Theorem icoopnst 21174

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 21008	. . . . 5 |- ( ( A - 1 ) (,) C ) e. ( topGen ` ran (,) )
2		simp1 996	. . . . . . . . . . 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 10378	. . . . . . . . . . . . . . . 16 |- ( A e. RR -> ( A - 1 ) < A )
5	4	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( A e. RR /\ v e. RR ) -> ( A - 1 ) < A )
6		peano2rem 9882	. . . . . . . . . . . . . . . . 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 9672	. . . . . . . . . . . . . . . . 17 |- ( ( ( A - 1 ) e. RR /\ A e. RR /\ v e. RR ) -> ( ( ( A - 1 ) < A /\ A <_ v ) -> ( A - 1 ) < v ) )
9	8	3expb 1197	. . . . . . . . . . . . . . . 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 1157	. . . . . . . . . . . 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 998	. . . . . . . . . . 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 1177	. . . . . . . . 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 997	. . . . . . . . . . 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 9635	. . . . . . . . . . . . 13 |- ( A e. RR -> A e. RR* )
22		elioc2 11583	. . . . . . . . . . . . 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 9672	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( v e. RR /\ C e. RR /\ B e. RR ) -> ( ( v < C /\ C <_ B ) -> v < B ) )
26		ltle 9669	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( v e. RR /\ B e. RR ) -> ( v < B -> v <_ B ) )
27	26	3adant2 1015	. . . . . . . . . . . . . . . . . . . . . 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 1196	. . . . . . . . . . . . . . . . . . . 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 824	. . . . . . . . . . . . . . . 16 |- ( ( ( ( B e. RR /\ C e. RR ) /\ C <_ B ) /\ ( v e. RR /\ v < C ) ) -> v <_ B )
34	33	3adantr2 1156	. . . . . . . . . . . . . . 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 1156	. . . . . . . . . . . 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 1177	. . . . . . . . 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 999	. . . . . . . . 9 |- ( ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) -> v e. RR )
43		simpr2 1003	. . . . . . . . 9 |- ( ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) -> A <_ v )
44		simpl3 1001	. . . . . . . . 9 |- ( ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) -> v < C )
45	42, 43, 44	3jca 1176	. . . . . . . 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 1008	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> C e. RR )
49		rexr 9635	. . . . . . . . 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 11584	. . . . . . . 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 3687	. . . . . . . 8 |- ( v e. ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) <-> ( v e. ( ( A - 1 ) (,) C ) /\ v e. ( A [,] B ) ) )
54		rexr 9635	. . . . . . . . . . . 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 11566	. . . . . . . . . 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 11585	. . . . . . . . . 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 2464	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A [,) C ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) )
65		ineq1 3693	. . . . . . 7 |- ( v = ( ( A - 1 ) (,) C ) -> ( v i^i ( A [,] B ) ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) )
66	65	eqeq2d 2481	. . . . . 6 |- ( v = ( ( A - 1 ) (,) C ) -> ( ( A [,) C ) = ( v i^i ( A [,] B ) ) <-> ( A [,) C ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) ) )
67	66	rspcev 3214	. . . . 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 21003	. . . . 5 |- ( topGen ` ran (,) ) e. Top
70		ovex 6307	. . . . 5 |- ( A [,] B ) e. _V
71		elrest 14679	. . . . 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 11602	. . . . 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 2467	. . . . 5 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
77		icoopnst.1	. . . . 5 |- J = ( MetOpen ` ( ( abs o. - ) |` ( ( A [,] B ) X. ( A [,] B ) ) ) )
78	76, 77	resubmet 21042	. . . 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 2558	. 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 973   = wceq 1379   e. wcel 1767   E.wrex 2815   _Vcvv 3113   i^i cin 3475   C_ wss 3476   class class class wbr 4447   X. cxp 4997   ran crn 5000   |` cres 5001   o. ccom 5003   ` cfv 5586  (class class class)co 6282   RRcr 9487   1c1 9489   RR*cxr 9623   < clt 9624   <_ cle 9625   - cmin 9801   (,)cioo 11525   (,]cioc 11526   [,)cico 11527   [,]cicc 11528   abscabs 13026   ↾_t crest 14672   topGenctg 14689   MetOpencmopn 18179   Topctop 19161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-rest 14674  df-topgen 14695  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-top 19166  df-bases 19168  df-topon 19169

This theorem is referenced by: (None)