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Theorem phtpyco2 21784
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 20062 . . 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 20062 . . 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 21776 . . . 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 3445 . . 3  |-  ( ph  ->  H  e.  ( F ( II Htpy  J ) G ) )
111, 5, 2, 10htpyco2 21773 . 2  |-  ( ph  ->  ( P  o.  H
)  e.  ( ( P  o.  F ) ( II Htpy  K ) ( P  o.  G
) ) )
121, 5, 9phtpyi 21778 . . . . 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 5855 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( P `  ( 0 H s ) )  =  ( P `  ( F `  0 ) ) )
15 0elunit 11694 . . . . . 6  |-  0  e.  ( 0 [,] 1
)
16 simpr 461 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
17 opelxpi 4857 . . . . . 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 21677 . . . . . . . . 9  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
20 txtopon 20386 . . . . . . . . 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 20037 . . . . . . . . 9  |-  ( F  e.  ( II  Cn  J )  ->  J  e.  Top )
241, 23syl 17 . . . . . . . 8  |-  ( ph  ->  J  e.  Top )
25 eqid 2404 . . . . . . . . 9  |-  U. J  =  U. J
2625toptopon 19728 . . . . . . . 8  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
2724, 26sylib 198 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
281, 5phtpycn 21777 . . . . . . . 8  |-  ( ph  ->  ( F ( PHtpy `  J ) G ) 
C_  ( ( II 
tX  II )  Cn  J ) )
2928, 9sseldd 3445 . . . . . . 7  |-  ( ph  ->  H  e.  ( ( II  tX  II )  Cn  J ) )
30 cnf2 20045 . . . . . . 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 1232 . . . . . 6  |-  ( ph  ->  H : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> U. J )
32 fvco3 5928 . . . . . 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 6283 . . . 4  |-  ( 0 ( P  o.  H
) s )  =  ( ( P  o.  H ) `  <. 0 ,  s >. )
36 df-ov 6283 . . . . 5  |-  ( 0 H s )  =  ( H `  <. 0 ,  s >. )
3736fveq2i 5854 . . . 4  |-  ( P `
 ( 0 H s ) )  =  ( P `  ( H `  <. 0 ,  s >. ) )
3834, 35, 373eqtr4g 2470 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 ( P  o.  H ) s )  =  ( P `  ( 0 H s ) ) )
39 iiuni 21679 . . . . . . 7  |-  ( 0 [,] 1 )  = 
U. II
4039, 25cnf 20042 . . . . . 6  |-  ( F  e.  ( II  Cn  J )  ->  F : ( 0 [,] 1 ) --> U. J
)
411, 40syl 17 . . . . 5  |-  ( ph  ->  F : ( 0 [,] 1 ) --> U. J )
4241adantr 465 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  F : ( 0 [,] 1 ) --> U. J
)
43 fvco3 5928 . . . 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 2455 . 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 5855 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( P `  ( 1 H s ) )  =  ( P `  ( F `  1 ) ) )
48 1elunit 11695 . . . . . 6  |-  1  e.  ( 0 [,] 1
)
49 opelxpi 4857 . . . . . 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 5928 . . . . . 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 6283 . . . 4  |-  ( 1 ( P  o.  H
) s )  =  ( ( P  o.  H ) `  <. 1 ,  s >. )
55 df-ov 6283 . . . . 5  |-  ( 1 H s )  =  ( H `  <. 1 ,  s >. )
5655fveq2i 5854 . . . 4  |-  ( P `
 ( 1 H s ) )  =  ( P `  ( H `  <. 1 ,  s >. ) )
5753, 54, 563eqtr4g 2470 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 ( P  o.  H ) s )  =  ( P `  ( 1 H s ) ) )
58 fvco3 5928 . . . 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 2455 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 ( P  o.  H ) s )  =  ( ( P  o.  F ) ` 
1 ) )
614, 7, 11, 45, 60isphtpyd 21780 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 1407    e. wcel 1844   <.cop 3980   U.cuni 4193    X. cxp 4823    o. ccom 4829   -->wf 5567   ` cfv 5571  (class class class)co 6280   0cc0 9524   1c1 9525   [,]cicc 11587   Topctop 19688  TopOnctopon 19689    Cn ccn 20020    tX ctx 20355   IIcii 21673   Htpy chtpy 21761   PHtpycphtpy 21762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-sup 7937  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-n0 10839  df-z 10908  df-uz 11130  df-q 11230  df-rp 11268  df-xneg 11373  df-xadd 11374  df-xmul 11375  df-icc 11591  df-seq 12154  df-exp 12213  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-topgen 15060  df-psmet 18733  df-xmet 18734  df-met 18735  df-bl 18736  df-mopn 18737  df-top 19693  df-bases 19695  df-topon 19696  df-cn 20023  df-tx 20357  df-ii 21675  df-htpy 21764  df-phtpy 21765
This theorem is referenced by:  phtpcco2  21793
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