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Theorem copco 21363
Description: The composition of a concatenation of paths with a continuous function. (Contributed by Mario Carneiro, 9-Jul-2015.)
Hypotheses
Ref Expression
pcoval.2  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
pcoval.3  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
pcoval2.4  |-  ( ph  ->  ( F `  1
)  =  ( G `
 0 ) )
copco.6  |-  ( ph  ->  H  e.  ( J  Cn  K ) )
Assertion
Ref Expression
copco  |-  ( ph  ->  ( H  o.  ( F ( *p `  J ) G ) )  =  ( ( H  o.  F ) ( *p `  K
) ( H  o.  G ) ) )

Proof of Theorem copco
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pcoval.2 . . . . . . . 8  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
2 iiuni 21230 . . . . . . . . 9  |-  ( 0 [,] 1 )  = 
U. II
3 eqid 2467 . . . . . . . . 9  |-  U. J  =  U. J
42, 3cnf 19592 . . . . . . . 8  |-  ( F  e.  ( II  Cn  J )  ->  F : ( 0 [,] 1 ) --> U. J
)
51, 4syl 16 . . . . . . 7  |-  ( ph  ->  F : ( 0 [,] 1 ) --> U. J )
6 elii1 21280 . . . . . . . 8  |-  ( x  e.  ( 0 [,] ( 1  /  2
) )  <->  ( x  e.  ( 0 [,] 1
)  /\  x  <_  ( 1  /  2 ) ) )
7 iihalf1 21276 . . . . . . . 8  |-  ( x  e.  ( 0 [,] ( 1  /  2
) )  ->  (
2  x.  x )  e.  ( 0 [,] 1 ) )
86, 7sylbir 213 . . . . . . 7  |-  ( ( x  e.  ( 0 [,] 1 )  /\  x  <_  ( 1  / 
2 ) )  -> 
( 2  x.  x
)  e.  ( 0 [,] 1 ) )
9 fvco3 5950 . . . . . . 7  |-  ( ( F : ( 0 [,] 1 ) --> U. J  /\  ( 2  x.  x )  e.  ( 0 [,] 1
) )  ->  (
( H  o.  F
) `  ( 2  x.  x ) )  =  ( H `  ( F `  ( 2  x.  x ) ) ) )
105, 8, 9syl2an 477 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  x  <_  ( 1  /  2 ) ) )  ->  (
( H  o.  F
) `  ( 2  x.  x ) )  =  ( H `  ( F `  ( 2  x.  x ) ) ) )
1110anassrs 648 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( 0 [,] 1
) )  /\  x  <_  ( 1  /  2
) )  ->  (
( H  o.  F
) `  ( 2  x.  x ) )  =  ( H `  ( F `  ( 2  x.  x ) ) ) )
1211ifeq1da 3974 . . . 4  |-  ( (
ph  /\  x  e.  ( 0 [,] 1
) )  ->  if ( x  <_  ( 1  /  2 ) ,  ( ( H  o.  F ) `  (
2  x.  x ) ) ,  ( ( H  o.  G ) `
 ( ( 2  x.  x )  - 
1 ) ) )  =  if ( x  <_  ( 1  / 
2 ) ,  ( H `  ( F `
 ( 2  x.  x ) ) ) ,  ( ( H  o.  G ) `  ( ( 2  x.  x )  -  1 ) ) ) )
13 pcoval.3 . . . . . . . 8  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
142, 3cnf 19592 . . . . . . . 8  |-  ( G  e.  ( II  Cn  J )  ->  G : ( 0 [,] 1 ) --> U. J
)
1513, 14syl 16 . . . . . . 7  |-  ( ph  ->  G : ( 0 [,] 1 ) --> U. J )
16 elii2 21281 . . . . . . . 8  |-  ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  <_  ( 1  /  2 ) )  ->  x  e.  ( ( 1  /  2
) [,] 1 ) )
17 iihalf2 21278 . . . . . . . 8  |-  ( x  e.  ( ( 1  /  2 ) [,] 1 )  ->  (
( 2  x.  x
)  -  1 )  e.  ( 0 [,] 1 ) )
1816, 17syl 16 . . . . . . 7  |-  ( ( x  e.  ( 0 [,] 1 )  /\  -.  x  <_  ( 1  /  2 ) )  ->  ( ( 2  x.  x )  - 
1 )  e.  ( 0 [,] 1 ) )
19 fvco3 5950 . . . . . . 7  |-  ( ( G : ( 0 [,] 1 ) --> U. J  /\  ( ( 2  x.  x )  -  1 )  e.  ( 0 [,] 1
) )  ->  (
( H  o.  G
) `  ( (
2  x.  x )  -  1 ) )  =  ( H `  ( G `  ( ( 2  x.  x )  -  1 ) ) ) )
2015, 18, 19syl2an 477 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  -.  x  <_  ( 1  /  2
) ) )  -> 
( ( H  o.  G ) `  (
( 2  x.  x
)  -  1 ) )  =  ( H `
 ( G `  ( ( 2  x.  x )  -  1 ) ) ) )
2120anassrs 648 . . . . 5  |-  ( ( ( ph  /\  x  e.  ( 0 [,] 1
) )  /\  -.  x  <_  ( 1  / 
2 ) )  -> 
( ( H  o.  G ) `  (
( 2  x.  x
)  -  1 ) )  =  ( H `
 ( G `  ( ( 2  x.  x )  -  1 ) ) ) )
2221ifeq2da 3975 . . . 4  |-  ( (
ph  /\  x  e.  ( 0 [,] 1
) )  ->  if ( x  <_  ( 1  /  2 ) ,  ( H `  ( F `  ( 2  x.  x ) ) ) ,  ( ( H  o.  G ) `  ( ( 2  x.  x )  -  1 ) ) )  =  if ( x  <_ 
( 1  /  2
) ,  ( H `
 ( F `  ( 2  x.  x
) ) ) ,  ( H `  ( G `  ( (
2  x.  x )  -  1 ) ) ) ) )
2312, 22eqtrd 2508 . . 3  |-  ( (
ph  /\  x  e.  ( 0 [,] 1
) )  ->  if ( x  <_  ( 1  /  2 ) ,  ( ( H  o.  F ) `  (
2  x.  x ) ) ,  ( ( H  o.  G ) `
 ( ( 2  x.  x )  - 
1 ) ) )  =  if ( x  <_  ( 1  / 
2 ) ,  ( H `  ( F `
 ( 2  x.  x ) ) ) ,  ( H `  ( G `  ( ( 2  x.  x )  -  1 ) ) ) ) )
2423mpteq2dva 4538 . 2  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( ( H  o.  F ) `
 ( 2  x.  x ) ) ,  ( ( H  o.  G ) `  (
( 2  x.  x
)  -  1 ) ) ) )  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( H `
 ( F `  ( 2  x.  x
) ) ) ,  ( H `  ( G `  ( (
2  x.  x )  -  1 ) ) ) ) ) )
25 copco.6 . . . 4  |-  ( ph  ->  H  e.  ( J  Cn  K ) )
26 cnco 19612 . . . 4  |-  ( ( F  e.  ( II 
Cn  J )  /\  H  e.  ( J  Cn  K ) )  -> 
( H  o.  F
)  e.  ( II 
Cn  K ) )
271, 25, 26syl2anc 661 . . 3  |-  ( ph  ->  ( H  o.  F
)  e.  ( II 
Cn  K ) )
28 cnco 19612 . . . 4  |-  ( ( G  e.  ( II 
Cn  J )  /\  H  e.  ( J  Cn  K ) )  -> 
( H  o.  G
)  e.  ( II 
Cn  K ) )
2913, 25, 28syl2anc 661 . . 3  |-  ( ph  ->  ( H  o.  G
)  e.  ( II 
Cn  K ) )
3027, 29pcoval 21356 . 2  |-  ( ph  ->  ( ( H  o.  F ) ( *p
`  K ) ( H  o.  G ) )  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( ( H  o.  F ) `  (
2  x.  x ) ) ,  ( ( H  o.  G ) `
 ( ( 2  x.  x )  - 
1 ) ) ) ) )
311, 13pcoval 21356 . . . . . 6  |-  ( ph  ->  ( F ( *p
`  J ) G )  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) ) )
32 pcoval2.4 . . . . . . 7  |-  ( ph  ->  ( F `  1
)  =  ( G `
 0 ) )
331, 13, 32pcocn 21362 . . . . . 6  |-  ( ph  ->  ( F ( *p
`  J ) G )  e.  ( II 
Cn  J ) )
3431, 33eqeltrrd 2556 . . . . 5  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( F `
 ( 2  x.  x ) ) ,  ( G `  (
( 2  x.  x
)  -  1 ) ) ) )  e.  ( II  Cn  J
) )
352, 3cnf 19592 . . . . 5  |-  ( ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  (
1  /  2 ) ,  ( F `  ( 2  x.  x
) ) ,  ( G `  ( ( 2  x.  x )  -  1 ) ) ) )  e.  ( II  Cn  J )  ->  ( x  e.  ( 0 [,] 1
)  |->  if ( x  <_  ( 1  / 
2 ) ,  ( F `  ( 2  x.  x ) ) ,  ( G `  ( ( 2  x.  x )  -  1 ) ) ) ) : ( 0 [,] 1 ) --> U. J
)
3634, 35syl 16 . . . 4  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( F `
 ( 2  x.  x ) ) ,  ( G `  (
( 2  x.  x
)  -  1 ) ) ) ) : ( 0 [,] 1
) --> U. J )
37 eqid 2467 . . . . 5  |-  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) )  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) )
3837fmpt 6052 . . . 4  |-  ( A. x  e.  ( 0 [,] 1 ) if ( x  <_  (
1  /  2 ) ,  ( F `  ( 2  x.  x
) ) ,  ( G `  ( ( 2  x.  x )  -  1 ) ) )  e.  U. J  <->  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  (
1  /  2 ) ,  ( F `  ( 2  x.  x
) ) ,  ( G `  ( ( 2  x.  x )  -  1 ) ) ) ) : ( 0 [,] 1 ) --> U. J )
3936, 38sylibr 212 . . 3  |-  ( ph  ->  A. x  e.  ( 0 [,] 1 ) if ( x  <_ 
( 1  /  2
) ,  ( F `
 ( 2  x.  x ) ) ,  ( G `  (
( 2  x.  x
)  -  1 ) ) )  e.  U. J )
40 eqid 2467 . . . . . 6  |-  U. K  =  U. K
413, 40cnf 19592 . . . . 5  |-  ( H  e.  ( J  Cn  K )  ->  H : U. J --> U. K
)
4225, 41syl 16 . . . 4  |-  ( ph  ->  H : U. J --> U. K )
4342feqmptd 5926 . . 3  |-  ( ph  ->  H  =  ( y  e.  U. J  |->  ( H `  y ) ) )
44 fveq2 5871 . . . 4  |-  ( y  =  if ( x  <_  ( 1  / 
2 ) ,  ( F `  ( 2  x.  x ) ) ,  ( G `  ( ( 2  x.  x )  -  1 ) ) )  -> 
( H `  y
)  =  ( H `
 if ( x  <_  ( 1  / 
2 ) ,  ( F `  ( 2  x.  x ) ) ,  ( G `  ( ( 2  x.  x )  -  1 ) ) ) ) )
45 fvif 5882 . . . 4  |-  ( H `
 if ( x  <_  ( 1  / 
2 ) ,  ( F `  ( 2  x.  x ) ) ,  ( G `  ( ( 2  x.  x )  -  1 ) ) ) )  =  if ( x  <_  ( 1  / 
2 ) ,  ( H `  ( F `
 ( 2  x.  x ) ) ) ,  ( H `  ( G `  ( ( 2  x.  x )  -  1 ) ) ) )
4644, 45syl6eq 2524 . . 3  |-  ( y  =  if ( x  <_  ( 1  / 
2 ) ,  ( F `  ( 2  x.  x ) ) ,  ( G `  ( ( 2  x.  x )  -  1 ) ) )  -> 
( H `  y
)  =  if ( x  <_  ( 1  /  2 ) ,  ( H `  ( F `  ( 2  x.  x ) ) ) ,  ( H `  ( G `  ( ( 2  x.  x )  -  1 ) ) ) ) )
4739, 31, 43, 46fmptcof 6065 . 2  |-  ( ph  ->  ( H  o.  ( F ( *p `  J ) G ) )  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( H `  ( F `  ( 2  x.  x ) ) ) ,  ( H `  ( G `  ( ( 2  x.  x )  -  1 ) ) ) ) ) )
4824, 30, 473eqtr4rd 2519 1  |-  ( ph  ->  ( H  o.  ( F ( *p `  J ) G ) )  =  ( ( H  o.  F ) ( *p `  K
) ( H  o.  G ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   ifcif 3944   U.cuni 4250   class class class wbr 4452    |-> cmpt 4510    o. ccom 5008   -->wf 5589   ` cfv 5593  (class class class)co 6294   0cc0 9502   1c1 9503    x. cmul 9507    <_ cle 9639    - cmin 9815    / cdiv 10216   2c2 10595   [,]cicc 11542    Cn ccn 19570   IIcii 21224   *pcpco 21345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580  ax-mulf 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-of 6534  df-om 6695  df-1st 6794  df-2nd 6795  df-supp 6912  df-recs 7052  df-rdg 7086  df-1o 7140  df-2o 7141  df-oadd 7144  df-er 7321  df-map 7432  df-ixp 7480  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-fsupp 7840  df-fi 7881  df-sup 7911  df-oi 7945  df-card 8330  df-cda 8558  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-5 10607  df-6 10608  df-7 10609  df-8 10610  df-9 10611  df-10 10612  df-n0 10806  df-z 10875  df-dec 10987  df-uz 11093  df-q 11193  df-rp 11231  df-xneg 11328  df-xadd 11329  df-xmul 11330  df-ioo 11543  df-icc 11546  df-fz 11683  df-fzo 11803  df-seq 12086  df-exp 12145  df-hash 12384  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-starv 14582  df-sca 14583  df-vsca 14584  df-ip 14585  df-tset 14586  df-ple 14587  df-ds 14589  df-unif 14590  df-hom 14591  df-cco 14592  df-rest 14690  df-topn 14691  df-0g 14709  df-gsum 14710  df-topgen 14711  df-pt 14712  df-prds 14715  df-xrs 14769  df-qtop 14774  df-imas 14775  df-xps 14777  df-mre 14853  df-mrc 14854  df-acs 14856  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-submnd 15820  df-mulg 15909  df-cntz 16204  df-cmn 16650  df-psmet 18258  df-xmet 18259  df-met 18260  df-bl 18261  df-mopn 18262  df-cnfld 18268  df-top 19245  df-bases 19247  df-topon 19248  df-topsp 19249  df-cld 19365  df-cn 19573  df-cnp 19574  df-tx 19908  df-hmeo 20101  df-xms 20668  df-ms 20669  df-tms 20670  df-ii 21226  df-pco 21350
This theorem is referenced by:  pi1coghm  21406  cvmlift3lem6  28562
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