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Theorem isphtpyd 18500
Description: Deduction for membership in the class of path homotopies. (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 ) )
isphtpyd.1  |-  ( ph  ->  H  e.  ( F ( II Htpy  J ) G ) )
isphtpyd.2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 H s )  =  ( F ` 
0 ) )
isphtpyd.3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 H s )  =  ( F ` 
1 ) )
Assertion
Ref Expression
isphtpyd  |-  ( ph  ->  H  e.  ( F ( PHtpy `  J ) G ) )
Distinct variable groups:    F, s    G, s    H, s    J, s    ph, s

Proof of Theorem isphtpyd
StepHypRef Expression
1 isphtpyd.1 . 2  |-  ( ph  ->  H  e.  ( F ( II Htpy  J ) G ) )
2 isphtpyd.2 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 H s )  =  ( F ` 
0 ) )
3 isphtpyd.3 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 H s )  =  ( F ` 
1 ) )
42, 3jca 518 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0 H s )  =  ( F `
 0 )  /\  ( 1 H s )  =  ( F `
 1 ) ) )
54ralrimiva 2639 . 2  |-  ( ph  ->  A. s  e.  ( 0 [,] 1 ) ( ( 0 H s )  =  ( F `  0 )  /\  ( 1 H s )  =  ( F `  1 ) ) )
6 isphtpy.2 . . 3  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
7 isphtpy.3 . . 3  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
86, 7isphtpy 18495 . 2  |-  ( 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
) ) ) ) )
91, 5, 8mpbir2and 888 1  |-  ( ph  ->  H  e.  ( F ( PHtpy `  J ) G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754   [,]cicc 10675    Cn ccn 16970   IIcii 18395   Htpy chtpy 18481   PHtpycphtpy 18482
This theorem is referenced by:  isphtpy2d  18501  phtpycom  18502  phtpyid  18503  phtpyco2  18504  phtpycc  18505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-top 16652  df-topon 16655  df-cn 16973  df-phtpy 18485
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