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Theorem phtpyco2 18586
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 17095 . . 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 17095 . . 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 18578 . . . 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 3257 . . 3  |-  ( ph  ->  H  e.  ( F ( II Htpy  J ) G ) )
111, 5, 2, 10htpyco2 18575 . 2  |-  ( ph  ->  ( P  o.  H
)  e.  ( ( P  o.  F ) ( II Htpy  K ) ( P  o.  G
) ) )
121, 5, 9phtpyi 18580 . . . . 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 5609 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( P `  ( 0 H s ) )  =  ( P `  ( F `  0 ) ) )
15 0elunit 10843 . . . . . 6  |-  0  e.  ( 0 [,] 1
)
16 simpr 447 . . . . . 6  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
17 opelxpi 4800 . . . . . 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 18480 . . . . . . . . 9  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
20 txtopon 17386 . . . . . . . . 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 17071 . . . . . . . . 9  |-  ( F  e.  ( II  Cn  J )  ->  J  e.  Top )
241, 23syl 15 . . . . . . . 8  |-  ( ph  ->  J  e.  Top )
25 eqid 2358 . . . . . . . . 9  |-  U. J  =  U. J
2625toptopon 16771 . . . . . . . 8  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
2724, 26sylib 188 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
281, 5phtpycn 18579 . . . . . . . 8  |-  ( ph  ->  ( F ( PHtpy `  J ) G ) 
C_  ( ( II 
tX  II )  Cn  J ) )
2928, 9sseldd 3257 . . . . . . 7  |-  ( ph  ->  H  e.  ( ( II  tX  II )  Cn  J ) )
30 cnf2 17079 . . . . . . 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 5676 . . . . . 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 5945 . . . 4  |-  ( 0 ( P  o.  H
) s )  =  ( ( P  o.  H ) `  <. 0 ,  s >. )
36 df-ov 5945 . . . . 5  |-  ( 0 H s )  =  ( H `  <. 0 ,  s >. )
3736fveq2i 5608 . . . 4  |-  ( P `
 ( 0 H s ) )  =  ( P `  ( H `  <. 0 ,  s >. ) )
3834, 35, 373eqtr4g 2415 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 ( P  o.  H ) s )  =  ( P `  ( 0 H s ) ) )
39 iiuni 18482 . . . . . . 7  |-  ( 0 [,] 1 )  = 
U. II
4039, 25cnf 17076 . . . . . 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 5676 . . . 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 2400 . 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 5609 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  ( P `  ( 1 H s ) )  =  ( P `  ( F `  1 ) ) )
48 1elunit 10844 . . . . . 6  |-  1  e.  ( 0 [,] 1
)
49 opelxpi 4800 . . . . . 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 5676 . . . . . 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 5945 . . . 4  |-  ( 1 ( P  o.  H
) s )  =  ( ( P  o.  H ) `  <. 1 ,  s >. )
55 df-ov 5945 . . . . 5  |-  ( 1 H s )  =  ( H `  <. 1 ,  s >. )
5655fveq2i 5608 . . . 4  |-  ( P `
 ( 1 H s ) )  =  ( P `  ( H `  <. 1 ,  s >. ) )
5753, 54, 563eqtr4g 2415 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 ( P  o.  H ) s )  =  ( P `  ( 1 H s ) ) )
58 fvco3 5676 . . . 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 2400 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 ( P  o.  H ) s )  =  ( ( P  o.  F ) ` 
1 ) )
614, 7, 11, 45, 60isphtpyd 18582 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 1642    e. wcel 1710   <.cop 3719   U.cuni 3906    X. cxp 4766    o. ccom 4772   -->wf 5330   ` cfv 5334  (class class class)co 5942   0cc0 8824   1c1 8825   [,]cicc 10748   Topctop 16731  TopOnctopon 16732    Cn ccn 17054    tX ctx 17355   IIcii 18476   Htpy chtpy 18563   PHtpycphtpy 18564
This theorem is referenced by:  phtpcco2  18595
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-map 6859  df-en 6949  df-dom 6950  df-sdom 6951  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-icc 10752  df-seq 11136  df-exp 11195  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-topgen 13437  df-xmet 16469  df-met 16470  df-bl 16471  df-mopn 16472  df-top 16736  df-bases 16738  df-topon 16739  df-cn 17057  df-tx 17357  df-ii 18478  df-htpy 18566  df-phtpy 18567
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