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Theorem pcocn 20548
Description: The concatenation of two paths is a path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)
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 ) )
Assertion
Ref Expression
pcocn  |-  ( ph  ->  ( F ( *p
`  J ) G )  e.  ( II 
Cn  J ) )

Proof of Theorem pcocn
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pcoval.2 . . 3  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
2 pcoval.3 . . 3  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
31, 2pcoval 20542 . 2  |-  ( ph  ->  ( F ( *p
`  J ) G )  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) ) )
4 iitopon 20414 . . . 4  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
54a1i 11 . . 3  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
65cnmptid 19193 . . 3  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( II  Cn  II ) )
7 0elunit 11399 . . . . 5  |-  0  e.  ( 0 [,] 1
)
87a1i 11 . . . 4  |-  ( ph  ->  0  e.  ( 0 [,] 1 ) )
95, 5, 8cnmptc 19194 . . 3  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  0 )  e.  ( II  Cn  II ) )
10 eqid 2441 . . . 4  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
11 eqid 2441 . . . 4  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )
12 eqid 2441 . . . 4  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )
13 dfii2 20417 . . . 4  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
14 0re 9382 . . . . 5  |-  0  e.  RR
1514a1i 11 . . . 4  |-  ( ph  ->  0  e.  RR )
16 1re 9381 . . . . 5  |-  1  e.  RR
1716a1i 11 . . . 4  |-  ( ph  ->  1  e.  RR )
18 halfre 10536 . . . . . 6  |-  ( 1  /  2 )  e.  RR
19 halfgt0 10538 . . . . . . 7  |-  0  <  ( 1  /  2
)
2014, 18, 19ltleii 9493 . . . . . 6  |-  0  <_  ( 1  /  2
)
21 halflt1 10539 . . . . . . 7  |-  ( 1  /  2 )  <  1
2218, 16, 21ltleii 9493 . . . . . 6  |-  ( 1  /  2 )  <_ 
1
2314, 16elicc2i 11357 . . . . . 6  |-  ( ( 1  /  2 )  e.  ( 0 [,] 1 )  <->  ( (
1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_ 
1 ) )
2418, 20, 22, 23mpbir3an 1165 . . . . 5  |-  ( 1  /  2 )  e.  ( 0 [,] 1
)
2524a1i 11 . . . 4  |-  ( ph  ->  ( 1  /  2
)  e.  ( 0 [,] 1 ) )
26 pcoval2.4 . . . . . 6  |-  ( ph  ->  ( F `  1
)  =  ( G `
 0 ) )
2726adantr 462 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( F `  1
)  =  ( G `
 0 ) )
28 simprl 750 . . . . . . . 8  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
y  =  ( 1  /  2 ) )
2928oveq2d 6106 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( 2  x.  y
)  =  ( 2  x.  ( 1  / 
2 ) ) )
30 2cn 10388 . . . . . . . 8  |-  2  e.  CC
31 2ne0 10410 . . . . . . . 8  |-  2  =/=  0
3230, 31recidi 10058 . . . . . . 7  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
3329, 32syl6eq 2489 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( 2  x.  y
)  =  1 )
3433fveq2d 5692 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( F `  (
2  x.  y ) )  =  ( F `
 1 ) )
3533oveq1d 6105 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( ( 2  x.  y )  -  1 )  =  ( 1  -  1 ) )
36 1m1e0 10386 . . . . . . 7  |-  ( 1  -  1 )  =  0
3735, 36syl6eq 2489 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( ( 2  x.  y )  -  1 )  =  0 )
3837fveq2d 5692 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( G `  (
( 2  x.  y
)  -  1 ) )  =  ( G `
 0 ) )
3927, 34, 383eqtr4d 2483 . . . 4  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( F `  (
2  x.  y ) )  =  ( G `
 ( ( 2  x.  y )  - 
1 ) ) )
40 retopon 20301 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
41 iccssre 11373 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( 0 [,] ( 1  /  2
) )  C_  RR )
4214, 18, 41mp2an 667 . . . . . . 7  |-  ( 0 [,] ( 1  / 
2 ) )  C_  RR
43 resttopon 18724 . . . . . . 7  |-  ( ( ( topGen `  ran  (,) )  e.  (TopOn `  RR )  /\  ( 0 [,] (
1  /  2 ) )  C_  RR )  ->  ( ( topGen `  ran  (,) )t  ( 0 [,] (
1  /  2 ) ) )  e.  (TopOn `  ( 0 [,] (
1  /  2 ) ) ) )
4440, 42, 43mp2an 667 . . . . . 6  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  e.  (TopOn `  ( 0 [,] (
1  /  2 ) ) )
4544a1i 11 . . . . 5  |-  ( ph  ->  ( ( topGen `  ran  (,) )t  ( 0 [,] (
1  /  2 ) ) )  e.  (TopOn `  ( 0 [,] (
1  /  2 ) ) ) )
4645, 5cnmpt1st 19200 . . . . . 6  |-  ( ph  ->  ( y  e.  ( 0 [,] ( 1  /  2 ) ) ,  z  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  II )  Cn  (
( topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) ) ) )
4711iihalf1cn 20463 . . . . . . 7  |-  ( x  e.  ( 0 [,] ( 1  /  2
) )  |->  ( 2  x.  x ) )  e.  ( ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  Cn  II )
4847a1i 11 . . . . . 6  |-  ( ph  ->  ( x  e.  ( 0 [,] ( 1  /  2 ) ) 
|->  ( 2  x.  x
) )  e.  ( ( ( topGen `  ran  (,) )t  ( 0 [,] (
1  /  2 ) ) )  Cn  II ) )
49 oveq2 6098 . . . . . 6  |-  ( x  =  y  ->  (
2  x.  x )  =  ( 2  x.  y ) )
5045, 5, 46, 45, 48, 49cnmpt21 19203 . . . . 5  |-  ( ph  ->  ( y  e.  ( 0 [,] ( 1  /  2 ) ) ,  z  e.  ( 0 [,] 1 ) 
|->  ( 2  x.  y
) )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  II )  Cn  II ) )
5145, 5, 50, 1cnmpt21f 19204 . . . 4  |-  ( ph  ->  ( y  e.  ( 0 [,] ( 1  /  2 ) ) ,  z  e.  ( 0 [,] 1 ) 
|->  ( F `  (
2  x.  y ) ) )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( 0 [,] ( 1  /  2
) ) )  tX  II )  Cn  J
) )
52 iccssre 11373 . . . . . . . 8  |-  ( ( ( 1  /  2
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1  / 
2 ) [,] 1
)  C_  RR )
5318, 16, 52mp2an 667 . . . . . . 7  |-  ( ( 1  /  2 ) [,] 1 )  C_  RR
54 resttopon 18724 . . . . . . 7  |-  ( ( ( topGen `  ran  (,) )  e.  (TopOn `  RR )  /\  ( ( 1  / 
2 ) [,] 1
)  C_  RR )  ->  ( ( topGen `  ran  (,) )t  ( ( 1  / 
2 ) [,] 1
) )  e.  (TopOn `  ( ( 1  / 
2 ) [,] 1
) ) )
5540, 53, 54mp2an 667 . . . . . 6  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  e.  (TopOn `  ( ( 1  / 
2 ) [,] 1
) )
5655a1i 11 . . . . 5  |-  ( ph  ->  ( ( topGen `  ran  (,) )t  ( ( 1  / 
2 ) [,] 1
) )  e.  (TopOn `  ( ( 1  / 
2 ) [,] 1
) ) )
5756, 5cnmpt1st 19200 . . . . . 6  |-  ( ph  ->  ( y  e.  ( ( 1  /  2
) [,] 1 ) ,  z  e.  ( 0 [,] 1 ) 
|->  y )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( ( 1  /  2 ) [,] 1 ) )  tX  II )  Cn  (
( topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) ) ) )
5812iihalf2cn 20465 . . . . . . 7  |-  ( x  e.  ( ( 1  /  2 ) [,] 1 )  |->  ( ( 2  x.  x )  -  1 ) )  e.  ( ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  Cn  II )
5958a1i 11 . . . . . 6  |-  ( ph  ->  ( x  e.  ( ( 1  /  2
) [,] 1 ) 
|->  ( ( 2  x.  x )  -  1 ) )  e.  ( ( ( topGen `  ran  (,) )t  ( ( 1  / 
2 ) [,] 1
) )  Cn  II ) )
6049oveq1d 6105 . . . . . 6  |-  ( x  =  y  ->  (
( 2  x.  x
)  -  1 )  =  ( ( 2  x.  y )  - 
1 ) )
6156, 5, 57, 56, 59, 60cnmpt21 19203 . . . . 5  |-  ( ph  ->  ( y  e.  ( ( 1  /  2
) [,] 1 ) ,  z  e.  ( 0 [,] 1 ) 
|->  ( ( 2  x.  y )  -  1 ) )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( ( 1  /  2 ) [,] 1 ) )  tX  II )  Cn  II ) )
6256, 5, 61, 2cnmpt21f 19204 . . . 4  |-  ( ph  ->  ( y  e.  ( ( 1  /  2
) [,] 1 ) ,  z  e.  ( 0 [,] 1 ) 
|->  ( G `  (
( 2  x.  y
)  -  1 ) ) )  e.  ( ( ( ( topGen ` 
ran  (,) )t  ( ( 1  /  2 ) [,] 1 ) )  tX  II )  Cn  J
) )
6310, 11, 12, 13, 15, 17, 25, 5, 39, 51, 62cnmpt2pc 20459 . . 3  |-  ( ph  ->  ( y  e.  ( 0 [,] 1 ) ,  z  e.  ( 0 [,] 1 ) 
|->  if ( y  <_ 
( 1  /  2
) ,  ( F `
 ( 2  x.  y ) ) ,  ( G `  (
( 2  x.  y
)  -  1 ) ) ) )  e.  ( ( II  tX  II )  Cn  J
) )
64 breq1 4292 . . . . 5  |-  ( y  =  x  ->  (
y  <_  ( 1  /  2 )  <->  x  <_  ( 1  /  2 ) ) )
65 oveq2 6098 . . . . . 6  |-  ( y  =  x  ->  (
2  x.  y )  =  ( 2  x.  x ) )
6665fveq2d 5692 . . . . 5  |-  ( y  =  x  ->  ( F `  ( 2  x.  y ) )  =  ( F `  (
2  x.  x ) ) )
6765oveq1d 6105 . . . . . 6  |-  ( y  =  x  ->  (
( 2  x.  y
)  -  1 )  =  ( ( 2  x.  x )  - 
1 ) )
6867fveq2d 5692 . . . . 5  |-  ( y  =  x  ->  ( G `  ( (
2  x.  y )  -  1 ) )  =  ( G `  ( ( 2  x.  x )  -  1 ) ) )
6964, 66, 68ifbieq12d 3813 . . . 4  |-  ( y  =  x  ->  if ( y  <_  (
1  /  2 ) ,  ( F `  ( 2  x.  y
) ) ,  ( G `  ( ( 2  x.  y )  -  1 ) ) )  =  if ( x  <_  ( 1  /  2 ) ,  ( F `  (
2  x.  x ) ) ,  ( G `
 ( ( 2  x.  x )  - 
1 ) ) ) )
7069adantr 462 . . 3  |-  ( ( y  =  x  /\  z  =  0 )  ->  if ( y  <_  ( 1  / 
2 ) ,  ( F `  ( 2  x.  y ) ) ,  ( G `  ( ( 2  x.  y )  -  1 ) ) )  =  if ( x  <_ 
( 1  /  2
) ,  ( F `
 ( 2  x.  x ) ) ,  ( G `  (
( 2  x.  x
)  -  1 ) ) ) )
715, 6, 9, 5, 5, 63, 70cnmpt12 19199 . 2  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  <_ 
( 1  /  2
) ,  ( F `
 ( 2  x.  x ) ) ,  ( G `  (
( 2  x.  x
)  -  1 ) ) ) )  e.  ( II  Cn  J
) )
723, 71eqeltrd 2515 1  |-  ( ph  ->  ( F ( *p
`  J ) G )  e.  ( II 
Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    C_ wss 3325   ifcif 3788   class class class wbr 4289    e. cmpt 4347   ran crn 4837   ` cfv 5415  (class class class)co 6090   RRcr 9277   0cc0 9278   1c1 9279    x. cmul 9283    <_ cle 9415    - cmin 9591    / cdiv 9989   2c2 10367   (,)cioo 11296   [,]cicc 11299   ↾t crest 14355   topGenctg 14372  TopOnctopon 18458    Cn ccn 18787   IIcii 20410   *pcpco 20531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-cn 18790  df-cnp 18791  df-tx 19094  df-hmeo 19287  df-xms 19854  df-ms 19855  df-tms 19856  df-ii 20412  df-pco 20536
This theorem is referenced by:  copco  20549  pcohtpylem  20550  pcohtpy  20551  pcoass  20555  pcorevlem  20557  om1addcl  20564  pi1xfrf  20584  pi1xfr  20586  pi1xfrcnvlem  20587  pi1coghm  20592  conpcon  27054  sconpht2  27057  cvmlift3lem6  27143
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