Theorem blssioo 15654

Description: The balls of the standard real metric space are included in the open real intervals. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

Hypothesis

Ref	Expression
remet.1	|- D = ( ( abs o. - ) |` ( RR X. RR ) )

Assertion

Ref	Expression
blssioo	|- ran ( ball ` D ) C_ ran (,)

Proof of Theorem blssioo

Dummy variables r y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		remet.1	. . . . 5 |- D = ( ( abs o. - ) |` ( RR X. RR ) )
2	1	rexmet 15650	. . . 4 |- D e. ( *Met ` RR )
3		blrn 15513	. . . 4 |- ( D e. ( *Met ` RR ) -> ( z e. ran ( ball ` D ) <-> E. y e. RR E. r e. RR* z = ( y ( ball ` D ) r ) ) )
4	2, 3	ax-mp 5	. . 3 |- ( z e. ran ( ball ` D ) <-> E. y e. RR E. r e. RR* z = ( y ( ball ` D ) r ) )
5		elxr 10178	. . . . . 6 |- ( r e. RR* <-> ( r e. RR \/ r = +oo \/ r = -oo ) )
6	1	bl2ioo 15651	. . . . . . . 8 |- ( ( y e. RR /\ r e. RR ) -> ( y ( ball ` D ) r ) = ( ( y - r ) (,) ( y + r ) ) )
7		resubcl 8590	. . . . . . . . 9 |- ( ( y e. RR /\ r e. RR ) -> ( y - r ) e. RR )
8		readdcl 8305	. . . . . . . . 9 |- ( ( y e. RR /\ r e. RR ) -> ( y + r ) e. RR )
9		rexr 8371	. . . . . . . . . 10 |- ( ( y - r ) e. RR -> ( y - r ) e. RR* )
10		rexr 8371	. . . . . . . . . 10 |- ( ( y + r ) e. RR -> ( y + r ) e. RR* )
11		ioorebasg 10377	. . . . . . . . . 10 |- ( ( ( y - r ) e. RR* /\ ( y + r ) e. RR* ) -> ( ( y - r ) (,) ( y + r ) ) e. ran (,) )
12	9, 10, 11	syl2an 289	. . . . . . . . 9 |- ( ( ( y - r ) e. RR /\ ( y + r ) e. RR ) -> ( ( y - r ) (,) ( y + r ) ) e. ran (,) )
13	7, 8, 12	syl2anc 415	. . . . . . . 8 |- ( ( y e. RR /\ r e. RR ) -> ( ( y - r ) (,) ( y + r ) ) e. ran (,) )
14	6, 13	eqeltrd 2315	. . . . . . 7 |- ( ( y e. RR /\ r e. RR ) -> ( y ( ball ` D ) r ) e. ran (,) )
15		oveq2 6093	. . . . . . . . 9 |- ( r = +oo -> ( y ( ball ` D ) r ) = ( y ( ball ` D ) +oo ) )
16	1	remet 15649	. . . . . . . . . 10 |- D e. ( Met ` RR )
17		blpnf 15501	. . . . . . . . . 10 |- ( ( D e. ( Met ` RR ) /\ y e. RR ) -> ( y ( ball ` D ) +oo ) = RR )
18	16, 17	mpan 428	. . . . . . . . 9 |- ( y e. RR -> ( y ( ball ` D ) +oo ) = RR )
19	15, 18	sylan9eqr 2293	. . . . . . . 8 |- ( ( y e. RR /\ r = +oo ) -> ( y ( ball ` D ) r ) = RR )
20		ioomax 10350	. . . . . . . . 9 |- ( -oo (,) +oo ) = RR
21		mnfxr 8382	. . . . . . . . . 10 |- -oo e. RR*
22		pnfxr 8378	. . . . . . . . . 10 |- +oo e. RR*
23		ioorebasg 10377	. . . . . . . . . 10 |- ( ( -oo e. RR* /\ +oo e. RR* ) -> ( -oo (,) +oo ) e. ran (,) )
24	21, 22, 23	mp2an 430	. . . . . . . . 9 |- ( -oo (,) +oo ) e. ran (,)
25	20, 24	eqeltrri 2312	. . . . . . . 8 |- RR e. ran (,)
26	19, 25	eqeltrdi 2329	. . . . . . 7 |- ( ( y e. RR /\ r = +oo ) -> ( y ( ball ` D ) r ) e. ran (,) )
27		oveq2 6093	. . . . . . . . 9 |- ( r = -oo -> ( y ( ball ` D ) r ) = ( y ( ball ` D ) -oo ) )
28		0xr 8372	. . . . . . . . . . . 12 |- 0 e. RR*
29		nltmnf 10190	. . . . . . . . . . . 12 |- ( 0 e. RR* -> -. 0 < -oo )
30	28, 29	ax-mp 5	. . . . . . . . . . 11 |- -. 0 < -oo
31		xblm 15518	. . . . . . . . . . . 12 |- ( ( D e. ( *Met ` RR ) /\ y e. RR /\ -oo e. RR* ) -> ( E. w w e. ( y ( ball ` D ) -oo ) <-> 0 < -oo ) )
32	2, 21, 31	mp3an13 1369	. . . . . . . . . . 11 |- ( y e. RR -> ( E. w w e. ( y ( ball ` D ) -oo ) <-> 0 < -oo ) )
33	30, 32	mtbiri 686	. . . . . . . . . 10 |- ( y e. RR -> -. E. w w e. ( y ( ball ` D ) -oo ) )
34		notm0 3542	. . . . . . . . . 10 |- ( -. E. w w e. ( y ( ball ` D ) -oo ) <-> ( y ( ball ` D ) -oo ) = (/) )
35	33, 34	sylib 122	. . . . . . . . 9 |- ( y e. RR -> ( y ( ball ` D ) -oo ) = (/) )
36	27, 35	sylan9eqr 2293	. . . . . . . 8 |- ( ( y e. RR /\ r = -oo ) -> ( y ( ball ` D ) r ) = (/) )
37		iooidg 10311	. . . . . . . . . 10 |- ( 0 e. RR* -> ( 0 (,) 0 ) = (/) )
38	28, 37	ax-mp 5	. . . . . . . . 9 |- ( 0 (,) 0 ) = (/)
39		ioorebasg 10377	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ 0 e. RR* ) -> ( 0 (,) 0 ) e. ran (,) )
40	28, 28, 39	mp2an 430	. . . . . . . . 9 |- ( 0 (,) 0 ) e. ran (,)
41	38, 40	eqeltrri 2312	. . . . . . . 8 |- (/) e. ran (,)
42	36, 41	eqeltrdi 2329	. . . . . . 7 |- ( ( y e. RR /\ r = -oo ) -> ( y ( ball ` D ) r ) e. ran (,) )
43	14, 26, 42	3jaodan 1347	. . . . . 6 |- ( ( y e. RR /\ ( r e. RR \/ r = +oo \/ r = -oo ) ) -> ( y ( ball ` D ) r ) e. ran (,) )
44	5, 43	sylan2b 287	. . . . 5 |- ( ( y e. RR /\ r e. RR* ) -> ( y ( ball ` D ) r ) e. ran (,) )
45		eleq1 2301	. . . . 5 |- ( z = ( y ( ball ` D ) r ) -> ( z e. ran (,) <-> ( y ( ball ` D ) r ) e. ran (,) ) )
46	44, 45	syl5ibrcom 157	. . . 4 |- ( ( y e. RR /\ r e. RR* ) -> ( z = ( y ( ball ` D ) r ) -> z e. ran (,) ) )
47	46	rexlimivv 2674	. . 3 |- ( E. y e. RR E. r e. RR* z = ( y ( ball ` D ) r ) -> z e. ran (,) )
48	4, 47	sylbi 121	. 2 |- ( z e. ran ( ball ` D ) -> z e. ran (,) )
49	48	ssriv 3252	1 |- ran ( ball ` D ) C_ ran (,)

This proof depends on syntax axioms:   -. wn 3   /\ wa 104   <-> wb 105   \/ w3o 1008   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   C_ wss 3220   (/)c0 3520   class class class wbr 4130   X. cxp 4772   ran crn 4775   |` cres 4776   o. ccom 4778   ` cfv 5377  (class class class)co 6085   RRcr 8178   0cc0 8179   + caddc 8182   +oocpnf 8357   -oocmnf 8358   RR*cxr 8359   < clt 8360   - cmin 8497   (,)cioo 10290   abscabs 11763   *Metcxmet 14873   Metcmet 14874   ballcbl 14875

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-map 6924  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-rp 10055  df-xadd 10175  df-ioo 10294  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-psmet 14880  df-xmet 14881  df-met 14882  df-bl 14883

This theorem is used by:  tgioo  15655