Theorem pcocn 18995

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 18989	. 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 18862	. . . 4 |- II e. (TopOn ` ( 0 [,] 1 ) )
5	4	a1i 11	. . 3 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
6	5	cnmptid 17646	. . 3 |- ( ph -> ( x e. ( 0 [,] 1 ) |-> x ) e. ( II Cn II ) )
7		0elunit 10971	. . . . 5 |- 0 e. ( 0 [,] 1 )
8	7	a1i 11	. . . 4 |- ( ph -> 0 e. ( 0 [,] 1 ) )
9	5, 5, 8	cnmptc 17647	. . 3 |- ( ph -> ( x e. ( 0 [,] 1 ) |-> 0 ) e. ( II Cn II ) )
10		eqid 2404	. . . 4 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
11		eqid 2404	. . . 4 |- ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) )
12		eqid 2404	. . . 4 |- ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) )
13		dfii2 18865	. . . 4 |- II = ( ( topGen ` ran (,) ) ↾t ( 0 [,] 1 ) )
14		0re 9047	. . . . 5 |- 0 e. RR
15	14	a1i 11	. . . 4 |- ( ph -> 0 e. RR )
16		1re 9046	. . . . 5 |- 1 e. RR
17	16	a1i 11	. . . 4 |- ( ph -> 1 e. RR )
18	16	rehalfcli 10172	. . . . . 6 |- ( 1 / 2 ) e. RR
19		halfgt0 10144	. . . . . . 7 |- 0 < ( 1 / 2 )
20	14, 18, 19	ltleii 9152	. . . . . 6 |- 0 <_ ( 1 / 2 )
21		halflt1 10145	. . . . . . 7 |- ( 1 / 2 ) < 1
22	18, 16, 21	ltleii 9152	. . . . . 6 |- ( 1 / 2 ) <_ 1
23	14, 16	elicc2i 10932	. . . . . 6 |- ( ( 1 / 2 ) e. ( 0 [,] 1 ) <-> ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ 1 ) )
24	18, 20, 22, 23	mpbir3an 1136	. . . . 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 452	. . . . 5 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( F ` 1 ) = ( G ` 0 ) )
28		simprl 733	. . . . . . . 8 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> y = ( 1 / 2 ) )
29	28	oveq2d 6056	. . . . . . 7 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( 2 x. y ) = ( 2 x. ( 1 / 2 ) ) )
30		2cn 10026	. . . . . . . 8 |- 2 e. CC
31		2ne0 10039	. . . . . . . 8 |- 2 =/= 0
32	30, 31	recidi 9701	. . . . . . 7 |- ( 2 x. ( 1 / 2 ) ) = 1
33	29, 32	syl6eq 2452	. . . . . 6 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( 2 x. y ) = 1 )
34	33	fveq2d 5691	. . . . 5 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( F ` ( 2 x. y ) ) = ( F ` 1 ) )
35	33	oveq1d 6055	. . . . . . 7 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( ( 2 x. y ) - 1 ) = ( 1 - 1 ) )
36		1m1e0 10024	. . . . . . 7 |- ( 1 - 1 ) = 0
37	35, 36	syl6eq 2452	. . . . . 6 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( ( 2 x. y ) - 1 ) = 0 )
38	37	fveq2d 5691	. . . . 5 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( G ` ( ( 2 x. y ) - 1 ) ) = ( G ` 0 ) )
39	27, 34, 38	3eqtr4d 2446	. . . 4 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( F ` ( 2 x. y ) ) = ( G ` ( ( 2 x. y ) - 1 ) ) )
40		retopon 18750	. . . . . . 7 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
41		iccssre 10948	. . . . . . . 8 |- ( ( 0 e. RR /\ ( 1 / 2 ) e. RR ) -> ( 0 [,] ( 1 / 2 ) ) C_ RR )
42	14, 18, 41	mp2an 654	. . . . . . 7 |- ( 0 [,] ( 1 / 2 ) ) C_ RR
43		resttopon 17179	. . . . . . 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 654	. . . . . 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 17653	. . . . . 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 18910	. . . . . . 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 6048	. . . . . 6 |- ( x = y -> ( 2 x. x ) = ( 2 x. y ) )
50	45, 5, 46, 45, 48, 49	cnmpt21 17656	. . . . 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 17657	. . . 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 10948	. . . . . . . 8 |- ( ( ( 1 / 2 ) e. RR /\ 1 e. RR ) -> ( ( 1 / 2 ) [,] 1 ) C_ RR )
53	18, 16, 52	mp2an 654	. . . . . . 7 |- ( ( 1 / 2 ) [,] 1 ) C_ RR
54		resttopon 17179	. . . . . . 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 654	. . . . . 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 17653	. . . . . 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 18912	. . . . . . 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 6055	. . . . . 6 |- ( x = y -> ( ( 2 x. x ) - 1 ) = ( ( 2 x. y ) - 1 ) )
61	56, 5, 57, 56, 59, 60	cnmpt21 17656	. . . . 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 17657	. . . 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 18906	. . 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 4175	. . . . 5 |- ( y = x -> ( y <_ ( 1 / 2 ) <-> x <_ ( 1 / 2 ) ) )
65		oveq2 6048	. . . . . 6 |- ( y = x -> ( 2 x. y ) = ( 2 x. x ) )
66	65	fveq2d 5691	. . . . 5 |- ( y = x -> ( F ` ( 2 x. y ) ) = ( F ` ( 2 x. x ) ) )
67	65	oveq1d 6055	. . . . . 6 |- ( y = x -> ( ( 2 x. y ) - 1 ) = ( ( 2 x. x ) - 1 ) )
68	67	fveq2d 5691	. . . . 5 |- ( y = x -> ( G ` ( ( 2 x. y ) - 1 ) ) = ( G ` ( ( 2 x. x ) - 1 ) ) )
69	64, 66, 68	ifbieq12d 3721	. . . 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 452	. . 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 17652	. 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 2478	1 |- ( ph -> ( F ( *p ` J ) G ) e. ( II Cn J ) )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   C_ wss 3280   ifcif 3699   class class class wbr 4172   e. cmpt 4226   ran crn 4838   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946   1c1 8947   x. cmul 8951   <_ cle 9077   - cmin 9247   / cdiv 9633   2c2 10005   (,)cioo 10872   [,]cicc 10875   ↾_t crest 13603   topGenctg 13620  TopOnctopon 16914   Cn ccn 17242   IIcii 18858   *pcpco 18978

This theorem is referenced by:  copco  18996  pcohtpylem  18997  pcohtpy  18998  pcoass  19002  pcorevlem  19004  om1addcl  19011  pi1xfrf  19031  pi1xfr  19033  pi1xfrcnvlem  19034  pi1coghm  19039  conpcon  24875  sconpht2  24878  cvmlift3lem6  24964

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-cn 17245  df-cnp 17246  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305  df-ii 18860  df-pco 18983