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Theorem htpyi 19001
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 18999 . . . 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 203 . . 3  |-  ( ph  ->  ( H  e.  ( ( J  tX  II )  Cn  K )  /\  A. s  e.  X  ( ( s H 0 )  =  ( F `
 s )  /\  ( s H 1 )  =  ( G `
 s ) ) ) )
76simprd 451 . 2  |-  ( ph  ->  A. s  e.  X  ( ( s H 0 )  =  ( F `  s )  /\  ( s H 1 )  =  ( G `  s ) ) )
8 oveq1 6090 . . . . 5  |-  ( s  =  A  ->  (
s H 0 )  =  ( A H 0 ) )
9 fveq2 5730 . . . . 5  |-  ( s  =  A  ->  ( F `  s )  =  ( F `  A ) )
108, 9eqeq12d 2452 . . . 4  |-  ( s  =  A  ->  (
( s H 0 )  =  ( F `
 s )  <->  ( A H 0 )  =  ( F `  A
) ) )
11 oveq1 6090 . . . . 5  |-  ( s  =  A  ->  (
s H 1 )  =  ( A H 1 ) )
12 fveq2 5730 . . . . 5  |-  ( s  =  A  ->  ( G `  s )  =  ( G `  A ) )
1311, 12eqeq12d 2452 . . . 4  |-  ( s  =  A  ->  (
( s H 1 )  =  ( G `
 s )  <->  ( A H 1 )  =  ( G `  A
) ) )
1410, 13anbi12d 693 . . 3  |-  ( s  =  A  ->  (
( ( s H 0 )  =  ( F `  s )  /\  ( s H 1 )  =  ( G `  s ) )  <->  ( ( A H 0 )  =  ( F `  A
)  /\  ( A H 1 )  =  ( G `  A
) ) ) )
1514rspccva 3053 . 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 459 1  |-  ( (
ph  /\  A  e.  X )  ->  (
( A H 0 )  =  ( F `
 A )  /\  ( A H 1 )  =  ( G `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   ` cfv 5456  (class class class)co 6083   0cc0 8992   1c1 8993  TopOnctopon 16961    Cn ccn 17290    tX ctx 17594   IIcii 18907   Htpy chtpy 18994
This theorem is referenced by:  htpycom  19003  htpyco1  19005  htpyco2  19006  htpycc  19007  phtpy01  19012  pcohtpylem  19046  txsconlem  24929  cvmliftphtlem  25006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-map 7022  df-top 16965  df-topon 16968  df-cn 17293  df-htpy 18997
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