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.

Step	Hyp	Ref	Expression
1		pcoval.2	. . 3 |- ( ph -> F e. ( II Cn J ) )
2		pcoval.3	. . 3 |- ( ph -> G e. ( II Cn J ) )
3	1, 2	pcoval 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 ) )
5	4	a1i 11	. . 3 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
6	5	cnmptid 19193	. . 3 |- ( ph -> ( x e. ( 0 [,] 1 ) |-> x ) e. ( II Cn II ) )
7		0elunit 11399	. . . . 5 |- 0 e. ( 0 [,] 1 )
8	7	a1i 11	. . . 4 |- ( ph -> 0 e. ( 0 [,] 1 ) )
9	5, 5, 8	cnmptc 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
15	14	a1i 11	. . . 4 |- ( ph -> 0 e. RR )
16		1re 9381	. . . . 5 |- 1 e. RR
17	16	a1i 11	. . . 4 |- ( ph -> 1 e. RR )
18		halfre 10536	. . . . . 6 |- ( 1 / 2 ) e. RR
19		halfgt0 10538	. . . . . . 7 |- 0 < ( 1 / 2 )
20	14, 18, 19	ltleii 9493	. . . . . 6 |- 0 <_ ( 1 / 2 )
21		halflt1 10539	. . . . . . 7 |- ( 1 / 2 ) < 1
22	18, 16, 21	ltleii 9493	. . . . . 6 |- ( 1 / 2 ) <_ 1
23	14, 16	elicc2i 11357	. . . . . 6 |- ( ( 1 / 2 ) e. ( 0 [,] 1 ) <-> ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ 1 ) )
24	18, 20, 22, 23	mpbir3an 1165	. . . . 5 |- ( 1 / 2 ) e. ( 0 [,] 1 )
25	24	a1i 11	. . . 4 |- ( ph -> ( 1 / 2 ) e. ( 0 [,] 1 ) )
26		pcoval2.4	. . . . . 6 |- ( ph -> ( F ` 1 ) = ( G ` 0 ) )
27	26	adantr 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 ) )
29	28	oveq2d 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
32	30, 31	recidi 10058	. . . . . . 7 |- ( 2 x. ( 1 / 2 ) ) = 1
33	29, 32	syl6eq 2489	. . . . . 6 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( 2 x. y ) = 1 )
34	33	fveq2d 5692	. . . . 5 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( F ` ( 2 x. y ) ) = ( F ` 1 ) )
35	33	oveq1d 6105	. . . . . . 7 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( ( 2 x. y ) - 1 ) = ( 1 - 1 ) )
36		1m1e0 10386	. . . . . . 7 |- ( 1 - 1 ) = 0
37	35, 36	syl6eq 2489	. . . . . 6 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( ( 2 x. y ) - 1 ) = 0 )
38	37	fveq2d 5692	. . . . 5 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( G ` ( ( 2 x. y ) - 1 ) ) = ( G ` 0 ) )
39	27, 34, 38	3eqtr4d 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 )
42	14, 18, 41	mp2an 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 ) ) ) )
44	40, 42, 43	mp2an 667	. . . . . 6 |- ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) e. (TopOn ` ( 0 [,] ( 1 / 2 ) ) )
45	44	a1i 11	. . . . 5 |- ( ph -> ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) e. (TopOn ` ( 0 [,] ( 1 / 2 ) ) ) )
46	45, 5	cnmpt1st 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 ) ) ) ) )
47	11	iihalf1cn 20463	. . . . . . 7 |- ( x e. ( 0 [,] ( 1 / 2 ) ) |-> ( 2 x. x ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) Cn II )
48	47	a1i 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 ) )
50	45, 5, 46, 45, 48, 49	cnmpt21 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 ) )
51	45, 5, 50, 1	cnmpt21f 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 )
53	18, 16, 52	mp2an 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 ) ) )
55	40, 53, 54	mp2an 667	. . . . . 6 |- ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) e. (TopOn ` ( ( 1 / 2 ) [,] 1 ) )
56	55	a1i 11	. . . . 5 |- ( ph -> ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) e. (TopOn ` ( ( 1 / 2 ) [,] 1 ) ) )
57	56, 5	cnmpt1st 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 ) ) ) )
58	12	iihalf2cn 20465	. . . . . . 7 |- ( x e. ( ( 1 / 2 ) [,] 1 ) |-> ( ( 2 x. x ) - 1 ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) Cn II )
59	58	a1i 11	. . . . . 6 |- ( ph -> ( x e. ( ( 1 / 2 ) [,] 1 ) |-> ( ( 2 x. x ) - 1 ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) Cn II ) )
60	49	oveq1d 6105	. . . . . 6 |- ( x = y -> ( ( 2 x. x ) - 1 ) = ( ( 2 x. y ) - 1 ) )
61	56, 5, 57, 56, 59, 60	cnmpt21 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 ) )
62	56, 5, 61, 2	cnmpt21f 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 ) )
63	10, 11, 12, 13, 15, 17, 25, 5, 39, 51, 62	cnmpt2pc 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 ) )
66	65	fveq2d 5692	. . . . 5 |- ( y = x -> ( F ` ( 2 x. y ) ) = ( F ` ( 2 x. x ) ) )
67	65	oveq1d 6105	. . . . . 6 |- ( y = x -> ( ( 2 x. y ) - 1 ) = ( ( 2 x. x ) - 1 ) )
68	67	fveq2d 5692	. . . . 5 |- ( y = x -> ( G ` ( ( 2 x. y ) - 1 ) ) = ( G ` ( ( 2 x. x ) - 1 ) ) )
69	64, 66, 68	ifbieq12d 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 ) ) ) )
70	69	adantr 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 ) ) ) )
71	5, 6, 9, 5, 5, 63, 70	cnmpt12 19199	. 2 |- ( ph -> ( x e. ( 0 [,] 1 ) |-> if ( x <_ ( 1 / 2 ) , ( F ` ( 2 x. x ) ) , ( G ` ( ( 2 x. x ) - 1 ) ) ) ) e. ( II Cn J ) )
72	3, 71	eqeltrd 2515	1 |- ( ph -> ( F ( *p ` J ) G ) e. ( II Cn J ) )

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