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.

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 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 ) )
5	4	a1i 11	. . 3 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
6	5	cnmptid 19350	. . 3 |- ( ph -> ( x e. ( 0 [,] 1 ) |-> x ) e. ( II Cn II ) )
7		0elunit 11504	. . . . 5 |- 0 e. ( 0 [,] 1 )
8	7	a1i 11	. . . 4 |- ( ph -> 0 e. ( 0 [,] 1 ) )
9	5, 5, 8	cnmptc 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
15	14	a1i 11	. . . 4 |- ( ph -> 0 e. RR )
16		1re 9486	. . . . 5 |- 1 e. RR
17	16	a1i 11	. . . 4 |- ( ph -> 1 e. RR )
18		halfre 10641	. . . . . 6 |- ( 1 / 2 ) e. RR
19		halfgt0 10643	. . . . . . 7 |- 0 < ( 1 / 2 )
20	14, 18, 19	ltleii 9598	. . . . . 6 |- 0 <_ ( 1 / 2 )
21		halflt1 10644	. . . . . . 7 |- ( 1 / 2 ) < 1
22	18, 16, 21	ltleii 9598	. . . . . 6 |- ( 1 / 2 ) <_ 1
23	14, 16	elicc2i 11462	. . . . . 6 |- ( ( 1 / 2 ) e. ( 0 [,] 1 ) <-> ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ 1 ) )
24	18, 20, 22, 23	mpbir3an 1170	. . . . 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 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 ) )
29	28	oveq2d 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
32	30, 31	recidi 10163	. . . . . . 7 |- ( 2 x. ( 1 / 2 ) ) = 1
33	29, 32	syl6eq 2508	. . . . . 6 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( 2 x. y ) = 1 )
34	33	fveq2d 5793	. . . . 5 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( F ` ( 2 x. y ) ) = ( F ` 1 ) )
35	33	oveq1d 6205	. . . . . . 7 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( ( 2 x. y ) - 1 ) = ( 1 - 1 ) )
36		1m1e0 10491	. . . . . . 7 |- ( 1 - 1 ) = 0
37	35, 36	syl6eq 2508	. . . . . 6 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( ( 2 x. y ) - 1 ) = 0 )
38	37	fveq2d 5793	. . . . 5 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( G ` ( ( 2 x. y ) - 1 ) ) = ( G ` 0 ) )
39	27, 34, 38	3eqtr4d 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 )
42	14, 18, 41	mp2an 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 ) ) ) )
44	40, 42, 43	mp2an 672	. . . . . 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 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 ) ) ) ) )
47	11	iihalf1cn 20620	. . . . . . 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 6198	. . . . . 6 |- ( x = y -> ( 2 x. x ) = ( 2 x. y ) )
50	45, 5, 46, 45, 48, 49	cnmpt21 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 ) )
51	45, 5, 50, 1	cnmpt21f 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 )
53	18, 16, 52	mp2an 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 ) ) )
55	40, 53, 54	mp2an 672	. . . . . 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 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 ) ) ) )
58	12	iihalf2cn 20622	. . . . . . 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 6205	. . . . . 6 |- ( x = y -> ( ( 2 x. x ) - 1 ) = ( ( 2 x. y ) - 1 ) )
61	56, 5, 57, 56, 59, 60	cnmpt21 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 ) )
62	56, 5, 61, 2	cnmpt21f 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 ) )
63	10, 11, 12, 13, 15, 17, 25, 5, 39, 51, 62	cnmpt2pc 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 ) )
66	65	fveq2d 5793	. . . . 5 |- ( y = x -> ( F ` ( 2 x. y ) ) = ( F ` ( 2 x. x ) ) )
67	65	oveq1d 6205	. . . . . 6 |- ( y = x -> ( ( 2 x. y ) - 1 ) = ( ( 2 x. x ) - 1 ) )
68	67	fveq2d 5793	. . . . 5 |- ( y = x -> ( G ` ( ( 2 x. y ) - 1 ) ) = ( G ` ( ( 2 x. x ) - 1 ) ) )
69	64, 66, 68	ifbieq12d 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 ) ) ) )
70	69	adantr 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 ) ) ) )
71	5, 6, 9, 5, 5, 63, 70	cnmpt12 19356	. 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 2539	1 |- ( ph -> ( F ( *p ` J ) G ) e. ( II Cn J ) )

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