Theorem opnrebl2 28357

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 2433	. . . . 5 |- ( ( abs o. - ) |` ( RR X. RR ) ) = ( ( abs o. - ) |` ( RR X. RR ) )
2	1	rexmet 20209	. . . 4 |- ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR )
3		eqid 2433	. . . . . 6 |- ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
4	1, 3	tgioo 20214	. . . . 5 |- ( topGen ` ran (,) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
5	4	mopnss 19862	. . . 4 |- ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) /\ A e. ( topGen ` ran (,) ) ) -> A C_ RR )
6	2, 5	mpan 663	. . 3 |- ( A e. ( topGen ` ran (,) ) -> A C_ RR )
7	4	mopni3 19910	. . . . . . . 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 1294	. . . . . 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 3344	. . . . . . 7 |- ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) -> x e. RR )
11		rpre 10984	. . . . . . . . . . . . 13 |- ( z e. RR+ -> z e. RR )
12	1	bl2ioo 20210	. . . . . . . . . . . . 13 |- ( ( x e. RR /\ z e. RR ) -> ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) = ( ( x - z ) (,) ( x + z ) ) )
13	11, 12	sylan2 471	. . . . . . . . . . . 12 |- ( ( x e. RR /\ z e. RR+ ) -> ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) = ( ( x - z ) (,) ( x + z ) ) )
14	13	sseq1d 3371	. . . . . . . . . . 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 696	. . . . . . . . . 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 2722	. . . . . . . . 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 10984	. . . . . . . . . . 11 |- ( y e. RR+ -> y e. RR )
19		ltle 9450	. . . . . . . . . . 11 |- ( ( z e. RR /\ y e. RR ) -> ( z < y -> z <_ y ) )
20	11, 18, 19	syl2anr 475	. . . . . . . . . 10 |- ( ( y e. RR+ /\ z e. RR+ ) -> ( z < y -> z <_ y ) )
21	20	anim1d 559	. . . . . . . . 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 2818	. . . . . . . 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 598	. . . 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 2797	. . 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 529	. 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 3339	. . . . . 6 |- ( ( A C_ RR /\ x e. A ) -> x e. RR )
30		1rp 10982	. . . . . . . 8 |- 1 e. RR+
31		simpr 458	. . . . . . . . . 10 |- ( ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) -> ( ( x - z ) (,) ( x + z ) ) C_ A )
32	31	reximi 2813	. . . . . . . . 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 2781	. . . . . . . 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 3058	. . . . . . . 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 2722	. . . . . . 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 2784	. . . 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 683	. . 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 19861	. . . 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 958   = wceq 1362   e. wcel 1755   A.wral 2705   E.wrex 2706   C_ wss 3316   class class class wbr 4280   X. cxp 4825   ran crn 4828   |` cres 4829   o. ccom 4831   ` cfv 5406  (class class class)co 6080   RRcr 9268   1c1 9270   + caddc 9272   < clt 9405   <_ cle 9406   - cmin 9582   RR+crp 10978   (,)cioo 11287   abscabs 12706   topGenctg 14358   *Metcxmt 17644   ballcbl 17646   MetOpencmopn 17649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-sup 7679  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-n0 10567  df-z 10634  df-uz 10849  df-q 10941  df-rp 10979  df-xneg 11076  df-xadd 11077  df-xmul 11078  df-ioo 11291  df-seq 11790  df-exp 11849  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-topgen 14364  df-psmet 17652  df-xmet 17653  df-met 17654  df-bl 17655  df-mopn 17656  df-top 18344  df-bases 18346  df-topon 18347

This theorem is referenced by: (None)