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

Theorem phtpyco2 18488
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 16995 . . 3  |-  ( ( F  e.  ( II 
Cn  J )  /\  P  e.  ( J  Cn  K ) )  -> 
( P  o.  F
)  e.  ( II 
Cn  K ) )
41, 2, 3syl2anc 642 . 2  |-  ( ph  ->  ( P  o.  F
)  e.  ( II 
Cn  K ) )
5 phtpyco2.g . . 3  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
6 cnco 16995 . . 3  |-  ( ( G  e.  ( II 
Cn  J )  /\  P  e.  ( J  Cn  K ) )  -> 
( P  o.  G
)  e.  ( II 
Cn  K ) )
75, 2, 6syl2anc 642 . 2  |-  ( ph  ->  ( P  o.  G
)  e.  ( II 
Cn  K ) )
81, 5phtpyhtpy 18480 . . . 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 3181 . . 3  |-  ( ph  ->  H  e.  ( F ( II Htpy  J ) G ) )
111, 5, 2, 10htpyco2 18477 . 2  |-  ( ph  ->  ( P  o.  H
)  e.  ( ( P  o.  F ) ( II Htpy  K ) ( P  o.  G
) ) )
121, 5, 9phtpyi 18482 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0 H s )  =  ( F `
 0 )  /\  ( 1 H s )  =  ( F `
 1 ) ) )
1312simpld 445 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 H s )  =  ( F ` 
0 ) )
1413fveq2d 5529 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( P `  ( 0 H s ) )  =  ( P `  ( F `  0 ) ) )
15 0elunit 10754 . . . . . 6  |-  0  e.  ( 0 [,] 1
)
16 simpr 447 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
17 opelxpi 4721 . . . . . 6  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  <. 0 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
1815, 16, 17sylancr 644 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  <. 0 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
19 iitopon 18383 . . . . . . . . 9  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
20 txtopon 17286 . . . . . . . . 9  |-  ( ( II  e.  (TopOn `  ( 0 [,] 1
) )  /\  II  e.  (TopOn `  ( 0 [,] 1 ) ) )  ->  ( II  tX  II )  e.  (TopOn `  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
2119, 19, 20mp2an 653 . . . . . . . 8  |-  ( II 
tX  II )  e.  (TopOn `  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
2221a1i 10 . . . . . . 7  |-  ( ph  ->  ( II  tX  II )  e.  (TopOn `  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) ) )
23 cntop2 16971 . . . . . . . . 9  |-  ( F  e.  ( II  Cn  J )  ->  J  e.  Top )
241, 23syl 15 . . . . . . . 8  |-  ( ph  ->  J  e.  Top )
25 eqid 2283 . . . . . . . . 9  |-  U. J  =  U. J
2625toptopon 16671 . . . . . . . 8  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
2724, 26sylib 188 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
281, 5phtpycn 18481 . . . . . . . 8  |-  ( ph  ->  ( F ( PHtpy `  J ) G ) 
C_  ( ( II 
tX  II )  Cn  J ) )
2928, 9sseldd 3181 . . . . . . 7  |-  ( ph  ->  H  e.  ( ( II  tX  II )  Cn  J ) )
30 cnf2 16979 . . . . . . 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 1182 . . . . . 6  |-  ( ph  ->  H : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> U. J )
32 fvco3 5596 . . . . . 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 457 . . . . 5  |-  ( (
ph  /\  <. 0 ,  s >.  e.  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  ->  ( ( P  o.  H ) `  <. 0 ,  s
>. )  =  ( P `  ( H `  <. 0 ,  s
>. ) ) )
3418, 33syldan 456 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( P  o.  H
) `  <. 0 ,  s >. )  =  ( P `  ( H `
 <. 0 ,  s
>. ) ) )
35 df-ov 5861 . . . 4  |-  ( 0 ( P  o.  H
) s )  =  ( ( P  o.  H ) `  <. 0 ,  s >. )
36 df-ov 5861 . . . . 5  |-  ( 0 H s )  =  ( H `  <. 0 ,  s >. )
3736fveq2i 5528 . . . 4  |-  ( P `
 ( 0 H s ) )  =  ( P `  ( H `  <. 0 ,  s >. ) )
3834, 35, 373eqtr4g 2340 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 ( P  o.  H ) s )  =  ( P `  ( 0 H s ) ) )
39 iiuni 18385 . . . . . . 7  |-  ( 0 [,] 1 )  = 
U. II
4039, 25cnf 16976 . . . . . 6  |-  ( F  e.  ( II  Cn  J )  ->  F : ( 0 [,] 1 ) --> U. J
)
411, 40syl 15 . . . . 5  |-  ( ph  ->  F : ( 0 [,] 1 ) --> U. J )
4241adantr 451 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  F : ( 0 [,] 1 ) --> U. J
)
43 fvco3 5596 . . . 4  |-  ( ( F : ( 0 [,] 1 ) --> U. J  /\  0  e.  ( 0 [,] 1
) )  ->  (
( P  o.  F
) `  0 )  =  ( P `  ( F `  0 ) ) )
4442, 15, 43sylancl 643 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( P  o.  F
) `  0 )  =  ( P `  ( F `  0 ) ) )
4514, 38, 443eqtr4d 2325 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 ( P  o.  H ) s )  =  ( ( P  o.  F ) ` 
0 ) )
4612simprd 449 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 H s )  =  ( F ` 
1 ) )
4746fveq2d 5529 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( P `  ( 1 H s ) )  =  ( P `  ( F `  1 ) ) )
48 1elunit 10755 . . . . . 6  |-  1  e.  ( 0 [,] 1
)
49 opelxpi 4721 . . . . . 6  |-  ( ( 1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  <. 1 ,  s
>.  e.  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) )
5048, 16, 49sylancr 644 . . . . 5  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  <. 1 ,  s >.  e.  ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )
51 fvco3 5596 . . . . . 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 457 . . . . 5  |-  ( (
ph  /\  <. 1 ,  s >.  e.  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  ->  ( ( P  o.  H ) `  <. 1 ,  s
>. )  =  ( P `  ( H `  <. 1 ,  s
>. ) ) )
5350, 52syldan 456 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( P  o.  H
) `  <. 1 ,  s >. )  =  ( P `  ( H `
 <. 1 ,  s
>. ) ) )
54 df-ov 5861 . . . 4  |-  ( 1 ( P  o.  H
) s )  =  ( ( P  o.  H ) `  <. 1 ,  s >. )
55 df-ov 5861 . . . . 5  |-  ( 1 H s )  =  ( H `  <. 1 ,  s >. )
5655fveq2i 5528 . . . 4  |-  ( P `
 ( 1 H s ) )  =  ( P `  ( H `  <. 1 ,  s >. ) )
5753, 54, 563eqtr4g 2340 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 ( P  o.  H ) s )  =  ( P `  ( 1 H s ) ) )
58 fvco3 5596 . . . 4  |-  ( ( F : ( 0 [,] 1 ) --> U. J  /\  1  e.  ( 0 [,] 1
) )  ->  (
( P  o.  F
) `  1 )  =  ( P `  ( F `  1 ) ) )
5942, 48, 58sylancl 643 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
( P  o.  F
) `  1 )  =  ( P `  ( F `  1 ) ) )
6047, 57, 593eqtr4d 2325 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 ( P  o.  H ) s )  =  ( ( P  o.  F ) ` 
1 ) )
614, 7, 11, 45, 60isphtpyd 18484 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 358    = wceq 1623    e. wcel 1684   <.cop 3643   U.cuni 3827    X. cxp 4687    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738   [,]cicc 10659   Topctop 16631  TopOnctopon 16632    Cn ccn 16954    tX ctx 17255   IIcii 18379   Htpy chtpy 18465   PHtpycphtpy 18466
This theorem is referenced by:  phtpcco2  18497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-tx 17257  df-ii 18381  df-htpy 18468  df-phtpy 18469
  Copyright terms: Public domain W3C validator