MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  phtpyco2 Unicode version

Theorem phtpyco2 18972
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 17288 . . 3  |-  ( ( F  e.  ( II 
Cn  J )  /\  P  e.  ( J  Cn  K ) )  -> 
( P  o.  F
)  e.  ( II 
Cn  K ) )
41, 2, 3syl2anc 643 . 2  |-  ( ph  ->  ( P  o.  F
)  e.  ( II 
Cn  K ) )
5 phtpyco2.g . . 3  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
6 cnco 17288 . . 3  |-  ( ( G  e.  ( II 
Cn  J )  /\  P  e.  ( J  Cn  K ) )  -> 
( P  o.  G
)  e.  ( II 
Cn  K ) )
75, 2, 6syl2anc 643 . 2  |-  ( ph  ->  ( P  o.  G
)  e.  ( II 
Cn  K ) )
81, 5phtpyhtpy 18964 . . . 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 3313 . . 3  |-  ( ph  ->  H  e.  ( F ( II Htpy  J ) G ) )
111, 5, 2, 10htpyco2 18961 . 2  |-  ( ph  ->  ( P  o.  H
)  e.  ( ( P  o.  F ) ( II Htpy  K ) ( P  o.  G
) ) )
121, 5, 9phtpyi 18966 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0 H s )  =  ( F `
 0 )  /\  ( 1 H s )  =  ( F `
 1 ) ) )
1312simpld 446 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 H s )  =  ( F ` 
0 ) )
1413fveq2d 5695 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( P `  ( 0 H s ) )  =  ( P `  ( F `  0 ) ) )
15 0elunit 10975 . . . . . 6  |-  0  e.  ( 0 [,] 1
)
16 simpr 448 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
17 opelxpi 4873 . . . . . 6  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  <. 0 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
1815, 16, 17sylancr 645 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  <. 0 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
19 iitopon 18866 . . . . . . . . 9  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
20 txtopon 17580 . . . . . . . . 9  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  II  e.  (TopOn `  ( 0 [,] 1 ) ) )  ->  ( II  tX  II )  e.  (TopOn `  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
2119, 19, 20mp2an 654 . . . . . . . 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 17263 . . . . . . . . 9  |-  ( F  e.  ( II  Cn  J )  ->  J  e.  Top )
241, 23syl 16 . . . . . . . 8  |-  ( ph  ->  J  e.  Top )
25 eqid 2408 . . . . . . . . 9  |-  U. J  =  U. J
2625toptopon 16957 . . . . . . . 8  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
2724, 26sylib 189 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
281, 5phtpycn 18965 . . . . . . . 8  |-  ( ph  ->  ( F ( PHtpy `  J ) G ) 
C_  ( ( II 
tX  II )  Cn  J ) )
2928, 9sseldd 3313 . . . . . . 7  |-  ( ph  ->  H  e.  ( ( II  tX  II )  Cn  J ) )
30 cnf2 17271 . . . . . . 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 1184 . . . . . 6  |-  ( ph  ->  H : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> U. J )
32 fvco3 5763 . . . . . 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 458 . . . . 5  |-  ( (
ph  /\  <. 0 ,  s >.  e.  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  ->  ( ( P  o.  H ) `  <. 0 ,  s
>. )  =  ( P `  ( H `  <. 0 ,  s
>. ) ) )
3418, 33syldan 457 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( P  o.  H
) `  <. 0 ,  s >. )  =  ( P `  ( H `
 <. 0 ,  s
>. ) ) )
35 df-ov 6047 . . . 4  |-  ( 0 ( P  o.  H
) s )  =  ( ( P  o.  H ) `  <. 0 ,  s >. )
36 df-ov 6047 . . . . 5  |-  ( 0 H s )  =  ( H `  <. 0 ,  s >. )
3736fveq2i 5694 . . . 4  |-  ( P `
 ( 0 H s ) )  =  ( P `  ( H `  <. 0 ,  s >. ) )
3834, 35, 373eqtr4g 2465 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 ( P  o.  H ) s )  =  ( P `  ( 0 H s ) ) )
39 iiuni 18868 . . . . . . 7  |-  ( 0 [,] 1 )  = 
U. II
4039, 25cnf 17268 . . . . . 6  |-  ( F  e.  ( II  Cn  J )  ->  F : ( 0 [,] 1 ) --> U. J
)
411, 40syl 16 . . . . 5  |-  ( ph  ->  F : ( 0 [,] 1 ) --> U. J )
4241adantr 452 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  F : ( 0 [,] 1 ) --> U. J
)
43 fvco3 5763 . . . 4  |-  ( ( F : ( 0 [,] 1 ) --> U. J  /\  0  e.  ( 0 [,] 1
) )  ->  (
( P  o.  F
) `  0 )  =  ( P `  ( F `  0 ) ) )
4442, 15, 43sylancl 644 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( P  o.  F
) `  0 )  =  ( P `  ( F `  0 ) ) )
4514, 38, 443eqtr4d 2450 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 ( P  o.  H ) s )  =  ( ( P  o.  F ) ` 
0 ) )
4612simprd 450 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 H s )  =  ( F ` 
1 ) )
4746fveq2d 5695 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( P `  ( 1 H s ) )  =  ( P `  ( F `  1 ) ) )
48 1elunit 10976 . . . . . 6  |-  1  e.  ( 0 [,] 1
)
49 opelxpi 4873 . . . . . 6  |-  ( ( 1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  <. 1 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
5048, 16, 49sylancr 645 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  <. 1 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
51 fvco3 5763 . . . . . 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 458 . . . . 5  |-  ( (
ph  /\  <. 1 ,  s >.  e.  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  ->  ( ( P  o.  H ) `  <. 1 ,  s
>. )  =  ( P `  ( H `  <. 1 ,  s
>. ) ) )
5350, 52syldan 457 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( P  o.  H
) `  <. 1 ,  s >. )  =  ( P `  ( H `
 <. 1 ,  s
>. ) ) )
54 df-ov 6047 . . . 4  |-  ( 1 ( P  o.  H
) s )  =  ( ( P  o.  H ) `  <. 1 ,  s >. )
55 df-ov 6047 . . . . 5  |-  ( 1 H s )  =  ( H `  <. 1 ,  s >. )
5655fveq2i 5694 . . . 4  |-  ( P `
 ( 1 H s ) )  =  ( P `  ( H `  <. 1 ,  s >. ) )
5753, 54, 563eqtr4g 2465 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 ( P  o.  H ) s )  =  ( P `  ( 1 H s ) ) )
58 fvco3 5763 . . . 4  |-  ( ( F : ( 0 [,] 1 ) --> U. J  /\  1  e.  ( 0 [,] 1
) )  ->  (
( P  o.  F
) `  1 )  =  ( P `  ( F `  1 ) ) )
5942, 48, 58sylancl 644 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( P  o.  F
) `  1 )  =  ( P `  ( F `  1 ) ) )
6047, 57, 593eqtr4d 2450 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 ( P  o.  H ) s )  =  ( ( P  o.  F ) ` 
1 ) )
614, 7, 11, 45, 60isphtpyd 18968 1  |-  ( ph  ->  ( P  o.  H
)  e.  ( ( P  o.  F ) ( PHtpy `  K )
( P  o.  G
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   <.cop 3781   U.cuni 3979    X. cxp 4839    o. ccom 4845   -->wf 5413   ` cfv 5417  (class class class)co 6044   0cc0 8950   1c1 8951   [,]cicc 10879   Topctop 16917  TopOnctopon 16918    Cn ccn 17246    tX ctx 17549   IIcii 18862   Htpy chtpy 18949   PHtpycphtpy 18950
This theorem is referenced by:  phtpcco2  18981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-sup 7408  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-q 10535  df-rp 10573  df-xneg 10670  df-xadd 10671  df-xmul 10672  df-icc 10883  df-seq 11283  df-exp 11342  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-topgen 13626  df-psmet 16653  df-xmet 16654  df-met 16655  df-bl 16656  df-mopn 16657  df-top 16922  df-bases 16924  df-topon 16925  df-cn 17249  df-tx 17551  df-ii 18864  df-htpy 18952  df-phtpy 18953
  Copyright terms: Public domain W3C validator