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Theorem htpyi 20505
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 20503 . . . 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 460 . 2  |-  ( ph  ->  A. s  e.  X  ( ( s H 0 )  =  ( F `  s )  /\  ( s H 1 )  =  ( G `  s ) ) )
8 oveq1 6097 . . . . 5  |-  ( s  =  A  ->  (
s H 0 )  =  ( A H 0 ) )
9 fveq2 5688 . . . . 5  |-  ( s  =  A  ->  ( F `  s )  =  ( F `  A ) )
108, 9eqeq12d 2455 . . . 4  |-  ( s  =  A  ->  (
( s H 0 )  =  ( F `
 s )  <->  ( A H 0 )  =  ( F `  A
) ) )
11 oveq1 6097 . . . . 5  |-  ( s  =  A  ->  (
s H 1 )  =  ( A H 1 ) )
12 fveq2 5688 . . . . 5  |-  ( s  =  A  ->  ( G `  s )  =  ( G `  A ) )
1311, 12eqeq12d 2455 . . . 4  |-  ( s  =  A  ->  (
( s H 1 )  =  ( G `
 s )  <->  ( A H 1 )  =  ( G `  A
) ) )
1410, 13anbi12d 705 . . 3  |-  ( s  =  A  ->  (
( ( s H 0 )  =  ( F `  s )  /\  ( s H 1 )  =  ( G `  s ) )  <->  ( ( A H 0 )  =  ( F `  A
)  /\  ( A H 1 )  =  ( G `  A
) ) ) )
1514rspccva 3069 . 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 468 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 1364    e. wcel 1761   A.wral 2713   ` cfv 5415  (class class class)co 6090   0cc0 9278   1c1 9279  TopOnctopon 18458    Cn ccn 18787    tX ctx 19092   IIcii 20410   Htpy chtpy 20498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-map 7212  df-top 18462  df-topon 18465  df-cn 18790  df-htpy 20501
This theorem is referenced by:  htpycom  20507  htpyco1  20509  htpyco2  20510  htpycc  20511  phtpy01  20516  pcohtpylem  20550  txsconlem  27059  cvmliftphtlem  27136
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