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Theorem phtpyco2 21220
Description: Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
Hypotheses
Ref Expression
phtpyco2.f  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
phtpyco2.g  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
phtpyco2.p  |-  ( ph  ->  P  e.  ( J  Cn  K ) )
phtpyco2.h  |-  ( ph  ->  H  e.  ( F ( PHtpy `  J ) G ) )
Assertion
Ref Expression
phtpyco2  |-  ( ph  ->  ( P  o.  H
)  e.  ( ( P  o.  F ) ( PHtpy `  K )
( P  o.  G
) ) )

Proof of Theorem phtpyco2
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 phtpyco2.f . . 3  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
2 phtpyco2.p . . 3  |-  ( ph  ->  P  e.  ( J  Cn  K ) )
3 cnco 19528 . . 3  |-  ( ( F  e.  ( II 
Cn  J )  /\  P  e.  ( J  Cn  K ) )  -> 
( P  o.  F
)  e.  ( II 
Cn  K ) )
41, 2, 3syl2anc 661 . 2  |-  ( ph  ->  ( P  o.  F
)  e.  ( II 
Cn  K ) )
5 phtpyco2.g . . 3  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
6 cnco 19528 . . 3  |-  ( ( G  e.  ( II 
Cn  J )  /\  P  e.  ( J  Cn  K ) )  -> 
( P  o.  G
)  e.  ( II 
Cn  K ) )
75, 2, 6syl2anc 661 . 2  |-  ( ph  ->  ( P  o.  G
)  e.  ( II 
Cn  K ) )
81, 5phtpyhtpy 21212 . . . 4  |-  ( ph  ->  ( F ( PHtpy `  J ) G ) 
C_  ( F ( II Htpy  J ) G ) )
9 phtpyco2.h . . . 4  |-  ( ph  ->  H  e.  ( F ( PHtpy `  J ) G ) )
108, 9sseldd 3500 . . 3  |-  ( ph  ->  H  e.  ( F ( II Htpy  J ) G ) )
111, 5, 2, 10htpyco2 21209 . 2  |-  ( ph  ->  ( P  o.  H
)  e.  ( ( P  o.  F ) ( II Htpy  K ) ( P  o.  G
) ) )
121, 5, 9phtpyi 21214 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0 H s )  =  ( F `
 0 )  /\  ( 1 H s )  =  ( F `
 1 ) ) )
1312simpld 459 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 H s )  =  ( F ` 
0 ) )
1413fveq2d 5863 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( P `  ( 0 H s ) )  =  ( P `  ( F `  0 ) ) )
15 0elunit 11629 . . . . . 6  |-  0  e.  ( 0 [,] 1
)
16 simpr 461 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
17 opelxpi 5025 . . . . . 6  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  <. 0 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
1815, 16, 17sylancr 663 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  <. 0 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
19 iitopon 21113 . . . . . . . . 9  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
20 txtopon 19822 . . . . . . . . 9  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  II  e.  (TopOn `  ( 0 [,] 1 ) ) )  ->  ( II  tX  II )  e.  (TopOn `  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
2119, 19, 20mp2an 672 . . . . . . . 8  |-  ( II 
tX  II )  e.  (TopOn `  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
2221a1i 11 . . . . . . 7  |-  ( ph  ->  ( II  tX  II )  e.  (TopOn `  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) ) )
23 cntop2 19503 . . . . . . . . 9  |-  ( F  e.  ( II  Cn  J )  ->  J  e.  Top )
241, 23syl 16 . . . . . . . 8  |-  ( ph  ->  J  e.  Top )
25 eqid 2462 . . . . . . . . 9  |-  U. J  =  U. J
2625toptopon 19196 . . . . . . . 8  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
2724, 26sylib 196 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
281, 5phtpycn 21213 . . . . . . . 8  |-  ( ph  ->  ( F ( PHtpy `  J ) G ) 
C_  ( ( II 
tX  II )  Cn  J ) )
2928, 9sseldd 3500 . . . . . . 7  |-  ( ph  ->  H  e.  ( ( II  tX  II )  Cn  J ) )
30 cnf2 19511 . . . . . . 7  |-  ( ( ( II  tX  II )  e.  (TopOn `  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  /\  J  e.  (TopOn `  U. J )  /\  H  e.  ( ( II  tX  II )  Cn  J ) )  ->  H : ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) --> U. J )
3122, 27, 29, 30syl3anc 1223 . . . . . 6  |-  ( ph  ->  H : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> U. J )
32 fvco3 5937 . . . . . 6  |-  ( ( H : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> U. J  /\  <. 0 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  ->  ( ( P  o.  H ) `  <. 0 ,  s
>. )  =  ( P `  ( H `  <. 0 ,  s
>. ) ) )
3331, 32sylan 471 . . . . 5  |-  ( (
ph  /\  <. 0 ,  s >.  e.  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  ->  ( ( P  o.  H ) `  <. 0 ,  s
>. )  =  ( P `  ( H `  <. 0 ,  s
>. ) ) )
3418, 33syldan 470 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( P  o.  H
) `  <. 0 ,  s >. )  =  ( P `  ( H `
 <. 0 ,  s
>. ) ) )
35 df-ov 6280 . . . 4  |-  ( 0 ( P  o.  H
) s )  =  ( ( P  o.  H ) `  <. 0 ,  s >. )
36 df-ov 6280 . . . . 5  |-  ( 0 H s )  =  ( H `  <. 0 ,  s >. )
3736fveq2i 5862 . . . 4  |-  ( P `
 ( 0 H s ) )  =  ( P `  ( H `  <. 0 ,  s >. ) )
3834, 35, 373eqtr4g 2528 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 ( P  o.  H ) s )  =  ( P `  ( 0 H s ) ) )
39 iiuni 21115 . . . . . . 7  |-  ( 0 [,] 1 )  = 
U. II
4039, 25cnf 19508 . . . . . 6  |-  ( F  e.  ( II  Cn  J )  ->  F : ( 0 [,] 1 ) --> U. J
)
411, 40syl 16 . . . . 5  |-  ( ph  ->  F : ( 0 [,] 1 ) --> U. J )
4241adantr 465 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  F : ( 0 [,] 1 ) --> U. J
)
43 fvco3 5937 . . . 4  |-  ( ( F : ( 0 [,] 1 ) --> U. J  /\  0  e.  ( 0 [,] 1
) )  ->  (
( P  o.  F
) `  0 )  =  ( P `  ( F `  0 ) ) )
4442, 15, 43sylancl 662 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( P  o.  F
) `  0 )  =  ( P `  ( F `  0 ) ) )
4514, 38, 443eqtr4d 2513 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 ( P  o.  H ) s )  =  ( ( P  o.  F ) ` 
0 ) )
4612simprd 463 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 H s )  =  ( F ` 
1 ) )
4746fveq2d 5863 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( P `  ( 1 H s ) )  =  ( P `  ( F `  1 ) ) )
48 1elunit 11630 . . . . . 6  |-  1  e.  ( 0 [,] 1
)
49 opelxpi 5025 . . . . . 6  |-  ( ( 1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  <. 1 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
5048, 16, 49sylancr 663 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  <. 1 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
51 fvco3 5937 . . . . . 6  |-  ( ( H : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> U. J  /\  <. 1 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  ->  ( ( P  o.  H ) `  <. 1 ,  s
>. )  =  ( P `  ( H `  <. 1 ,  s
>. ) ) )
5231, 51sylan 471 . . . . 5  |-  ( (
ph  /\  <. 1 ,  s >.  e.  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  ->  ( ( P  o.  H ) `  <. 1 ,  s
>. )  =  ( P `  ( H `  <. 1 ,  s
>. ) ) )
5350, 52syldan 470 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( P  o.  H
) `  <. 1 ,  s >. )  =  ( P `  ( H `
 <. 1 ,  s
>. ) ) )
54 df-ov 6280 . . . 4  |-  ( 1 ( P  o.  H
) s )  =  ( ( P  o.  H ) `  <. 1 ,  s >. )
55 df-ov 6280 . . . . 5  |-  ( 1 H s )  =  ( H `  <. 1 ,  s >. )
5655fveq2i 5862 . . . 4  |-  ( P `
 ( 1 H s ) )  =  ( P `  ( H `  <. 1 ,  s >. ) )
5753, 54, 563eqtr4g 2528 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 ( P  o.  H ) s )  =  ( P `  ( 1 H s ) ) )
58 fvco3 5937 . . . 4  |-  ( ( F : ( 0 [,] 1 ) --> U. J  /\  1  e.  ( 0 [,] 1
) )  ->  (
( P  o.  F
) `  1 )  =  ( P `  ( F `  1 ) ) )
5942, 48, 58sylancl 662 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( P  o.  F
) `  1 )  =  ( P `  ( F `  1 ) ) )
6047, 57, 593eqtr4d 2513 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 ( P  o.  H ) s )  =  ( ( P  o.  F ) ` 
1 ) )
614, 7, 11, 45, 60isphtpyd 21216 1  |-  ( ph  ->  ( P  o.  H
)  e.  ( ( P  o.  F ) ( PHtpy `  K )
( P  o.  G
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   <.cop 4028   U.cuni 4240    X. cxp 4992    o. ccom 4998   -->wf 5577   ` cfv 5581  (class class class)co 6277   0cc0 9483   1c1 9484   [,]cicc 11523   Topctop 19156  TopOnctopon 19157    Cn ccn 19486    tX ctx 19791   IIcii 21109   Htpy chtpy 21197   PHtpycphtpy 21198
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-icc 11527  df-seq 12066  df-exp 12125  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-topgen 14690  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-top 19161  df-bases 19163  df-topon 19164  df-cn 19489  df-tx 19793  df-ii 21111  df-htpy 21200  df-phtpy 21201
This theorem is referenced by:  phtpcco2  21229
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