Theorem opnrebl2 26214

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 2404	. . . . 5 |- ( ( abs o. - ) |` ( RR X. RR ) ) = ( ( abs o. - ) |` ( RR X. RR ) )
2	1	rexmet 18775	. . . 4 |- ( ( abs o. - ) |` ( RR X. RR ) ) e. ( * Met ` RR )
3		eqid 2404	. . . . . 6 |- ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
4	1, 3	tgioo 18780	. . . . 5 |- ( topGen ` ran (,) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) )
5	4	mopnss 18429	. . . 4 |- ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( * Met ` RR ) /\ A e. ( topGen ` ran (,) ) ) -> A C_ RR )
6	2, 5	mpan 652	. . 3 |- ( A e. ( topGen ` ran (,) ) -> A C_ RR )
7	4	mopni3 18477	. . . . . . . 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 424	. . . . . . 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 1266	. . . . . 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 3308	. . . . . . 7 |- ( ( A e. ( topGen ` ran (,) ) /\ x e. A ) -> x e. RR )
11		rpre 10574	. . . . . . . . . . . . 13 |- ( z e. RR+ -> z e. RR )
12	1	bl2ioo 18776	. . . . . . . . . . . . 13 |- ( ( x e. RR /\ z e. RR ) -> ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) = ( ( x - z ) (,) ( x + z ) ) )
13	11, 12	sylan2 461	. . . . . . . . . . . 12 |- ( ( x e. RR /\ z e. RR+ ) -> ( x ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) z ) = ( ( x - z ) (,) ( x + z ) ) )
14	13	sseq1d 3335	. . . . . . . . . . 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 685	. . . . . . . . . 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 2683	. . . . . . . . 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 199	. . . . . . . 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 10574	. . . . . . . . . . 11 |- ( y e. RR+ -> y e. RR )
19		ltle 9119	. . . . . . . . . . 11 |- ( ( z e. RR /\ y e. RR ) -> ( z < y -> z <_ y ) )
20	11, 18, 19	syl2anr 465	. . . . . . . . . 10 |- ( ( y e. RR+ /\ z e. RR+ ) -> ( z < y -> z <_ y ) )
21	20	anim1d 548	. . . . . . . . 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 2778	. . . . . . . 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 68	. . . . . . 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 38	. . . . 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 587	. . . 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 2757	. . 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 519	. 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 3303	. . . . . 6 |- ( ( A C_ RR /\ x e. A ) -> x e. RR )
30		1rp 10572	. . . . . . . 8 |- 1 e. RR+
31		simpr 448	. . . . . . . . . 10 |- ( ( z <_ y /\ ( ( x - z ) (,) ( x + z ) ) C_ A ) -> ( ( x - z ) (,) ( x + z ) ) C_ A )
32	31	reximi 2773	. . . . . . . . 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 2741	. . . . . . . 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 229	. . . . . . . . 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 3008	. . . . . . . 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 61	. . . . . . 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 2683	. . . . . . 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 213	. . . . . 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 2744	. . . 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 672	. . 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 18428	. . . 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 8	. . 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 204	. 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 181	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 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   A.wral 2666   E.wrex 2667   C_ wss 3280   class class class wbr 4172   X. cxp 4835   ran crn 4838   |` cres 4839   o. ccom 4841   ` cfv 5413  (class class class)co 6040   RRcr 8945   1c1 8947   + caddc 8949   < clt 9076   <_ cle 9077   - cmin 9247   RR+crp 10568   (,)cioo 10872   abscabs 11994   topGenctg 13620   * Metcxmt 16641   ballcbl 16643   MetOpencmopn 16646

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921