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Theorem htpyi 20549
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 20547 . . . 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 463 . 2  |-  ( ph  ->  A. s  e.  X  ( ( s H 0 )  =  ( F `  s )  /\  ( s H 1 )  =  ( G `  s ) ) )
8 oveq1 6101 . . . . 5  |-  ( s  =  A  ->  (
s H 0 )  =  ( A H 0 ) )
9 fveq2 5694 . . . . 5  |-  ( s  =  A  ->  ( F `  s )  =  ( F `  A ) )
108, 9eqeq12d 2457 . . . 4  |-  ( s  =  A  ->  (
( s H 0 )  =  ( F `
 s )  <->  ( A H 0 )  =  ( F `  A
) ) )
11 oveq1 6101 . . . . 5  |-  ( s  =  A  ->  (
s H 1 )  =  ( A H 1 ) )
12 fveq2 5694 . . . . 5  |-  ( s  =  A  ->  ( G `  s )  =  ( G `  A ) )
1311, 12eqeq12d 2457 . . . 4  |-  ( s  =  A  ->  (
( s H 1 )  =  ( G `
 s )  <->  ( A H 1 )  =  ( G `  A
) ) )
1410, 13anbi12d 710 . . 3  |-  ( s  =  A  ->  (
( ( s H 0 )  =  ( F `  s )  /\  ( s H 1 )  =  ( G `  s ) )  <->  ( ( A H 0 )  =  ( F `  A
)  /\  ( A H 1 )  =  ( G `  A
) ) ) )
1514rspccva 3075 . 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 471 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 369    = wceq 1369    e. wcel 1756   A.wral 2718   ` cfv 5421  (class class class)co 6094   0cc0 9285   1c1 9286  TopOnctopon 18502    Cn ccn 18831    tX ctx 19136   IIcii 20454   Htpy chtpy 20542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-1st 6580  df-2nd 6581  df-map 7219  df-top 18506  df-topon 18509  df-cn 18834  df-htpy 20545
This theorem is referenced by:  htpycom  20551  htpyco1  20553  htpyco2  20554  htpycc  20555  phtpy01  20560  pcohtpylem  20594  txsconlem  27132  cvmliftphtlem  27209
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