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Theorem pcocn 20705
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 20699 . 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 20571 . . . 4  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
54a1i 11 . . 3  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
65cnmptid 19350 . . 3  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( II  Cn  II ) )
7 0elunit 11504 . . . . 5  |-  0  e.  ( 0 [,] 1
)
87a1i 11 . . . 4  |-  ( ph  ->  0  e.  ( 0 [,] 1 ) )
95, 5, 8cnmptc 19351 . . 3  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  0 )  e.  ( II  Cn  II ) )
10 eqid 2451 . . . 4  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
11 eqid 2451 . . . 4  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )
12 eqid 2451 . . . 4  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )
13 dfii2 20574 . . . 4  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
14 0re 9487 . . . . 5  |-  0  e.  RR
1514a1i 11 . . . 4  |-  ( ph  ->  0  e.  RR )
16 1re 9486 . . . . 5  |-  1  e.  RR
1716a1i 11 . . . 4  |-  ( ph  ->  1  e.  RR )
18 halfre 10641 . . . . . 6  |-  ( 1  /  2 )  e.  RR
19 halfgt0 10643 . . . . . . 7  |-  0  <  ( 1  /  2
)
2014, 18, 19ltleii 9598 . . . . . 6  |-  0  <_  ( 1  /  2
)
21 halflt1 10644 . . . . . . 7  |-  ( 1  /  2 )  <  1
2218, 16, 21ltleii 9598 . . . . . 6  |-  ( 1  /  2 )  <_ 
1
2314, 16elicc2i 11462 . . . . . 6  |-  ( ( 1  /  2 )  e.  ( 0 [,] 1 )  <->  ( (
1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_ 
1 ) )
2418, 20, 22, 23mpbir3an 1170 . . . . 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 6206 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( 2  x.  y
)  =  ( 2  x.  ( 1  / 
2 ) ) )
30 2cn 10493 . . . . . . . 8  |-  2  e.  CC
31 2ne0 10515 . . . . . . . 8  |-  2  =/=  0
3230, 31recidi 10163 . . . . . . 7  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
3329, 32syl6eq 2508 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( 2  x.  y
)  =  1 )
3433fveq2d 5793 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( F `  (
2  x.  y ) )  =  ( F `
 1 ) )
3533oveq1d 6205 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( ( 2  x.  y )  -  1 )  =  ( 1  -  1 ) )
36 1m1e0 10491 . . . . . . 7  |-  ( 1  -  1 )  =  0
3735, 36syl6eq 2508 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( ( 2  x.  y )  -  1 )  =  0 )
3837fveq2d 5793 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( G `  (
( 2  x.  y
)  -  1 ) )  =  ( G `
 0 ) )
3927, 34, 383eqtr4d 2502 . . . 4  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( F `  (
2  x.  y ) )  =  ( G `
 ( ( 2  x.  y )  - 
1 ) ) )
40 retopon 20458 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
41 iccssre 11478 . . . . . . . 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 18881 . . . . . . 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 19357 . . . . . 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 20620 . . . . . . 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 6198 . . . . . 6  |-  ( x  =  y  ->  (
2  x.  x )  =  ( 2  x.  y ) )
5045, 5, 46, 45, 48, 49cnmpt21 19360 . . . . 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 19361 . . . 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 11478 . . . . . . . 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 18881 . . . . . . 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 19357 . . . . . 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 20622 . . . . . . 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 6205 . . . . . 6  |-  ( x  =  y  ->  (
( 2  x.  x
)  -  1 )  =  ( ( 2  x.  y )  - 
1 ) )
6156, 5, 57, 56, 59, 60cnmpt21 19360 . . . . 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 19361 . . . 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 20616 . . 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 4393 . . . . 5  |-  ( y  =  x  ->  (
y  <_  ( 1  /  2 )  <->  x  <_  ( 1  /  2 ) ) )
65 oveq2 6198 . . . . . 6  |-  ( y  =  x  ->  (
2  x.  y )  =  ( 2  x.  x ) )
6665fveq2d 5793 . . . . 5  |-  ( y  =  x  ->  ( F `  ( 2  x.  y ) )  =  ( F `  (
2  x.  x ) ) )
6765oveq1d 6205 . . . . . 6  |-  ( y  =  x  ->  (
( 2  x.  y
)  -  1 )  =  ( ( 2  x.  x )  - 
1 ) )
6867fveq2d 5793 . . . . 5  |-  ( y  =  x  ->  ( G `  ( (
2  x.  y )  -  1 ) )  =  ( G `  ( ( 2  x.  x )  -  1 ) ) )
6964, 66, 68ifbieq12d 3914 . . . 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 19356 . 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 2539 1  |-  ( ph  ->  ( F ( *p
`  J ) G )  e.  ( II 
Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3426   ifcif 3889   class class class wbr 4390    |-> cmpt 4448   ran crn 4939   ` cfv 5516  (class class class)co 6190   RRcr 9382   0cc0 9383   1c1 9384    x. cmul 9388    <_ cle 9520    - cmin 9696    / cdiv 10094   2c2 10472   (,)cioo 11401   [,]cicc 11404   ↾t crest 14461   topGenctg 14478  TopOnctopon 18615    Cn ccn 18944   IIcii 20567   *pcpco 20688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-mulf 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-fi 7762  df-sup 7792  df-oi 7825  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-q 11055  df-rp 11093  df-xneg 11190  df-xadd 11191  df-xmul 11192  df-ioo 11405  df-icc 11408  df-fz 11539  df-fzo 11650  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-starv 14355  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-unif 14363  df-hom 14364  df-cco 14365  df-rest 14463  df-topn 14464  df-0g 14482  df-gsum 14483  df-topgen 14484  df-pt 14485  df-prds 14488  df-xrs 14542  df-qtop 14547  df-imas 14548  df-xps 14550  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-submnd 15567  df-mulg 15650  df-cntz 15937  df-cmn 16383  df-psmet 17918  df-xmet 17919  df-met 17920  df-bl 17921  df-mopn 17922  df-cnfld 17928  df-top 18619  df-bases 18621  df-topon 18622  df-topsp 18623  df-cld 18739  df-cn 18947  df-cnp 18948  df-tx 19251  df-hmeo 19444  df-xms 20011  df-ms 20012  df-tms 20013  df-ii 20569  df-pco 20693
This theorem is referenced by:  copco  20706  pcohtpylem  20707  pcohtpy  20708  pcoass  20712  pcorevlem  20714  om1addcl  20721  pi1xfrf  20741  pi1xfr  20743  pi1xfrcnvlem  20744  pi1coghm  20749  conpcon  27258  sconpht2  27261  cvmlift3lem6  27347
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