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Theorem phtpyi 21908
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 21905 . . . 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 213 . . 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 464 . 2  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( ( 0 H s )  =  ( F `  0 )  /\  ( 1 H s )  =  ( F `  1 ) ) )
7 oveq2 6313 . . . . 5  |-  ( s  =  A  ->  (
0 H s )  =  ( 0 H A ) )
87eqeq1d 2431 . . . 4  |-  ( s  =  A  ->  (
( 0 H s )  =  ( F `
 0 )  <->  ( 0 H A )  =  ( F `  0
) ) )
9 oveq2 6313 . . . . 5  |-  ( s  =  A  ->  (
1 H s )  =  ( 1 H A ) )
109eqeq1d 2431 . . . 4  |-  ( s  =  A  ->  (
( 1 H s )  =  ( F `
 1 )  <->  ( 1 H A )  =  ( F `  1
) ) )
118, 10anbi12d 715 . . 3  |-  ( s  =  A  ->  (
( ( 0 H s )  =  ( F `  0 )  /\  ( 1 H s )  =  ( F `  1 ) )  <->  ( ( 0 H A )  =  ( F `  0
)  /\  ( 1 H A )  =  ( F `  1
) ) ) )
1211rspccva 3187 . 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 473 1  |-  ( (
ph  /\  A  e.  ( 0 [,] 1
) )  ->  (
( 0 H A )  =  ( F `
 0 )  /\  ( 1 H A )  =  ( F `
 1 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   ` cfv 5601  (class class class)co 6305   0cc0 9538   1c1 9539   [,]cicc 11638    Cn ccn 20171   IIcii 21803   Htpy chtpy 21891   PHtpycphtpy 21892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7482  df-top 19852  df-topon 19854  df-cn 20174  df-phtpy 21895
This theorem is referenced by:  phtpy01  21909  phtpycom  21912  phtpyco2  21914  phtpycc  21915  pcohtpylem  21943  txsconlem  29751  cvmliftphtlem  29828  cvmliftpht  29829
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