Theorem icoopnst 20636

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 20470	. . . . 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 10273	. . . . . . . . . . . . . . . 16 |- ( A e. RR -> ( A - 1 ) < A )
5	4	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( A e. RR /\ v e. RR ) -> ( A - 1 ) < A )
6		peano2rem 9779	. . . . . . . . . . . . . . . . 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 9570	. . . . . . . . . . . . . . . . 17 |- ( ( ( A - 1 ) e. RR /\ A e. RR /\ v e. RR ) -> ( ( ( A - 1 ) < A /\ A <_ v ) -> ( A - 1 ) < v ) )
9	8	3expb 1189	. . . . . . . . . . . . . . . 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 9533	. . . . . . . . . . . . 13 |- ( A e. RR -> A e. RR* )
22		elioc2 11462	. . . . . . . . . . . . 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 9570	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( v e. RR /\ C e. RR /\ B e. RR ) -> ( ( v < C /\ C <_ B ) -> v < B ) )
26		ltle 9567	. . . . . . . . . . . . . . . . . . . . . . 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 1188	. . . . . . . . . . . . . . . . . . . 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 9533	. . . . . . . . 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 11463	. . . . . . . 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 3640	. . . . . . . 8 |- ( v e. ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) <-> ( v e. ( ( A - 1 ) (,) C ) /\ v e. ( A [,] B ) ) )
54		rexr 9533	. . . . . . . . . . . 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 11445	. . . . . . . . . 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 11464	. . . . . . . . . 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 2448	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A [,) C ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) )
65		ineq1 3646	. . . . . . 7 |- ( v = ( ( A - 1 ) (,) C ) -> ( v i^i ( A [,] B ) ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) )
66	65	eqeq2d 2465	. . . . . 6 |- ( v = ( ( A - 1 ) (,) C ) -> ( ( A [,) C ) = ( v i^i ( A [,] B ) ) <-> ( A [,) C ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) ) )
67	66	rspcev 3172	. . . . 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 20465	. . . . 5 |- ( topGen ` ran (,) ) e. Top
70		ovex 6218	. . . . 5 |- ( A [,] B ) e. _V
71		elrest 14477	. . . . 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 11481	. . . . 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 2451	. . . . 5 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
77		icoopnst.1	. . . . 5 |- J = ( MetOpen ` ( ( abs o. - ) |` ( ( A [,] B ) X. ( A [,] B ) ) ) )
78	76, 77	resubmet 20504	. . . 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 2542	. 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 1370   e. wcel 1758   E.wrex 2796   _Vcvv 3071   i^i cin 3428   C_ wss 3429   class class class wbr 4393   X. cxp 4939   ran crn 4942   |` cres 4943   o. ccom 4945   ` cfv 5519  (class class class)co 6193   RRcr 9385   1c1 9387   RR*cxr 9521   < clt 9522   <_ cle 9523   - cmin 9699   (,)cioo 11404   (,]cioc 11405   [,)cico 11406   [,]cicc 11407   abscabs 12834   ↾_t crest 14470   topGenctg 14487   MetOpencmopn 17924   Topctop 18623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-ioo 11408  df-ioc 11409  df-ico 11410  df-icc 11411  df-seq 11917  df-exp 11976  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-rest 14472  df-topgen 14493  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-top 18628  df-bases 18630  df-topon 18631

This theorem is referenced by: (None)