Theorem opnrebl2 28659

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 2452	. . . . 5 |- ( ( abs o. - ) |` ( RR X. RR ) ) = ( ( abs o. - ) |` ( RR X. RR ) )
2	1	rexmet 20495	. . . 4 |- ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR )
3		eqid 2452	. . . . . 6 |- ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
4	1, 3	tgioo 20500	. . . . 5 |- ( topGen ` ran (,) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
5	4	mopnss 20148	. . . 4 |- ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) /\ A e. ( topGen ` ran (,) ) ) -> A C_ RR )
6	2, 5	mpan 670	. . 3 |- ( A e. ( topGen ` ran (,) ) -> A C_ RR )
7	4	mopni3 20196	. . . . . . . 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 434	. . . . . . 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 1302	. . . . . 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 3459	. . . . . . 7 |- ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) -> x e. RR )
11		rpre 11103	. . . . . . . . . . . . 13 |- ( z e. RR+ -> z e. RR )
12	1	bl2ioo 20496	. . . . . . . . . . . . 13 |- ( ( x e. RR /\ z e. RR ) -> ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) = ( ( x - z ) (,) ( x + z ) ) )
13	11, 12	sylan2 474	. . . . . . . . . . . 12 |- ( ( x e. RR /\ z e. RR+ ) -> ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) = ( ( x - z ) (,) ( x + z ) ) )
14	13	sseq1d 3486	. . . . . . . . . . 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 703	. . . . . . . . . 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 2854	. . . . . . . . 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 207	. . . . . . . 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 11103	. . . . . . . . . . 11 |- ( y e. RR+ -> y e. RR )
19		ltle 9569	. . . . . . . . . . 11 |- ( ( z e. RR /\ y e. RR ) -> ( z < y -> z <_ y ) )
20	11, 18, 19	syl2anr 478	. . . . . . . . . 10 |- ( ( y e. RR+ /\ z e. RR+ ) -> ( z < y -> z <_ y ) )
21	20	anim1d 564	. . . . . . . . 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 2928	. . . . . . . 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 71	. . . . . . 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 16	. . . . . 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 40	. . . . 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 603	. . . 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 2907	. . 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 532	. 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 3454	. . . . . 6 |- ( ( A C_ RR /\ x e. A ) -> x e. RR )
30		1rp 11101	. . . . . . . 8 |- 1 e. RR+
31		simpr 461	. . . . . . . . . 10 |- ( ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) -> ( ( x - z ) (,) ( x + z ) ) C_ A )
32	31	reximi 2923	. . . . . . . . 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 2816	. . . . . . . 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 237	. . . . . . . . 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 3169	. . . . . . . 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 63	. . . . . . 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 2854	. . . . . . 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 221	. . . . . 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 16	. . . . 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 2829	. . . 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 690	. . 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 20147	. . . 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 212	. 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 188	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 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   A.wral 2796   E.wrex 2797   C_ wss 3431   class class class wbr 4395   X. cxp 4941   ran crn 4944   |` cres 4945   o. ccom 4947   ` cfv 5521  (class class class)co 6195   RRcr 9387   1c1 9389   + caddc 9391   < clt 9524   <_ cle 9525   - cmin 9701   RR+crp 11097   (,)cioo 11406   abscabs 12836   topGenctg 14490   *Metcxmt 17921   ballcbl 17923   MetOpencmopn 17926

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 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466

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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-sup 7797  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-q 11060  df-rp 11098  df-xneg 11195  df-xadd 11196  df-xmul 11197  df-ioo 11410  df-seq 11919  df-exp 11978  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-topgen 14496  df-psmet 17929  df-xmet 17930  df-met 17931  df-bl 17932  df-mopn 17933  df-top 18630  df-bases 18632  df-topon 18633

This theorem is referenced by: (None)