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Theorem ishtpyd 21659
Description: Deduction for membership in the class of homotopies. (Contributed by Mario Carneiro, 22-Feb-2015.)
Hypotheses
Ref Expression
ishtpy.1  |-  ( ph  ->  J  e.  (TopOn `  X ) )
ishtpy.3  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
ishtpy.4  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
ishtpyd.1  |-  ( ph  ->  H  e.  ( ( J  tX  II )  Cn  K ) )
ishtpyd.2  |-  ( (
ph  /\  s  e.  X )  ->  (
s H 0 )  =  ( F `  s ) )
ishtpyd.3  |-  ( (
ph  /\  s  e.  X )  ->  (
s H 1 )  =  ( G `  s ) )
Assertion
Ref Expression
ishtpyd  |-  ( ph  ->  H  e.  ( F ( J Htpy  K ) G ) )
Distinct variable groups:    F, s    G, s    H, s    J, s    ph, s    X, s
Allowed substitution hint:    K( s)

Proof of Theorem ishtpyd
StepHypRef Expression
1 ishtpyd.1 . 2  |-  ( ph  ->  H  e.  ( ( J  tX  II )  Cn  K ) )
2 ishtpyd.2 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  (
s H 0 )  =  ( F `  s ) )
3 ishtpyd.3 . . . 4  |-  ( (
ph  /\  s  e.  X )  ->  (
s H 1 )  =  ( G `  s ) )
42, 3jca 530 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( s H 0 )  =  ( F `
 s )  /\  ( s H 1 )  =  ( G `
 s ) ) )
54ralrimiva 2817 . 2  |-  ( ph  ->  A. s  e.  X  ( ( s H 0 )  =  ( F `  s )  /\  ( s H 1 )  =  ( G `  s ) ) )
6 ishtpy.1 . . 3  |-  ( ph  ->  J  e.  (TopOn `  X ) )
7 ishtpy.3 . . 3  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
8 ishtpy.4 . . 3  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
96, 7, 8ishtpy 21656 . 2  |-  ( ph  ->  ( H  e.  ( F ( J Htpy  K
) G )  <->  ( H  e.  ( ( J  tX  II )  Cn  K
)  /\  A. s  e.  X  ( (
s H 0 )  =  ( F `  s )  /\  (
s H 1 )  =  ( G `  s ) ) ) ) )
101, 5, 9mpbir2and 923 1  |-  ( ph  ->  H  e.  ( F ( J Htpy  K ) G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   ` cfv 5525  (class class class)co 6234   0cc0 9442   1c1 9443  TopOnctopon 19579    Cn ccn 19910    tX ctx 20245   IIcii 21563   Htpy chtpy 21651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-map 7379  df-top 19583  df-topon 19586  df-cn 19913  df-htpy 21654
This theorem is referenced by:  htpycom  21660  htpyid  21661  htpyco1  21662  htpyco2  21663  htpycc  21664  isphtpy2d  21671
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