Theorem pcocn 21247

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 21241	. 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 21113	. . . 4 |- II e. (TopOn ` ( 0 [,] 1 ) )
5	4	a1i 11	. . 3 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
6	5	cnmptid 19892	. . 3 |- ( ph -> ( x e. ( 0 [,] 1 ) |-> x ) e. ( II Cn II ) )
7		0elunit 11629	. . . . 5 |- 0 e. ( 0 [,] 1 )
8	7	a1i 11	. . . 4 |- ( ph -> 0 e. ( 0 [,] 1 ) )
9	5, 5, 8	cnmptc 19893	. . 3 |- ( ph -> ( x e. ( 0 [,] 1 ) |-> 0 ) e. ( II Cn II ) )
10		eqid 2462	. . . 4 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
11		eqid 2462	. . . 4 |- ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) )
12		eqid 2462	. . . 4 |- ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) )
13		dfii2 21116	. . . 4 |- II = ( ( topGen ` ran (,) ) ↾t ( 0 [,] 1 ) )
14		0re 9587	. . . . 5 |- 0 e. RR
15	14	a1i 11	. . . 4 |- ( ph -> 0 e. RR )
16		1re 9586	. . . . 5 |- 1 e. RR
17	16	a1i 11	. . . 4 |- ( ph -> 1 e. RR )
18		halfre 10745	. . . . . 6 |- ( 1 / 2 ) e. RR
19		halfgt0 10747	. . . . . . 7 |- 0 < ( 1 / 2 )
20	14, 18, 19	ltleii 9698	. . . . . 6 |- 0 <_ ( 1 / 2 )
21		halflt1 10748	. . . . . . 7 |- ( 1 / 2 ) < 1
22	18, 16, 21	ltleii 9698	. . . . . 6 |- ( 1 / 2 ) <_ 1
23	14, 16	elicc2i 11581	. . . . . 6 |- ( ( 1 / 2 ) e. ( 0 [,] 1 ) <-> ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ 1 ) )
24	18, 20, 22, 23	mpbir3an 1173	. . . . 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 6293	. . . . . . 7 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( 2 x. y ) = ( 2 x. ( 1 / 2 ) ) )
30		2cn 10597	. . . . . . . 8 |- 2 e. CC
31		2ne0 10619	. . . . . . . 8 |- 2 =/= 0
32	30, 31	recidi 10266	. . . . . . 7 |- ( 2 x. ( 1 / 2 ) ) = 1
33	29, 32	syl6eq 2519	. . . . . 6 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( 2 x. y ) = 1 )
34	33	fveq2d 5863	. . . . 5 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( F ` ( 2 x. y ) ) = ( F ` 1 ) )
35	33	oveq1d 6292	. . . . . . 7 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( ( 2 x. y ) - 1 ) = ( 1 - 1 ) )
36		1m1e0 10595	. . . . . . 7 |- ( 1 - 1 ) = 0
37	35, 36	syl6eq 2519	. . . . . 6 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( ( 2 x. y ) - 1 ) = 0 )
38	37	fveq2d 5863	. . . . 5 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( G ` ( ( 2 x. y ) - 1 ) ) = ( G ` 0 ) )
39	27, 34, 38	3eqtr4d 2513	. . . 4 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( F ` ( 2 x. y ) ) = ( G ` ( ( 2 x. y ) - 1 ) ) )
40		retopon 21000	. . . . . . 7 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
41		iccssre 11597	. . . . . . . 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 19423	. . . . . . 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 19899	. . . . . 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 21162	. . . . . . 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 6285	. . . . . 6 |- ( x = y -> ( 2 x. x ) = ( 2 x. y ) )
50	45, 5, 46, 45, 48, 49	cnmpt21 19902	. . . . 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 19903	. . . 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 11597	. . . . . . . 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 19423	. . . . . . 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 19899	. . . . . 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 21164	. . . . . . 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 6292	. . . . . 6 |- ( x = y -> ( ( 2 x. x ) - 1 ) = ( ( 2 x. y ) - 1 ) )
61	56, 5, 57, 56, 59, 60	cnmpt21 19902	. . . . 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 19903	. . . 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 21158	. . 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 4445	. . . . 5 |- ( y = x -> ( y <_ ( 1 / 2 ) <-> x <_ ( 1 / 2 ) ) )
65		oveq2 6285	. . . . . 6 |- ( y = x -> ( 2 x. y ) = ( 2 x. x ) )
66	65	fveq2d 5863	. . . . 5 |- ( y = x -> ( F ` ( 2 x. y ) ) = ( F ` ( 2 x. x ) ) )
67	65	oveq1d 6292	. . . . . 6 |- ( y = x -> ( ( 2 x. y ) - 1 ) = ( ( 2 x. x ) - 1 ) )
68	67	fveq2d 5863	. . . . 5 |- ( y = x -> ( G ` ( ( 2 x. y ) - 1 ) ) = ( G ` ( ( 2 x. x ) - 1 ) ) )
69	64, 66, 68	ifbieq12d 3961	. . . 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 19898	. 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 2550	1 |- ( ph -> ( F ( *p ` J ) G ) e. ( II Cn J ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1374   e. wcel 1762   C_ wss 3471   ifcif 3934   class class class wbr 4442   |-> cmpt 4500   ran crn 4995   ` cfv 5581  (class class class)co 6277   RRcr 9482   0cc0 9483   1c1 9484   x. cmul 9488   <_ cle 9620   - cmin 9796   / cdiv 10197   2c2 10576   (,)cioo 11520   [,]cicc 11523   ↾_t crest 14667   topGenctg 14684  TopOnctopon 19157   Cn ccn 19486   IIcii 21109   *pcpco 21230

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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-mulf 9563

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-fi 7862  df-sup 7892  df-oi 7926  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ioo 11524  df-icc 11527  df-fz 11664  df-fzo 11784  df-seq 12066  df-exp 12125  df-hash 12363  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-hom 14570  df-cco 14571  df-rest 14669  df-topn 14670  df-0g 14688  df-gsum 14689  df-topgen 14690  df-pt 14691  df-prds 14694  df-xrs 14748  df-qtop 14753  df-imas 14754  df-xps 14756  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-submnd 15773  df-mulg 15856  df-cntz 16145  df-cmn 16591  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-cnfld 18187  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cld 19281  df-cn 19489  df-cnp 19490  df-tx 19793  df-hmeo 19986  df-xms 20553  df-ms 20554  df-tms 20555  df-ii 21111  df-pco 21235

This theorem is referenced by:  copco  21248  pcohtpylem  21249  pcohtpy  21250  pcoass  21254  pcorevlem  21256  om1addcl  21263  pi1xfrf  21283  pi1xfr  21285  pi1xfrcnvlem  21286  pi1coghm  21291  conpcon  28308  sconpht2  28311  cvmlift3lem6  28397