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Theorem htpyi 21764
Description: A homotopy evaluated at its endpoints. (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 ) )
htpyi.1  |-  ( ph  ->  H  e.  ( F ( J Htpy  K ) G ) )
Assertion
Ref Expression
htpyi  |-  ( (
ph  /\  A  e.  X )  ->  (
( A H 0 )  =  ( F `
 A )  /\  ( A H 1 )  =  ( G `  A ) ) )

Proof of Theorem htpyi
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 htpyi.1 . . . 4  |-  ( ph  ->  H  e.  ( F ( J Htpy  K ) G ) )
2 ishtpy.1 . . . . 5  |-  ( ph  ->  J  e.  (TopOn `  X ) )
3 ishtpy.3 . . . . 5  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
4 ishtpy.4 . . . . 5  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
52, 3, 4ishtpy 21762 . . . 4  |-  ( 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 ) ) ) ) )
61, 5mpbid 210 . . 3  |-  ( ph  ->  ( H  e.  ( ( J  tX  II )  Cn  K )  /\  A. s  e.  X  ( ( s H 0 )  =  ( F `
 s )  /\  ( s H 1 )  =  ( G `
 s ) ) ) )
76simprd 461 . 2  |-  ( ph  ->  A. s  e.  X  ( ( s H 0 )  =  ( F `  s )  /\  ( s H 1 )  =  ( G `  s ) ) )
8 oveq1 6284 . . . . 5  |-  ( s  =  A  ->  (
s H 0 )  =  ( A H 0 ) )
9 fveq2 5848 . . . . 5  |-  ( s  =  A  ->  ( F `  s )  =  ( F `  A ) )
108, 9eqeq12d 2424 . . . 4  |-  ( s  =  A  ->  (
( s H 0 )  =  ( F `
 s )  <->  ( A H 0 )  =  ( F `  A
) ) )
11 oveq1 6284 . . . . 5  |-  ( s  =  A  ->  (
s H 1 )  =  ( A H 1 ) )
12 fveq2 5848 . . . . 5  |-  ( s  =  A  ->  ( G `  s )  =  ( G `  A ) )
1311, 12eqeq12d 2424 . . . 4  |-  ( s  =  A  ->  (
( s H 1 )  =  ( G `
 s )  <->  ( A H 1 )  =  ( G `  A
) ) )
1410, 13anbi12d 709 . . 3  |-  ( s  =  A  ->  (
( ( s H 0 )  =  ( F `  s )  /\  ( s H 1 )  =  ( G `  s ) )  <->  ( ( A H 0 )  =  ( F `  A
)  /\  ( A H 1 )  =  ( G `  A
) ) ) )
1514rspccva 3158 . 2  |-  ( ( A. s  e.  X  ( ( s H 0 )  =  ( F `  s )  /\  ( s H 1 )  =  ( G `  s ) )  /\  A  e.  X )  ->  (
( A H 0 )  =  ( F `
 A )  /\  ( A H 1 )  =  ( G `  A ) ) )
167, 15sylan 469 1  |-  ( (
ph  /\  A  e.  X )  ->  (
( A H 0 )  =  ( F `
 A )  /\  ( A H 1 )  =  ( G `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   ` cfv 5568  (class class class)co 6277   0cc0 9521   1c1 9522  TopOnctopon 19685    Cn ccn 20016    tX ctx 20351   IIcii 21669   Htpy chtpy 21757
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 6573
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 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-map 7458  df-top 19689  df-topon 19692  df-cn 20019  df-htpy 21760
This theorem is referenced by:  htpycom  21766  htpyco1  21768  htpyco2  21769  htpycc  21770  phtpy01  21775  pcohtpylem  21809  txsconlem  29524  cvmliftphtlem  29601
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