IIcii 18905   Htpy chtpy 18992   PHtpycphtpy 18993
This theorem is referenced by:  phtpy01  19010  phtpycom  19013  phtpyco2  19015  phtpycc  19016  pcohtpylem  19044  txsconlem  24927  cvmliftphtlem  25004  cvmliftpht  25005
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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