Theorem icoopnst 22738

Description: A half-open interval starting at A is open in the closed interval from A to B. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)

Hypothesis

Ref	Expression
icoopnst.1	|- J = ( MetOpen ` ( ( abs o. - ) |` ( ( A [,] B ) X. ( A [,] B ) ) ) )

Assertion

Ref	Expression
icoopnst	|- ( ( A e. RR /\ B e. RR ) -> ( C e. ( A (,] B ) -> ( A [,) C ) e. J ) )

Proof of Theorem icoopnst

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iooretop 22569	. . . . 5 |- ( ( A - 1 ) (,) C ) e. ( topGen ` ran (,) )
2		simp1 1061	. . . . . . . . . . 11 |- ( ( v e. RR /\ A <_ v /\ v < C ) -> v e. RR )
3	2	a1i 11	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> v e. RR ) )
4		ltm1 10863	. . . . . . . . . . . . . . . 16 |- ( A e. RR -> ( A - 1 ) < A )
5	4	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( A e. RR /\ v e. RR ) -> ( A - 1 ) < A )
6		peano2rem 10348	. . . . . . . . . . . . . . . . 17 |- ( A e. RR -> ( A - 1 ) e. RR )
7	6	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( A e. RR /\ v e. RR ) -> ( A - 1 ) e. RR )
8		ltletr 10129	. . . . . . . . . . . . . . . . 17 |- ( ( ( A - 1 ) e. RR /\ A e. RR /\ v e. RR ) -> ( ( ( A - 1 ) < A /\ A <_ v ) -> ( A - 1 ) < v ) )
9	8	3expb 1266	. . . . . . . . . . . . . . . 16 |- ( ( ( A - 1 ) e. RR /\ ( A e. RR /\ v e. RR ) ) -> ( ( ( A - 1 ) < A /\ A <_ v ) -> ( A - 1 ) < v ) )
10	7, 9	mpancom 703	. . . . . . . . . . . . . . 15 |- ( ( A e. RR /\ v e. RR ) -> ( ( ( A - 1 ) < A /\ A <_ v ) -> ( A - 1 ) < v ) )
11	5, 10	mpand 711	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ v e. RR ) -> ( A <_ v -> ( A - 1 ) < v ) )
12	11	impr 649	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ ( v e. RR /\ A <_ v ) ) -> ( A - 1 ) < v )
13	12	3adantr3 1222	. . . . . . . . . . . 12 |- ( ( A e. RR /\ ( v e. RR /\ A <_ v /\ v < C ) ) -> ( A - 1 ) < v )
14	13	ex 450	. . . . . . . . . . 11 |- ( A e. RR -> ( ( v e. RR /\ A <_ v /\ v < C ) -> ( A - 1 ) < v ) )
15	14	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> ( A - 1 ) < v ) )
16		simp3 1063	. . . . . . . . . . 11 |- ( ( v e. RR /\ A <_ v /\ v < C ) -> v < C )
17	16	a1i 11	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> v < C ) )
18	3, 15, 17	3jcad 1243	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> ( v e. RR /\ ( A - 1 ) < v /\ v < C ) ) )
19		simp2 1062	. . . . . . . . . . 11 |- ( ( v e. RR /\ A <_ v /\ v < C ) -> A <_ v )
20	19	a1i 11	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> A <_ v ) )
21		rexr 10085	. . . . . . . . . . . . 13 |- ( A e. RR -> A e. RR* )
22		elioc2 12236	. . . . . . . . . . . . 13 |- ( ( A e. RR* /\ B e. RR ) -> ( C e. ( A (,] B ) <-> ( C e. RR /\ A < C /\ C <_ B ) ) )
23	21, 22	sylan 488	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( C e. ( A (,] B ) <-> ( C e. RR /\ A < C /\ C <_ B ) ) )
24	23	biimpa 501	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( C e. RR /\ A < C /\ C <_ B ) )
25		ltleletr 10130	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( v e. RR /\ C e. RR /\ B e. RR ) -> ( ( v < C /\ C <_ B ) -> v <_ B ) )
26	25	3expa 1265	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( v e. RR /\ C e. RR ) /\ B e. RR ) -> ( ( v < C /\ C <_ B ) -> v <_ B ) )
27	26	an31s 848	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( B e. RR /\ C e. RR ) /\ v e. RR ) -> ( ( v < C /\ C <_ B ) -> v <_ B ) )
28	27	imp 445	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( B e. RR /\ C e. RR ) /\ v e. RR ) /\ ( v < C /\ C <_ B ) ) -> v <_ B )
29	28	ancom2s 844	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( B e. RR /\ C e. RR ) /\ v e. RR ) /\ ( C <_ B /\ v < C ) ) -> v <_ B )
30	29	an4s 869	. . . . . . . . . . . . . . . 16 |- ( ( ( ( B e. RR /\ C e. RR ) /\ C <_ B ) /\ ( v e. RR /\ v < C ) ) -> v <_ B )
31	30	3adantr2 1221	. . . . . . . . . . . . . . 15 |- ( ( ( ( B e. RR /\ C e. RR ) /\ C <_ B ) /\ ( v e. RR /\ A <_ v /\ v < C ) ) -> v <_ B )
32	31	ex 450	. . . . . . . . . . . . . 14 |- ( ( ( B e. RR /\ C e. RR ) /\ C <_ B ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> v <_ B ) )
33	32	anasss 679	. . . . . . . . . . . . 13 |- ( ( B e. RR /\ ( C e. RR /\ C <_ B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> v <_ B ) )
34	33	3adantr2 1221	. . . . . . . . . . . 12 |- ( ( B e. RR /\ ( C e. RR /\ A < C /\ C <_ B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> v <_ B ) )
35	34	adantll 750	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A < C /\ C <_ B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> v <_ B ) )
36	24, 35	syldan 487	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> v <_ B ) )
37	3, 20, 36	3jcad 1243	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> ( v e. RR /\ A <_ v /\ v <_ B ) ) )
38	18, 37	jcad 555	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) ) )
39		simpl1 1064	. . . . . . . . 9 |- ( ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) -> v e. RR )
40		simpr2 1068	. . . . . . . . 9 |- ( ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) -> A <_ v )
41		simpl3 1066	. . . . . . . . 9 |- ( ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) -> v < C )
42	39, 40, 41	3jca 1242	. . . . . . . 8 |- ( ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) -> ( v e. RR /\ A <_ v /\ v < C ) )
43	38, 42	impbid1 215	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) <-> ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) ) )
44		simpll 790	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> A e. RR )
45	24	simp1d 1073	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> C e. RR )
46	45	rexrd 10089	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> C e. RR* )
47		elico2 12237	. . . . . . . 8 |- ( ( A e. RR /\ C e. RR* ) -> ( v e. ( A [,) C ) <-> ( v e. RR /\ A <_ v /\ v < C ) ) )
48	44, 46, 47	syl2anc 693	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( v e. ( A [,) C ) <-> ( v e. RR /\ A <_ v /\ v < C ) ) )
49		elin 3796	. . . . . . . 8 |- ( v e. ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) <-> ( v e. ( ( A - 1 ) (,) C ) /\ v e. ( A [,] B ) ) )
50	6	rexrd 10089	. . . . . . . . . . 11 |- ( A e. RR -> ( A - 1 ) e. RR* )
51	50	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A - 1 ) e. RR* )
52		elioo2 12216	. . . . . . . . . 10 |- ( ( ( A - 1 ) e. RR* /\ C e. RR* ) -> ( v e. ( ( A - 1 ) (,) C ) <-> ( v e. RR /\ ( A - 1 ) < v /\ v < C ) ) )
53	51, 46, 52	syl2anc 693	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( v e. ( ( A - 1 ) (,) C ) <-> ( v e. RR /\ ( A - 1 ) < v /\ v < C ) ) )
54		elicc2 12238	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( v e. ( A [,] B ) <-> ( v e. RR /\ A <_ v /\ v <_ B ) ) )
55	54	adantr 481	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( v e. ( A [,] B ) <-> ( v e. RR /\ A <_ v /\ v <_ B ) ) )
56	53, 55	anbi12d 747	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( ( v e. ( ( A - 1 ) (,) C ) /\ v e. ( A [,] B ) ) <-> ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) ) )
57	49, 56	syl5bb 272	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( v e. ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) <-> ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) ) )
58	43, 48, 57	3bitr4d 300	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( v e. ( A [,) C ) <-> v e. ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) ) )
59	58	eqrdv 2620	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A [,) C ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) )
60		ineq1 3807	. . . . . . 7 |- ( v = ( ( A - 1 ) (,) C ) -> ( v i^i ( A [,] B ) ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) )
61	60	eqeq2d 2632	. . . . . 6 |- ( v = ( ( A - 1 ) (,) C ) -> ( ( A [,) C ) = ( v i^i ( A [,] B ) ) <-> ( A [,) C ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) ) )
62	61	rspcev 3309	. . . . 5 |- ( ( ( ( A - 1 ) (,) C ) e. ( topGen ` ran (,) ) /\ ( A [,) C ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) ) -> E. v e. ( topGen ` ran (,) ) ( A [,) C ) = ( v i^i ( A [,] B ) ) )
63	1, 59, 62	sylancr 695	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> E. v e. ( topGen ` ran (,) ) ( A [,) C ) = ( v i^i ( A [,] B ) ) )
64		retop 22565	. . . . 5 |- ( topGen ` ran (,) ) e. Top
65		ovex 6678	. . . . 5 |- ( A [,] B ) e. _V
66		elrest 16088	. . . . 5 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( A [,] B ) e. _V ) -> ( ( A [,) C ) e. ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) <-> E. v e. ( topGen ` ran (,) ) ( A [,) C ) = ( v i^i ( A [,] B ) ) ) )
67	64, 65, 66	mp2an 708	. . . 4 |- ( ( A [,) C ) e. ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) <-> E. v e. ( topGen ` ran (,) ) ( A [,) C ) = ( v i^i ( A [,] B ) ) )
68	63, 67	sylibr 224	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A [,) C ) e. ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
69		iccssre 12255	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
70	69	adantr 481	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A [,] B ) C_ RR )
71		eqid 2622	. . . . 5 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
72		icoopnst.1	. . . . 5 |- J = ( MetOpen ` ( ( abs o. - ) |` ( ( A [,] B ) X. ( A [,] B ) ) ) )
73	71, 72	resubmet 22605	. . . 4 |- ( ( A [,] B ) C_ RR -> J = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
74	70, 73	syl 17	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> J = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
75	68, 74	eleqtrrd 2704	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A [,) C ) e. J )
76	75	ex 450	1 |- ( ( A e. RR /\ B e. RR ) -> ( C e. ( A (,] B ) -> ( A [,) C ) e. J ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   class class class wbr 4653   X. cxp 5112   ran crn 5115   |` cres 5116   o. ccom 5118   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   (,)cioo 12175   (,]cioc 12176   [,)cico 12177   [,]cicc 12178   abscabs 13974   ↾_t crest 16081   topGenctg 16098   MetOpencmopn 19736   Topctop 20698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750

This theorem is referenced by: (None)