Theorem opnrebl2 32316

Description: A set is open in the standard topology of the reals precisely when every point can be enclosed in an arbitrarily small ball. (Contributed by Jeff Hankins, 22-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)

Assertion

Ref	Expression
opnrebl2	|- ( A e. ( topGen ` ran (,) ) <-> ( A C_ RR /\ A. x e. A A. y e. RR+ E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) ) )

Distinct variable group:   x, y, z, A

Proof of Theorem opnrebl2

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( ( abs o. - ) |` ( RR X. RR ) ) = ( ( abs o. - ) |` ( RR X. RR ) )
2	1	rexmet 22594	. . . 4 |- ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR )
3		eqid 2622	. . . . . 6 |- ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
4	1, 3	tgioo 22599	. . . . 5 |- ( topGen ` ran (,) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
5	4	mopnss 22251	. . . 4 |- ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) /\ A e. ( topGen ` ran (,) ) ) -> A C_ RR )
6	2, 5	mpan 706	. . 3 |- ( A e. ( topGen ` ran (,) ) -> A C_ RR )
7	4	mopni3 22299	. . . . . . . 8 |- ( ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) /\ A e. ( topGen ` ran (,) ) /\ x e. A ) /\ y e. RR+ ) -> E. z e. RR+ ( z < y /\ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A ) )
8	7	ex 450	. . . . . . 7 |- ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) /\ A e. ( topGen ` ran (,) ) /\ x e. A ) -> ( y e. RR+ -> E. z e. RR+ ( z < y /\ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A ) ) )
9	2, 8	mp3an1 1411	. . . . . 6 |- ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) -> ( y e. RR+ -> E. z e. RR+ ( z < y /\ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A ) ) )
10	6	sselda 3603	. . . . . . 7 |- ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) -> x e. RR )
11		rpre 11839	. . . . . . . . . . . . 13 |- ( z e. RR+ -> z e. RR )
12	1	bl2ioo 22595	. . . . . . . . . . . . 13 |- ( ( x e. RR /\ z e. RR ) -> ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) = ( ( x - z ) (,) ( x + z ) ) )
13	11, 12	sylan2 491	. . . . . . . . . . . 12 |- ( ( x e. RR /\ z e. RR+ ) -> ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) = ( ( x - z ) (,) ( x + z ) ) )
14	13	sseq1d 3632	. . . . . . . . . . 11 |- ( ( x e. RR /\ z e. RR+ ) -> ( ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A <-> ( ( x - z ) (,) ( x + z ) ) C_ A ) )
15	14	anbi2d 740	. . . . . . . . . 10 |- ( ( x e. RR /\ z e. RR+ ) -> ( ( z < y /\ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A ) <-> ( z < y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) ) )
16	15	rexbidva 3049	. . . . . . . . 9 |- ( x e. RR -> ( E. z e. RR+ ( z < y /\ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A ) <-> E. z e. RR+ ( z < y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) ) )
17	16	biimpd 219	. . . . . . . 8 |- ( x e. RR -> ( E. z e. RR+ ( z < y /\ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A ) -> E. z e. RR+ ( z < y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) ) )
18		rpre 11839	. . . . . . . . . . 11 |- ( y e. RR+ -> y e. RR )
19		ltle 10126	. . . . . . . . . . 11 |- ( ( z e. RR /\ y e. RR ) -> ( z < y -> z <_ y ) )
20	11, 18, 19	syl2anr 495	. . . . . . . . . 10 |- ( ( y e. RR+ /\ z e. RR+ ) -> ( z < y -> z <_ y ) )
21	20	anim1d 588	. . . . . . . . 9 |- ( ( y e. RR+ /\ z e. RR+ ) -> ( ( z < y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) -> ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) ) )
22	21	reximdva 3017	. . . . . . . 8 |- ( y e. RR+ -> ( E. z e. RR+ ( z < y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) -> E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) ) )
23	17, 22	syl9 77	. . . . . . 7 |- ( x e. RR -> ( y e. RR+ -> ( E. z e. RR+ ( z < y /\ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A ) -> E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) ) ) )
24	10, 23	syl 17	. . . . . 6 |- ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) -> ( y e. RR+ -> ( E. z e. RR+ ( z < y /\ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A ) -> E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) ) ) )
25	9, 24	mpdd 43	. . . . 5 |- ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) -> ( y e. RR+ -> E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) ) )
26	25	expimpd 629	. . . 4 |- ( A e. ( topGen ` ran (,) ) -> ( ( x e. A /\ y e. RR+ ) -> E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) ) )
27	26	ralrimivv 2970	. . 3 |- ( A e. ( topGen ` ran (,) ) -> A. x e. A A. y e. RR+ E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) )
28	6, 27	jca 554	. 2 |- ( A e. ( topGen ` ran (,) ) -> ( A C_ RR /\ A. x e. A A. y e. RR+ E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) ) )
29		ssel2 3598	. . . . . 6 |- ( ( A C_ RR /\ x e. A ) -> x e. RR )
30		1rp 11836	. . . . . . . 8 |- 1 e. RR+
31		simpr 477	. . . . . . . . . 10 |- ( ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) -> ( ( x - z ) (,) ( x + z ) ) C_ A )
32	31	reximi 3011	. . . . . . . . 9 |- ( E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) -> E. z e. RR+ ( ( x - z ) (,) ( x + z ) ) C_ A )
33	32	ralimi 2952	. . . . . . . 8 |- ( A. y e. RR+ E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) -> A. y e. RR+ E. z e. RR+ ( ( x - z ) (,) ( x + z ) ) C_ A )
34		biidd 252	. . . . . . . . 9 |- ( y = 1 -> ( E. z e. RR+ ( ( x - z ) (,) ( x + z ) ) C_ A <-> E. z e. RR+ ( ( x - z ) (,) ( x + z ) ) C_ A ) )
35	34	rspcv 3305	. . . . . . . 8 |- ( 1 e. RR+ -> ( A. y e. RR+ E. z e. RR+ ( ( x - z ) (,) ( x + z ) ) C_ A -> E. z e. RR+ ( ( x - z ) (,) ( x + z ) ) C_ A ) )
36	30, 33, 35	mpsyl 68	. . . . . . 7 |- ( A. y e. RR+ E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) -> E. z e. RR+ ( ( x - z ) (,) ( x + z ) ) C_ A )
37	14	rexbidva 3049	. . . . . . 7 |- ( x e. RR -> ( E. z e. RR+ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A <-> E. z e. RR+ ( ( x - z ) (,) ( x + z ) ) C_ A ) )
38	36, 37	syl5ibr 236	. . . . . 6 |- ( x e. RR -> ( A. y e. RR+ E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) -> E. z e. RR+ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A ) )
39	29, 38	syl 17	. . . . 5 |- ( ( A C_ RR /\ x e. A ) -> ( A. y e. RR+ E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) -> E. z e. RR+ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A ) )
40	39	ralimdva 2962	. . . 4 |- ( A C_ RR -> ( A. x e. A A. y e. RR+ E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) -> A. x e. A E. z e. RR+ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A ) )
41	40	imdistani 726	. . 3 |- ( ( A C_ RR /\ A. x e. A A. y e. RR+ E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) ) -> ( A C_ RR /\ A. x e. A E. z e. RR+ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A ) )
42	4	elmopn2 22250	. . . 4 |- ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) -> ( A e. ( topGen ` ran (,) ) <-> ( A C_ RR /\ A. x e. A E. z e. RR+ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A ) ) )
43	2, 42	ax-mp 5	. . 3 |- ( A e. ( topGen ` ran (,) ) <-> ( A C_ RR /\ A. x e. A E. z e. RR+ ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) C_ A ) )
44	41, 43	sylibr 224	. 2 |- ( ( A C_ RR /\ A. x e. A A. y e. RR+ E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) ) -> A e. ( topGen ` ran (,) ) )
45	28, 44	impbii 199	1 |- ( A e. ( topGen ` ran (,) ) <-> ( A C_ RR /\ A. x e. A A. y e. RR+ E. z e. RR+ ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   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   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   RR+crp 11832   (,)cioo 12175   abscabs 13974   topGenctg 16098   *Metcxmt 19731   ballcbl 19733   MetOpencmopn 19736

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-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-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  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)