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Theorem htpyi 18488
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 18486 . . . 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 201 . . 3  |-  ( ph  ->  ( H  e.  ( ( J  tX  II )  Cn  K )  /\  A. s  e.  X  ( ( s H 0 )  =  ( F `
 s )  /\  ( s H 1 )  =  ( G `
 s ) ) ) )
76simprd 449 . 2  |-  ( ph  ->  A. s  e.  X  ( ( s H 0 )  =  ( F `  s )  /\  ( s H 1 )  =  ( G `  s ) ) )
8 oveq1 5881 . . . . 5  |-  ( s  =  A  ->  (
s H 0 )  =  ( A H 0 ) )
9 fveq2 5541 . . . . 5  |-  ( s  =  A  ->  ( F `  s )  =  ( F `  A ) )
108, 9eqeq12d 2310 . . . 4  |-  ( s  =  A  ->  (
( s H 0 )  =  ( F `
 s )  <->  ( A H 0 )  =  ( F `  A
) ) )
11 oveq1 5881 . . . . 5  |-  ( s  =  A  ->  (
s H 1 )  =  ( A H 1 ) )
12 fveq2 5541 . . . . 5  |-  ( s  =  A  ->  ( G `  s )  =  ( G `  A ) )
1311, 12eqeq12d 2310 . . . 4  |-  ( s  =  A  ->  (
( s H 1 )  =  ( G `
 s )  <->  ( A H 1 )  =  ( G `  A
) ) )
1410, 13anbi12d 691 . . 3  |-  ( s  =  A  ->  (
( ( s H 0 )  =  ( F `  s )  /\  ( s H 1 )  =  ( G `  s ) )  <->  ( ( A H 0 )  =  ( F `  A
)  /\  ( A H 1 )  =  ( G `  A
) ) ) )
1514rspccva 2896 . 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 457 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 358    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754  TopOnctopon 16648    Cn ccn 16970    tX ctx 17271   IIcii 18395   Htpy chtpy 18481
This theorem is referenced by:  htpycom  18490  htpyco1  18492  htpyco2  18493  htpycc  18494  phtpy01  18499  pcohtpylem  18533  txsconlem  23786  cvmliftphtlem  23863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-top 16652  df-topon 16655  df-cn 16973  df-htpy 18484
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