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Theorem pcocn 21683
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 21677 . 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 21549 . . . 4  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
54a1i 11 . . 3  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
65cnmptid 20328 . . 3  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  x )  e.  ( II  Cn  II ) )
7 0elunit 11641 . . . . 5  |-  0  e.  ( 0 [,] 1
)
87a1i 11 . . . 4  |-  ( ph  ->  0  e.  ( 0 [,] 1 ) )
95, 5, 8cnmptc 20329 . . 3  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) 
|->  0 )  e.  ( II  Cn  II ) )
10 eqid 2454 . . . 4  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
11 eqid 2454 . . . 4  |-  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )  =  ( (
topGen `  ran  (,) )t  (
0 [,] ( 1  /  2 ) ) )
12 eqid 2454 . . . 4  |-  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )  =  ( (
topGen `  ran  (,) )t  (
( 1  /  2
) [,] 1 ) )
13 dfii2 21552 . . . 4  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
14 0re 9585 . . . . 5  |-  0  e.  RR
1514a1i 11 . . . 4  |-  ( ph  ->  0  e.  RR )
16 1re 9584 . . . . 5  |-  1  e.  RR
1716a1i 11 . . . 4  |-  ( ph  ->  1  e.  RR )
18 halfre 10750 . . . . . 6  |-  ( 1  /  2 )  e.  RR
19 halfgt0 10752 . . . . . . 7  |-  0  <  ( 1  /  2
)
2014, 18, 19ltleii 9696 . . . . . 6  |-  0  <_  ( 1  /  2
)
21 halflt1 10753 . . . . . . 7  |-  ( 1  /  2 )  <  1
2218, 16, 21ltleii 9696 . . . . . 6  |-  ( 1  /  2 )  <_ 
1
2314, 16elicc2i 11593 . . . . . 6  |-  ( ( 1  /  2 )  e.  ( 0 [,] 1 )  <->  ( (
1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
)  /\  ( 1  /  2 )  <_ 
1 ) )
2418, 20, 22, 23mpbir3an 1176 . . . . 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 463 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( F `  1
)  =  ( G `
 0 ) )
28 simprl 754 . . . . . . . 8  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
y  =  ( 1  /  2 ) )
2928oveq2d 6286 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( 2  x.  y
)  =  ( 2  x.  ( 1  / 
2 ) ) )
30 2cn 10602 . . . . . . . 8  |-  2  e.  CC
31 2ne0 10624 . . . . . . . 8  |-  2  =/=  0
3230, 31recidi 10271 . . . . . . 7  |-  ( 2  x.  ( 1  / 
2 ) )  =  1
3329, 32syl6eq 2511 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( 2  x.  y
)  =  1 )
3433fveq2d 5852 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( F `  (
2  x.  y ) )  =  ( F `
 1 ) )
3533oveq1d 6285 . . . . . . 7  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( ( 2  x.  y )  -  1 )  =  ( 1  -  1 ) )
36 1m1e0 10600 . . . . . . 7  |-  ( 1  -  1 )  =  0
3735, 36syl6eq 2511 . . . . . 6  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( ( 2  x.  y )  -  1 )  =  0 )
3837fveq2d 5852 . . . . 5  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( G `  (
( 2  x.  y
)  -  1 ) )  =  ( G `
 0 ) )
3927, 34, 383eqtr4d 2505 . . . 4  |-  ( (
ph  /\  ( y  =  ( 1  / 
2 )  /\  z  e.  ( 0 [,] 1
) ) )  -> 
( F `  (
2  x.  y ) )  =  ( G `
 ( ( 2  x.  y )  - 
1 ) ) )
40 retopon 21436 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
41 iccssre 11609 . . . . . . . 8  |-  ( ( 0  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( 0 [,] ( 1  /  2
) )  C_  RR )
4214, 18, 41mp2an 670 . . . . . . 7  |-  ( 0 [,] ( 1  / 
2 ) )  C_  RR
43 resttopon 19829 . . . . . . 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 670 . . . . . 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 20335 . . . . . 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 21598 . . . . . . 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 6278 . . . . . 6  |-  ( x  =  y  ->  (
2  x.  x )  =  ( 2  x.  y ) )
5045, 5, 46, 45, 48, 49cnmpt21 20338 . . . . 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 20339 . . . 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 11609 . . . . . . . 8  |-  ( ( ( 1  /  2
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1  / 
2 ) [,] 1
)  C_  RR )
5318, 16, 52mp2an 670 . . . . . . 7  |-  ( ( 1  /  2 ) [,] 1 )  C_  RR
54 resttopon 19829 . . . . . . 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 670 . . . . . 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 20335 . . . . . 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 21600 . . . . . . 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 6285 . . . . . 6  |-  ( x  =  y  ->  (
( 2  x.  x
)  -  1 )  =  ( ( 2  x.  y )  - 
1 ) )
6156, 5, 57, 56, 59, 60cnmpt21 20338 . . . . 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 20339 . . . 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 21594 . . 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 4442 . . . . 5  |-  ( y  =  x  ->  (
y  <_  ( 1  /  2 )  <->  x  <_  ( 1  /  2 ) ) )
65 oveq2 6278 . . . . . 6  |-  ( y  =  x  ->  (
2  x.  y )  =  ( 2  x.  x ) )
6665fveq2d 5852 . . . . 5  |-  ( y  =  x  ->  ( F `  ( 2  x.  y ) )  =  ( F `  (
2  x.  x ) ) )
6765oveq1d 6285 . . . . . 6  |-  ( y  =  x  ->  (
( 2  x.  y
)  -  1 )  =  ( ( 2  x.  x )  - 
1 ) )
6867fveq2d 5852 . . . . 5  |-  ( y  =  x  ->  ( G `  ( (
2  x.  y )  -  1 ) )  =  ( G `  ( ( 2  x.  x )  -  1 ) ) )
6964, 66, 68ifbieq12d 3956 . . . 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 463 . . 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 20334 . 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 2542 1  |-  ( ph  ->  ( F ( *p
`  J ) G )  e.  ( II 
Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    C_ wss 3461   ifcif 3929   class class class wbr 4439    |-> cmpt 4497   ran crn 4989   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481   1c1 9482    x. cmul 9486    <_ cle 9618    - cmin 9796    / cdiv 10202   2c2 10581   (,)cioo 11532   [,]cicc 11535   ↾t crest 14910   topGenctg 14927  TopOnctopon 19562    Cn ccn 19892   IIcii 21545   *pcpco 21666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-cn 19895  df-cnp 19896  df-tx 20229  df-hmeo 20422  df-xms 20989  df-ms 20990  df-tms 20991  df-ii 21547  df-pco 21671
This theorem is referenced by:  copco  21684  pcohtpylem  21685  pcohtpy  21686  pcoass  21690  pcorevlem  21692  om1addcl  21699  pi1xfrf  21719  pi1xfr  21721  pi1xfrcnvlem  21722  pi1coghm  21727  conpcon  28944  sconpht2  28947  cvmlift3lem6  29033
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