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Theorem pcocn 19044
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 19038 . 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 18911 . . . 4  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
54a1i 11 . . 3  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
65cnmptid 17695 . . 3  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( II  Cn  II ) )
7 0elunit 11017 . . . . 5  |-  0  e.  ( 0 [,] 1
)
87a1i 11 . . . 4  |-  ( ph  ->  0  e.  ( 0 [,] 1 ) )
95, 5, 8cnmptc 17696 . . 3  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  0 )  e.  ( II  Cn  II ) )
10 eqid 2438 . . . 4  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
11 eqid 2438 . . . 4  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )
12 eqid 2438 . . . 4  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )
13 dfii2 18914 . . . 4  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
14 0re 9093 . . . . 5  |-  0  e.  RR
1514a1i 11 . . . 4  |-  ( ph  ->  0  e.  RR )
16 1re 9092 . . . . 5  |-  1  e.  RR
1716a1i 11 . . . 4  |-  ( ph  ->  1  e.  RR )
1816rehalfcli 10218 . . . . . 6  |-  ( 1  /  2 )  e.  RR
19 halfgt0 10190 . . . . . . 7  |-  0  <  ( 1  /  2
)
2014, 18, 19ltleii 9198 . . . . . 6  |-  0  <_  ( 1  /  2
)
21 halflt1 10191 . . . . . . 7  |-  ( 1  /  2 )  <  1
2218, 16, 21ltleii 9198 . . . . . 6  |-  ( 1  /  2 )  <_ 
1
2314, 16elicc2i 10978 . . . . . 6  |-  ( ( 1  /  2 )  e.  ( 0 [,] 1 )  <->  ( (
1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_ 
1 ) )
2418, 20, 22, 23mpbir3an 1137 . . . . 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 453 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( F `  1
)  =  ( G `
 0 ) )
28 simprl 734 . . . . . . . 8  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
y  =  ( 1  /  2 ) )
2928oveq2d 6099 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( 2  x.  y
)  =  ( 2  x.  ( 1  / 
2 ) ) )
30 2cn 10072 . . . . . . . 8  |-  2  e.  CC
31 2ne0 10085 . . . . . . . 8  |-  2  =/=  0
3230, 31recidi 9747 . . . . . . 7  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
3329, 32syl6eq 2486 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( 2  x.  y
)  =  1 )
3433fveq2d 5734 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( F `  (
2  x.  y ) )  =  ( F `
 1 ) )
3533oveq1d 6098 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( ( 2  x.  y )  -  1 )  =  ( 1  -  1 ) )
36 1m1e0 10070 . . . . . . 7  |-  ( 1  -  1 )  =  0
3735, 36syl6eq 2486 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( ( 2  x.  y )  -  1 )  =  0 )
3837fveq2d 5734 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( G `  (
( 2  x.  y
)  -  1 ) )  =  ( G `
 0 ) )
3927, 34, 383eqtr4d 2480 . . . 4  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( F `  (
2  x.  y ) )  =  ( G `
 ( ( 2  x.  y )  - 
1 ) ) )
40 retopon 18799 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
41 iccssre 10994 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( 0 [,] ( 1  /  2
) )  C_  RR )
4214, 18, 41mp2an 655 . . . . . . 7  |-  ( 0 [,] ( 1  / 
2 ) )  C_  RR
43 resttopon 17227 . . . . . . 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 655 . . . . . 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 17702 . . . . . 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 18959 . . . . . . 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 6091 . . . . . 6  |-  ( x  =  y  ->  (
2  x.  x )  =  ( 2  x.  y ) )
5045, 5, 46, 45, 48, 49cnmpt21 17705 . . . . 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 17706 . . . 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 10994 . . . . . . . 8  |-  ( ( ( 1  /  2
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1  / 
2 ) [,] 1
)  C_  RR )
5318, 16, 52mp2an 655 . . . . . . 7  |-  ( ( 1  /  2 ) [,] 1 )  C_  RR
54 resttopon 17227 . . . . . . 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 655 . . . . . 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 17702 . . . . . 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 18961 . . . . . . 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 6098 . . . . . 6  |-  ( x  =  y  ->  (
( 2  x.  x
)  -  1 )  =  ( ( 2  x.  y )  - 
1 ) )
6156, 5, 57, 56, 59, 60cnmpt21 17705 . . . . 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 17706 . . . 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 18955 . . 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 4217 . . . . 5  |-  ( y  =  x  ->  (
y  <_  ( 1  /  2 )  <->  x  <_  ( 1  /  2 ) ) )
65 oveq2 6091 . . . . . 6  |-  ( y  =  x  ->  (
2  x.  y )  =  ( 2  x.  x ) )
6665fveq2d 5734 . . . . 5  |-  ( y  =  x  ->  ( F `  ( 2  x.  y ) )  =  ( F `  (
2  x.  x ) ) )
6765oveq1d 6098 . . . . . 6  |-  ( y  =  x  ->  (
( 2  x.  y
)  -  1 )  =  ( ( 2  x.  x )  - 
1 ) )
6867fveq2d 5734 . . . . 5  |-  ( y  =  x  ->  ( G `  ( (
2  x.  y )  -  1 ) )  =  ( G `  ( ( 2  x.  x )  -  1 ) ) )
6964, 66, 68ifbieq12d 3763 . . . 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 453 . . 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 17701 . 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 2512 1  |-  ( ph  ->  ( F ( *p
`  J ) G )  e.  ( II 
Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322   ifcif 3741   class class class wbr 4214    e. cmpt 4268   ran crn 4881   ` cfv 5456  (class class class)co 6083   RRcr 8991   0cc0 8992   1c1 8993    x. cmul 8997    <_ cle 9123    - cmin 9293    / cdiv 9679   2c2 10051   (,)cioo 10918   [,]cicc 10921   ↾t crest 13650   topGenctg 13667  TopOnctopon 16961    Cn ccn 17290   IIcii 18907   *pcpco 19027
This theorem is referenced by:  copco  19045  pcohtpylem  19046  pcohtpy  19047  pcoass  19051  pcorevlem  19053  om1addcl  19060  pi1xfrf  19080  pi1xfr  19082  pi1xfrcnvlem  19083  pi1coghm  19088  conpcon  24924  sconpht2  24927  cvmlift3lem6  25013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-icc 10925  df-fz 11046  df-fzo 11138  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-cn 17293  df-cnp 17294  df-tx 17596  df-hmeo 17789  df-xms 18352  df-ms 18353  df-tms 18354  df-ii 18909  df-pco 19032
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