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Theorem pcocn 21247
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 21241 . 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 21113 . . . 4  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
54a1i 11 . . 3  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
65cnmptid 19892 . . 3  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( II  Cn  II ) )
7 0elunit 11629 . . . . 5  |-  0  e.  ( 0 [,] 1
)
87a1i 11 . . . 4  |-  ( ph  ->  0  e.  ( 0 [,] 1 ) )
95, 5, 8cnmptc 19893 . . 3  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  0 )  e.  ( II  Cn  II ) )
10 eqid 2462 . . . 4  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
11 eqid 2462 . . . 4  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )
12 eqid 2462 . . . 4  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )
13 dfii2 21116 . . . 4  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
14 0re 9587 . . . . 5  |-  0  e.  RR
1514a1i 11 . . . 4  |-  ( ph  ->  0  e.  RR )
16 1re 9586 . . . . 5  |-  1  e.  RR
1716a1i 11 . . . 4  |-  ( ph  ->  1  e.  RR )
18 halfre 10745 . . . . . 6  |-  ( 1  /  2 )  e.  RR
19 halfgt0 10747 . . . . . . 7  |-  0  <  ( 1  /  2
)
2014, 18, 19ltleii 9698 . . . . . 6  |-  0  <_  ( 1  /  2
)
21 halflt1 10748 . . . . . . 7  |-  ( 1  /  2 )  <  1
2218, 16, 21ltleii 9698 . . . . . 6  |-  ( 1  /  2 )  <_ 
1
2314, 16elicc2i 11581 . . . . . 6  |-  ( ( 1  /  2 )  e.  ( 0 [,] 1 )  <->  ( (
1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_ 
1 ) )
2418, 20, 22, 23mpbir3an 1173 . . . . 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 465 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( F `  1
)  =  ( G `
 0 ) )
28 simprl 755 . . . . . . . 8  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
y  =  ( 1  /  2 ) )
2928oveq2d 6293 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( 2  x.  y
)  =  ( 2  x.  ( 1  / 
2 ) ) )
30 2cn 10597 . . . . . . . 8  |-  2  e.  CC
31 2ne0 10619 . . . . . . . 8  |-  2  =/=  0
3230, 31recidi 10266 . . . . . . 7  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
3329, 32syl6eq 2519 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( 2  x.  y
)  =  1 )
3433fveq2d 5863 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( F `  (
2  x.  y ) )  =  ( F `
 1 ) )
3533oveq1d 6292 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( ( 2  x.  y )  -  1 )  =  ( 1  -  1 ) )
36 1m1e0 10595 . . . . . . 7  |-  ( 1  -  1 )  =  0
3735, 36syl6eq 2519 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( ( 2  x.  y )  -  1 )  =  0 )
3837fveq2d 5863 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( G `  (
( 2  x.  y
)  -  1 ) )  =  ( G `
 0 ) )
3927, 34, 383eqtr4d 2513 . . . 4  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( F `  (
2  x.  y ) )  =  ( G `
 ( ( 2  x.  y )  - 
1 ) ) )
40 retopon 21000 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
41 iccssre 11597 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( 0 [,] ( 1  /  2
) )  C_  RR )
4214, 18, 41mp2an 672 . . . . . . 7  |-  ( 0 [,] ( 1  / 
2 ) )  C_  RR
43 resttopon 19423 . . . . . . 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 672 . . . . . 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 19899 . . . . . 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 21162 . . . . . . 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 6285 . . . . . 6  |-  ( x  =  y  ->  (
2  x.  x )  =  ( 2  x.  y ) )
5045, 5, 46, 45, 48, 49cnmpt21 19902 . . . . 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 19903 . . . 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 11597 . . . . . . . 8  |-  ( ( ( 1  /  2
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1  / 
2 ) [,] 1
)  C_  RR )
5318, 16, 52mp2an 672 . . . . . . 7  |-  ( ( 1  /  2 ) [,] 1 )  C_  RR
54 resttopon 19423 . . . . . . 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 672 . . . . . 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 19899 . . . . . 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 21164 . . . . . . 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 6292 . . . . . 6  |-  ( x  =  y  ->  (
( 2  x.  x
)  -  1 )  =  ( ( 2  x.  y )  - 
1 ) )
6156, 5, 57, 56, 59, 60cnmpt21 19902 . . . . 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 19903 . . . 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 21158 . . 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 4445 . . . . 5  |-  ( y  =  x  ->  (
y  <_  ( 1  /  2 )  <->  x  <_  ( 1  /  2 ) ) )
65 oveq2 6285 . . . . . 6  |-  ( y  =  x  ->  (
2  x.  y )  =  ( 2  x.  x ) )
6665fveq2d 5863 . . . . 5  |-  ( y  =  x  ->  ( F `  ( 2  x.  y ) )  =  ( F `  (
2  x.  x ) ) )
6765oveq1d 6292 . . . . . 6  |-  ( y  =  x  ->  (
( 2  x.  y
)  -  1 )  =  ( ( 2  x.  x )  - 
1 ) )
6867fveq2d 5863 . . . . 5  |-  ( y  =  x  ->  ( G `  ( (
2  x.  y )  -  1 ) )  =  ( G `  ( ( 2  x.  x )  -  1 ) ) )
6964, 66, 68ifbieq12d 3961 . . . 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 465 . . 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 19898 . 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 2550 1  |-  ( ph  ->  ( F ( *p
`  J ) G )  e.  ( II 
Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    C_ wss 3471   ifcif 3934   class class class wbr 4442    |-> cmpt 4500   ran crn 4995   ` cfv 5581  (class class class)co 6277   RRcr 9482   0cc0 9483   1c1 9484    x. cmul 9488    <_ cle 9620    - cmin 9796    / cdiv 10197   2c2 10576   (,)cioo 11520   [,]cicc 11523   ↾t crest 14667   topGenctg 14684  TopOnctopon 19157    Cn ccn 19486   IIcii 21109   *pcpco 21230
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-mulf 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-fi 7862  df-sup 7892  df-oi 7926  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ioo 11524  df-icc 11527  df-fz 11664  df-fzo 11784  df-seq 12066  df-exp 12125  df-hash 12363  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-hom 14570  df-cco 14571  df-rest 14669  df-topn 14670  df-0g 14688  df-gsum 14689  df-topgen 14690  df-pt 14691  df-prds 14694  df-xrs 14748  df-qtop 14753  df-imas 14754  df-xps 14756  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-submnd 15773  df-mulg 15856  df-cntz 16145  df-cmn 16591  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-cnfld 18187  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cld 19281  df-cn 19489  df-cnp 19490  df-tx 19793  df-hmeo 19986  df-xms 20553  df-ms 20554  df-tms 20555  df-ii 21111  df-pco 21235
This theorem is referenced by:  copco  21248  pcohtpylem  21249  pcohtpy  21250  pcoass  21254  pcorevlem  21256  om1addcl  21263  pi1xfrf  21283  pi1xfr  21285  pi1xfrcnvlem  21286  pi1coghm  21291  conpcon  28308  sconpht2  28311  cvmlift3lem6  28397
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