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Theorem ishtpyd 22006
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 535 . . 3  |-  ( (
ph  /\  s  e.  X )  ->  (
( s H 0 )  =  ( F `
 s )  /\  ( s H 1 )  =  ( G `
 s ) ) )
54ralrimiva 2802 . 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 22003 . 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 933 1  |-  ( ph  ->  H  e.  ( F ( J Htpy  K ) G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   ` cfv 5582  (class class class)co 6290   0cc0 9539   1c1 9540  TopOnctopon 19918    Cn ccn 20240    tX ctx 20575   IIcii 21907   Htpy chtpy 21998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-map 7474  df-top 19921  df-topon 19923  df-cn 20243  df-htpy 22001
This theorem is referenced by:  htpycom  22007  htpyid  22008  htpyco1  22009  htpyco2  22010  htpycc  22011  isphtpy2d  22018
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