Theorem icoopnst 20486

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 20320	. . . . 5 |- ( ( A - 1 ) (,) C ) e. ( topGen ` ran (,) )
2		simp1 988	. . . . . . . . . . 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 10161	. . . . . . . . . . . . . . . 16 |- ( A e. RR -> ( A - 1 ) < A )
5	4	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( A e. RR /\ v e. RR ) -> ( A - 1 ) < A )
6		peano2rem 9667	. . . . . . . . . . . . . . . . 17 |- ( A e. RR -> ( A - 1 ) e. RR )
7	6	adantr 465	. . . . . . . . . . . . . . . 16 |- ( ( A e. RR /\ v e. RR ) -> ( A - 1 ) e. RR )
8		ltletr 9458	. . . . . . . . . . . . . . . . 17 |- ( ( ( A - 1 ) e. RR /\ A e. RR /\ v e. RR ) -> ( ( ( A - 1 ) < A /\ A <_ v ) -> ( A - 1 ) < v ) )
9	8	3expb 1188	. . . . . . . . . . . . . . . 16 |- ( ( ( A - 1 ) e. RR /\ ( A e. RR /\ v e. RR ) ) -> ( ( ( A - 1 ) < A /\ A <_ v ) -> ( A - 1 ) < v ) )
10	7, 9	mpancom 669	. . . . . . . . . . . . . . 15 |- ( ( A e. RR /\ v e. RR ) -> ( ( ( A - 1 ) < A /\ A <_ v ) -> ( A - 1 ) < v ) )
11	5, 10	mpand 675	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ v e. RR ) -> ( A <_ v -> ( A - 1 ) < v ) )
12	11	impr 619	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ ( v e. RR /\ A <_ v ) ) -> ( A - 1 ) < v )
13	12	3adantr3 1149	. . . . . . . . . . . 12 |- ( ( A e. RR /\ ( v e. RR /\ A <_ v /\ v < C ) ) -> ( A - 1 ) < v )
14	13	ex 434	. . . . . . . . . . 11 |- ( A e. RR -> ( ( v e. RR /\ A <_ v /\ v < C ) -> ( A - 1 ) < v ) )
15	14	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> ( A - 1 ) < v ) )
16		simp3 990	. . . . . . . . . . 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 1169	. . . . . . . . 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 989	. . . . . . . . . . 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 9421	. . . . . . . . . . . . 13 |- ( A e. RR -> A e. RR* )
22		elioc2 11350	. . . . . . . . . . . . 13 |- ( ( A e. RR* /\ B e. RR ) -> ( C e. ( A (,] B ) <-> ( C e. RR /\ A < C /\ C <_ B ) ) )
23	21, 22	sylan 471	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR ) -> ( C e. ( A (,] B ) <-> ( C e. RR /\ A < C /\ C <_ B ) ) )
24	23	biimpa 484	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( C e. RR /\ A < C /\ C <_ B ) )
25		ltletr 9458	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( v e. RR /\ C e. RR /\ B e. RR ) -> ( ( v < C /\ C <_ B ) -> v < B ) )
26		ltle 9455	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( v e. RR /\ B e. RR ) -> ( v < B -> v <_ B ) )
27	26	3adant2 1007	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( v e. RR /\ C e. RR /\ B e. RR ) -> ( v < B -> v <_ B ) )
28	25, 27	syld 44	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( v e. RR /\ C e. RR /\ B e. RR ) -> ( ( v < C /\ C <_ B ) -> v <_ B ) )
29	28	3expa 1187	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( v e. RR /\ C e. RR ) /\ B e. RR ) -> ( ( v < C /\ C <_ B ) -> v <_ B ) )
30	29	an31s 804	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( B e. RR /\ C e. RR ) /\ v e. RR ) -> ( ( v < C /\ C <_ B ) -> v <_ B ) )
31	30	imp 429	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( B e. RR /\ C e. RR ) /\ v e. RR ) /\ ( v < C /\ C <_ B ) ) -> v <_ B )
32	31	ancom2s 800	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( B e. RR /\ C e. RR ) /\ v e. RR ) /\ ( C <_ B /\ v < C ) ) -> v <_ B )
33	32	an4s 822	. . . . . . . . . . . . . . . 16 |- ( ( ( ( B e. RR /\ C e. RR ) /\ C <_ B ) /\ ( v e. RR /\ v < C ) ) -> v <_ B )
34	33	3adantr2 1148	. . . . . . . . . . . . . . 15 |- ( ( ( ( B e. RR /\ C e. RR ) /\ C <_ B ) /\ ( v e. RR /\ A <_ v /\ v < C ) ) -> v <_ B )
35	34	ex 434	. . . . . . . . . . . . . 14 |- ( ( ( B e. RR /\ C e. RR ) /\ C <_ B ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> v <_ B ) )
36	35	anasss 647	. . . . . . . . . . . . 13 |- ( ( B e. RR /\ ( C e. RR /\ C <_ B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> v <_ B ) )
37	36	3adantr2 1148	. . . . . . . . . . . 12 |- ( ( B e. RR /\ ( C e. RR /\ A < C /\ C <_ B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> v <_ B ) )
38	37	adantll 713	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A < C /\ C <_ B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> v <_ B ) )
39	24, 38	syldan 470	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> v <_ B ) )
40	3, 20, 39	3jcad 1169	. . . . . . . . 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 ) ) )
41	18, 40	jcad 533	. . . . . . . 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 ) ) ) )
42		simpl1 991	. . . . . . . . 9 |- ( ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) -> v e. RR )
43		simpr2 995	. . . . . . . . 9 |- ( ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) -> A <_ v )
44		simpl3 993	. . . . . . . . 9 |- ( ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) -> v < C )
45	42, 43, 44	3jca 1168	. . . . . . . 8 |- ( ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) -> ( v e. RR /\ A <_ v /\ v < C ) )
46	41, 45	impbid1 203	. . . . . . 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 ) ) ) )
47		simpll 753	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> A e. RR )
48	24	simp1d 1000	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> C e. RR )
49		rexr 9421	. . . . . . . . 9 |- ( C e. RR -> C e. RR* )
50	48, 49	syl 16	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> C e. RR* )
51		elico2 11351	. . . . . . . 8 |- ( ( A e. RR /\ C e. RR* ) -> ( v e. ( A [,) C ) <-> ( v e. RR /\ A <_ v /\ v < C ) ) )
52	47, 50, 51	syl2anc 661	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( v e. ( A [,) C ) <-> ( v e. RR /\ A <_ v /\ v < C ) ) )
53		elin 3534	. . . . . . . 8 |- ( v e. ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) <-> ( v e. ( ( A - 1 ) (,) C ) /\ v e. ( A [,] B ) ) )
54		rexr 9421	. . . . . . . . . . . 12 |- ( ( A - 1 ) e. RR -> ( A - 1 ) e. RR* )
55	6, 54	syl 16	. . . . . . . . . . 11 |- ( A e. RR -> ( A - 1 ) e. RR* )
56	55	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A - 1 ) e. RR* )
57		elioo2 11333	. . . . . . . . . 10 |- ( ( ( A - 1 ) e. RR* /\ C e. RR* ) -> ( v e. ( ( A - 1 ) (,) C ) <-> ( v e. RR /\ ( A - 1 ) < v /\ v < C ) ) )
58	56, 50, 57	syl2anc 661	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( v e. ( ( A - 1 ) (,) C ) <-> ( v e. RR /\ ( A - 1 ) < v /\ v < C ) ) )
59		elicc2 11352	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( v e. ( A [,] B ) <-> ( v e. RR /\ A <_ v /\ v <_ B ) ) )
60	59	adantr 465	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( v e. ( A [,] B ) <-> ( v e. RR /\ A <_ v /\ v <_ B ) ) )
61	58, 60	anbi12d 710	. . . . . . . 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 ) ) ) )
62	53, 61	syl5bb 257	. . . . . . 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 ) ) ) )
63	46, 52, 62	3bitr4d 285	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( v e. ( A [,) C ) <-> v e. ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) ) )
64	63	eqrdv 2436	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A [,) C ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) )
65		ineq1 3540	. . . . . . 7 |- ( v = ( ( A - 1 ) (,) C ) -> ( v i^i ( A [,] B ) ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) )
66	65	eqeq2d 2449	. . . . . 6 |- ( v = ( ( A - 1 ) (,) C ) -> ( ( A [,) C ) = ( v i^i ( A [,] B ) ) <-> ( A [,) C ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) ) )
67	66	rspcev 3068	. . . . 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 ) ) )
68	1, 64, 67	sylancr 663	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> E. v e. ( topGen ` ran (,) ) ( A [,) C ) = ( v i^i ( A [,] B ) ) )
69		retop 20315	. . . . 5 |- ( topGen ` ran (,) ) e. Top
70		ovex 6111	. . . . 5 |- ( A [,] B ) e. _V
71		elrest 14358	. . . . 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 ) ) ) )
72	69, 70, 71	mp2an 672	. . . 4 |- ( ( A [,) C ) e. ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) <-> E. v e. ( topGen ` ran (,) ) ( A [,) C ) = ( v i^i ( A [,] B ) ) )
73	68, 72	sylibr 212	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A [,) C ) e. ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
74		iccssre 11369	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
75	74	adantr 465	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A [,] B ) C_ RR )
76		eqid 2438	. . . . 5 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
77		icoopnst.1	. . . . 5 |- J = ( MetOpen ` ( ( abs o. - ) |` ( ( A [,] B ) X. ( A [,] B ) ) ) )
78	76, 77	resubmet 20354	. . . 4 |- ( ( A [,] B ) C_ RR -> J = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
79	75, 78	syl 16	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> J = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
80	73, 79	eleqtrrd 2515	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A [,) C ) e. J )
81	80	ex 434	1 |- ( ( A e. RR /\ B e. RR ) -> ( C e. ( A (,] B ) -> ( A [,) C ) e. J ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   E.wrex 2711   _Vcvv 2967   i^i cin 3322   C_ wss 3323   class class class wbr 4287   X. cxp 4833   ran crn 4836   |` cres 4837   o. ccom 4839   ` cfv 5413  (class class class)co 6086   RRcr 9273   1c1 9275   RR*cxr 9409   < clt 9410   <_ cle 9411   - cmin 9587   (,)cioo 11292   (,]cioc 11293   [,)cico 11294   [,]cicc 11295   abscabs 12715   ↾_t crest 14351   topGenctg 14368   MetOpencmopn 17781   Topctop 18473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-rest 14353  df-topgen 14374  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-top 18478  df-bases 18480  df-topon 18481

This theorem is referenced by: (None)