Theorem opnrebl2 30379

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 2454	. . . . 5 |- ( ( abs o. - ) |` ( RR X. RR ) ) = ( ( abs o. - ) |` ( RR X. RR ) )
2	1	rexmet 21462	. . . 4 |- ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR )
3		eqid 2454	. . . . . 6 |- ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
4	1, 3	tgioo 21467	. . . . 5 |- ( topGen ` ran (,) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
5	4	mopnss 21115	. . . 4 |- ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) /\ A e. ( topGen ` ran (,) ) ) -> A C_ RR )
6	2, 5	mpan 668	. . 3 |- ( A e. ( topGen ` ran (,) ) -> A C_ RR )
7	4	mopni3 21163	. . . . . . . 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 432	. . . . . . 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 1309	. . . . . 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 3489	. . . . . . 7 |- ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) -> x e. RR )
11		rpre 11227	. . . . . . . . . . . . 13 |- ( z e. RR+ -> z e. RR )
12	1	bl2ioo 21463	. . . . . . . . . . . . 13 |- ( ( x e. RR /\ z e. RR ) -> ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) = ( ( x - z ) (,) ( x + z ) ) )
13	11, 12	sylan2 472	. . . . . . . . . . . 12 |- ( ( x e. RR /\ z e. RR+ ) -> ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) = ( ( x - z ) (,) ( x + z ) ) )
14	13	sseq1d 3516	. . . . . . . . . . 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 701	. . . . . . . . . 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 2962	. . . . . . . . 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 11227	. . . . . . . . . . 11 |- ( y e. RR+ -> y e. RR )
19		ltle 9662	. . . . . . . . . . 11 |- ( ( z e. RR /\ y e. RR ) -> ( z < y -> z <_ y ) )
20	11, 18, 19	syl2anr 476	. . . . . . . . . 10 |- ( ( y e. RR+ /\ z e. RR+ ) -> ( z < y -> z <_ y ) )
21	20	anim1d 562	. . . . . . . . 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 2929	. . . . . . . 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 601	. . . 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 2874	. . 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 530	. 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 3484	. . . . . 6 |- ( ( A C_ RR /\ x e. A ) -> x e. RR )
30		1rp 11225	. . . . . . . 8 |- 1 e. RR+
31		simpr 459	. . . . . . . . . 10 |- ( ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) -> ( ( x - z ) (,) ( x + z ) ) C_ A )
32	31	reximi 2922	. . . . . . . . 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 2847	. . . . . . . 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 3203	. . . . . . . 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 2962	. . . . . . 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 2862	. . . 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 688	. . 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 21114	. . . 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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   A.wral 2804   E.wrex 2805   C_ wss 3461   class class class wbr 4439   X. cxp 4986   ran crn 4989   |` cres 4990   o. ccom 4992   ` cfv 5570  (class class class)co 6270   RRcr 9480   1c1 9482   + caddc 9484   < clt 9617   <_ cle 9618   - cmin 9796   RR+crp 11221   (,)cioo 11532   abscabs 13149   topGenctg 14927   *Metcxmt 18598   ballcbl 18600   MetOpencmopn 18603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-topgen 14933  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-top 19566  df-bases 19568  df-topon 19569

This theorem is referenced by: (None)