Theorem pcocn 22060

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 22054	. 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 21923	. . . 4 |- II e. (TopOn ` ( 0 [,] 1 ) )
5	4	a1i 11	. . 3 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
6	5	cnmptid 20688	. . 3 |- ( ph -> ( x e. ( 0 [,] 1 ) |-> x ) e. ( II Cn II ) )
7		0elunit 11757	. . . . 5 |- 0 e. ( 0 [,] 1 )
8	7	a1i 11	. . . 4 |- ( ph -> 0 e. ( 0 [,] 1 ) )
9	5, 5, 8	cnmptc 20689	. . 3 |- ( ph -> ( x e. ( 0 [,] 1 ) |-> 0 ) e. ( II Cn II ) )
10		eqid 2453	. . . 4 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
11		eqid 2453	. . . 4 |- ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) )
12		eqid 2453	. . . 4 |- ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) )
13		dfii2 21926	. . . 4 |- II = ( ( topGen ` ran (,) ) ↾t ( 0 [,] 1 ) )
14		0re 9648	. . . . 5 |- 0 e. RR
15	14	a1i 11	. . . 4 |- ( ph -> 0 e. RR )
16		1re 9647	. . . . 5 |- 1 e. RR
17	16	a1i 11	. . . 4 |- ( ph -> 1 e. RR )
18		halfre 10835	. . . . . 6 |- ( 1 / 2 ) e. RR
19		halfgt0 10837	. . . . . . 7 |- 0 < ( 1 / 2 )
20	14, 18, 19	ltleii 9762	. . . . . 6 |- 0 <_ ( 1 / 2 )
21		halflt1 10838	. . . . . . 7 |- ( 1 / 2 ) < 1
22	18, 16, 21	ltleii 9762	. . . . . 6 |- ( 1 / 2 ) <_ 1
23	14, 16	elicc2i 11707	. . . . . 6 |- ( ( 1 / 2 ) e. ( 0 [,] 1 ) <-> ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ 1 ) )
24	18, 20, 22, 23	mpbir3an 1191	. . . . 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 467	. . . . 5 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( F ` 1 ) = ( G ` 0 ) )
28		simprl 765	. . . . . . . 8 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> y = ( 1 / 2 ) )
29	28	oveq2d 6311	. . . . . . 7 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( 2 x. y ) = ( 2 x. ( 1 / 2 ) ) )
30		2cn 10687	. . . . . . . 8 |- 2 e. CC
31		2ne0 10709	. . . . . . . 8 |- 2 =/= 0
32	30, 31	recidi 10345	. . . . . . 7 |- ( 2 x. ( 1 / 2 ) ) = 1
33	29, 32	syl6eq 2503	. . . . . 6 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( 2 x. y ) = 1 )
34	33	fveq2d 5874	. . . . 5 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( F ` ( 2 x. y ) ) = ( F ` 1 ) )
35	33	oveq1d 6310	. . . . . . 7 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( ( 2 x. y ) - 1 ) = ( 1 - 1 ) )
36		1m1e0 10685	. . . . . . 7 |- ( 1 - 1 ) = 0
37	35, 36	syl6eq 2503	. . . . . 6 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( ( 2 x. y ) - 1 ) = 0 )
38	37	fveq2d 5874	. . . . 5 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( G ` ( ( 2 x. y ) - 1 ) ) = ( G ` 0 ) )
39	27, 34, 38	3eqtr4d 2497	. . . 4 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ z e. ( 0 [,] 1 ) ) ) -> ( F ` ( 2 x. y ) ) = ( G ` ( ( 2 x. y ) - 1 ) ) )
40		retopon 21796	. . . . . . 7 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
41		iccssre 11723	. . . . . . . 8 |- ( ( 0 e. RR /\ ( 1 / 2 ) e. RR ) -> ( 0 [,] ( 1 / 2 ) ) C_ RR )
42	14, 18, 41	mp2an 679	. . . . . . 7 |- ( 0 [,] ( 1 / 2 ) ) C_ RR
43		resttopon 20189	. . . . . . 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 679	. . . . . 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 20695	. . . . . 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 21972	. . . . . . 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 6303	. . . . . 6 |- ( x = y -> ( 2 x. x ) = ( 2 x. y ) )
50	45, 5, 46, 45, 48, 49	cnmpt21 20698	. . . . 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 20699	. . . 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 11723	. . . . . . . 8 |- ( ( ( 1 / 2 ) e. RR /\ 1 e. RR ) -> ( ( 1 / 2 ) [,] 1 ) C_ RR )
53	18, 16, 52	mp2an 679	. . . . . . 7 |- ( ( 1 / 2 ) [,] 1 ) C_ RR
54		resttopon 20189	. . . . . . 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 679	. . . . . 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 20695	. . . . . 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 21974	. . . . . . 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 6310	. . . . . 6 |- ( x = y -> ( ( 2 x. x ) - 1 ) = ( ( 2 x. y ) - 1 ) )
61	56, 5, 57, 56, 59, 60	cnmpt21 20698	. . . . 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 20699	. . . 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 21968	. . 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 4408	. . . . 5 |- ( y = x -> ( y <_ ( 1 / 2 ) <-> x <_ ( 1 / 2 ) ) )
65		oveq2 6303	. . . . . 6 |- ( y = x -> ( 2 x. y ) = ( 2 x. x ) )
66	65	fveq2d 5874	. . . . 5 |- ( y = x -> ( F ` ( 2 x. y ) ) = ( F ` ( 2 x. x ) ) )
67	65	oveq1d 6310	. . . . . 6 |- ( y = x -> ( ( 2 x. y ) - 1 ) = ( ( 2 x. x ) - 1 ) )
68	67	fveq2d 5874	. . . . 5 |- ( y = x -> ( G ` ( ( 2 x. y ) - 1 ) ) = ( G ` ( ( 2 x. x ) - 1 ) ) )
69	64, 66, 68	ifbieq12d 3910	. . . 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 467	. . 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 20694	. 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 2531	1 |- ( ph -> ( F ( *p ` J ) G ) e. ( II Cn J ) )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1446   e. wcel 1889   C_ wss 3406   ifcif 3883   class class class wbr 4405   |-> cmpt 4464   ran crn 4838   ` cfv 5585  (class class class)co 6295   RRcr 9543   0cc0 9544   1c1 9545   x. cmul 9549   <_ cle 9681   - cmin 9865   / cdiv 10276   2c2 10666   (,)cioo 11642   [,]cicc 11645   ↾_t crest 15331   topGenctg 15348  TopOnctopon 19930   Cn ccn 20252   IIcii 21919   *pcpco 22043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622  ax-mulf 9624

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  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-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-fi 7930  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-icc 11649  df-fz 11792  df-fzo 11923  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-starv 15217  df-sca 15218  df-vsca 15219  df-ip 15220  df-tset 15221  df-ple 15222  df-ds 15224  df-unif 15225  df-hom 15226  df-cco 15227  df-rest 15333  df-topn 15334  df-0g 15352  df-gsum 15353  df-topgen 15354  df-pt 15355  df-prds 15358  df-xrs 15412  df-qtop 15418  df-imas 15419  df-xps 15422  df-mre 15504  df-mrc 15505  df-acs 15507  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-submnd 16595  df-mulg 16688  df-cntz 16983  df-cmn 17444  df-psmet 18974  df-xmet 18975  df-met 18976  df-bl 18977  df-mopn 18978  df-cnfld 18983  df-top 19933  df-bases 19934  df-topon 19935  df-topsp 19936  df-cld 20046  df-cn 20255  df-cnp 20256  df-tx 20589  df-hmeo 20782  df-xms 21347  df-ms 21348  df-tms 21349  df-ii 21921  df-pco 22048

This theorem is referenced by:  copco  22061  pcohtpylem  22062  pcohtpy  22063  pcoass  22067  pcorevlem  22069  om1addcl  22076  pi1xfrf  22096  pi1xfr  22098  pi1xfrcnvlem  22099  pi1coghm  22104  conpcon  29970  sconpht2  29973  cvmlift3lem6  30059