Theorem icoopnst 21565

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 21399	. . . . 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 10403	. . . . . . . . . . . . . . . 16 |- ( A e. RR -> ( A - 1 ) < A )
5	4	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( A e. RR /\ v e. RR ) -> ( A - 1 ) < A )
6		peano2rem 9905	. . . . . . . . . . . . . . . . 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 9693	. . . . . . . . . . . . . . . . 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 9656	. . . . . . . . . . . . 13 |- ( A e. RR -> A e. RR* )
22		elioc2 11612	. . . . . . . . . . . . 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		ltleletr 9694	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( v e. RR /\ C e. RR /\ B e. RR ) -> ( ( v < C /\ C <_ B ) -> v <_ B ) )
26	25	3expa 1196	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( v e. RR /\ C e. RR ) /\ B e. RR ) -> ( ( v < C /\ C <_ B ) -> v <_ B ) )
27	26	an31s 806	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( B e. RR /\ C e. RR ) /\ v e. RR ) -> ( ( v < C /\ C <_ B ) -> v <_ B ) )
28	27	imp 429	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( B e. RR /\ C e. RR ) /\ v e. RR ) /\ ( v < C /\ C <_ B ) ) -> v <_ B )
29	28	ancom2s 802	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( B e. RR /\ C e. RR ) /\ v e. RR ) /\ ( C <_ B /\ v < C ) ) -> v <_ B )
30	29	an4s 826	. . . . . . . . . . . . . . . 16 |- ( ( ( ( B e. RR /\ C e. RR ) /\ C <_ B ) /\ ( v e. RR /\ v < C ) ) -> v <_ B )
31	30	3adantr2 1156	. . . . . . . . . . . . . . 15 |- ( ( ( ( B e. RR /\ C e. RR ) /\ C <_ B ) /\ ( v e. RR /\ A <_ v /\ v < C ) ) -> v <_ B )
32	31	ex 434	. . . . . . . . . . . . . 14 |- ( ( ( B e. RR /\ C e. RR ) /\ C <_ B ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> v <_ B ) )
33	32	anasss 647	. . . . . . . . . . . . 13 |- ( ( B e. RR /\ ( C e. RR /\ C <_ B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> v <_ B ) )
34	33	3adantr2 1156	. . . . . . . . . . . 12 |- ( ( B e. RR /\ ( C e. RR /\ A < C /\ C <_ B ) ) -> ( ( v e. RR /\ A <_ v /\ v < C ) -> v <_ B ) )
35	34	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 ) )
36	24, 35	syldan 470	. . . . . . . . . 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 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 ) ) )
38	18, 37	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 ) ) ) )
39		simpl1 999	. . . . . . . . 9 |- ( ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) -> v e. RR )
40		simpr2 1003	. . . . . . . . 9 |- ( ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) -> A <_ v )
41		simpl3 1001	. . . . . . . . 9 |- ( ( ( v e. RR /\ ( A - 1 ) < v /\ v < C ) /\ ( v e. RR /\ A <_ v /\ v <_ B ) ) -> v < C )
42	39, 40, 41	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 ) )
43	38, 42	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 ) ) ) )
44		simpll 753	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> A e. RR )
45	24	simp1d 1008	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> C e. RR )
46	45	rexrd 9660	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> C e. RR* )
47		elico2 11613	. . . . . . . 8 |- ( ( A e. RR /\ C e. RR* ) -> ( v e. ( A [,) C ) <-> ( v e. RR /\ A <_ v /\ v < C ) ) )
48	44, 46, 47	syl2anc 661	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( v e. ( A [,) C ) <-> ( v e. RR /\ A <_ v /\ v < C ) ) )
49		elin 3683	. . . . . . . 8 |- ( v e. ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) <-> ( v e. ( ( A - 1 ) (,) C ) /\ v e. ( A [,] B ) ) )
50	6	rexrd 9660	. . . . . . . . . . 11 |- ( A e. RR -> ( A - 1 ) e. RR* )
51	50	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A - 1 ) e. RR* )
52		elioo2 11595	. . . . . . . . . 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 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 ) ) )
54		elicc2 11614	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( v e. ( A [,] B ) <-> ( v e. RR /\ A <_ v /\ v <_ B ) ) )
55	54	adantr 465	. . . . . . . . 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 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 ) ) ) )
57	49, 56	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 ) ) ) )
58	43, 48, 57	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 ) ) ) )
59	58	eqrdv 2454	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A [,) C ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) )
60		ineq1 3689	. . . . . . 7 |- ( v = ( ( A - 1 ) (,) C ) -> ( v i^i ( A [,] B ) ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) )
61	60	eqeq2d 2471	. . . . . 6 |- ( v = ( ( A - 1 ) (,) C ) -> ( ( A [,) C ) = ( v i^i ( A [,] B ) ) <-> ( A [,) C ) = ( ( ( A - 1 ) (,) C ) i^i ( A [,] B ) ) ) )
62	61	rspcev 3210	. . . . 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 663	. . . 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 21394	. . . . 5 |- ( topGen ` ran (,) ) e. Top
65		ovex 6324	. . . . 5 |- ( A [,] B ) e. _V
66		elrest 14845	. . . . 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 672	. . . 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 212	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A [,) C ) e. ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
69		iccssre 11631	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
70	69	adantr 465	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A [,] B ) C_ RR )
71		eqid 2457	. . . . 5 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
72		icoopnst.1	. . . . 5 |- J = ( MetOpen ` ( ( abs o. - ) |` ( ( A [,] B ) X. ( A [,] B ) ) ) )
73	71, 72	resubmet 21433	. . . 4 |- ( ( A [,] B ) C_ RR -> J = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
74	70, 73	syl 16	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> J = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
75	68, 74	eleqtrrd 2548	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. ( A (,] B ) ) -> ( A [,) C ) e. J )
76	75	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 1395   e. wcel 1819   E.wrex 2808   _Vcvv 3109   i^i cin 3470   C_ wss 3471   class class class wbr 4456   X. cxp 5006   ran crn 5009   |` cres 5010   o. ccom 5012   ` cfv 5594  (class class class)co 6296   RRcr 9508   1c1 9510   RR*cxr 9644   < clt 9645   <_ cle 9646   - cmin 9824   (,)cioo 11554   (,]cioc 11555   [,)cico 11556   [,]cicc 11557   abscabs 13079   ↾_t crest 14838   topGenctg 14855   MetOpencmopn 18535   Topctop 19521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-rest 14840  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-top 19526  df-bases 19528  df-topon 19529

This theorem is referenced by: (None)