Theorem blssioo 15544

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 15540	. . . 4 |- D e. ( *Met ` RR )
3		blrn 15403	. . . 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 10128	. . . . . 6 |- ( r e. RR* <-> ( r e. RR \/ r = +oo \/ r = -oo ) )
6	1	bl2ioo 15541	. . . . . . . 8 |- ( ( y e. RR /\ r e. RR ) -> ( y ( ball ` D ) r ) = ( ( y - r ) (,) ( y + r ) ) )
7		resubcl 8553	. . . . . . . . 9 |- ( ( y e. RR /\ r e. RR ) -> ( y - r ) e. RR )
8		readdcl 8269	. . . . . . . . 9 |- ( ( y e. RR /\ r e. RR ) -> ( y + r ) e. RR )
9		rexr 8335	. . . . . . . . . 10 |- ( ( y - r ) e. RR -> ( y - r ) e. RR* )
10		rexr 8335	. . . . . . . . . 10 |- ( ( y + r ) e. RR -> ( y + r ) e. RR* )
11		ioorebasg 10327	. . . . . . . . . 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 411	. . . . . . . 8 |- ( ( y e. RR /\ r e. RR ) -> ( ( y - r ) (,) ( y + r ) ) e. ran (,) )
14	6, 13	eqeltrd 2311	. . . . . . 7 |- ( ( y e. RR /\ r e. RR ) -> ( y ( ball ` D ) r ) e. ran (,) )
15		oveq2 6066	. . . . . . . . 9 |- ( r = +oo -> ( y ( ball ` D ) r ) = ( y ( ball ` D ) +oo ) )
16	1	remet 15539	. . . . . . . . . 10 |- D e. ( Met ` RR )
17		blpnf 15391	. . . . . . . . . 10 |- ( ( D e. ( Met ` RR ) /\ y e. RR ) -> ( y ( ball ` D ) +oo ) = RR )
18	16, 17	mpan 424	. . . . . . . . 9 |- ( y e. RR -> ( y ( ball ` D ) +oo ) = RR )
19	15, 18	sylan9eqr 2289	. . . . . . . 8 |- ( ( y e. RR /\ r = +oo ) -> ( y ( ball ` D ) r ) = RR )
20		ioomax 10300	. . . . . . . . 9 |- ( -oo (,) +oo ) = RR
21		mnfxr 8346	. . . . . . . . . 10 |- -oo e. RR*
22		pnfxr 8342	. . . . . . . . . 10 |- +oo e. RR*
23		ioorebasg 10327	. . . . . . . . . 10 |- ( ( -oo e. RR* /\ +oo e. RR* ) -> ( -oo (,) +oo ) e. ran (,) )
24	21, 22, 23	mp2an 426	. . . . . . . . 9 |- ( -oo (,) +oo ) e. ran (,)
25	20, 24	eqeltrri 2308	. . . . . . . 8 |- RR e. ran (,)
26	19, 25	eqeltrdi 2325	. . . . . . 7 |- ( ( y e. RR /\ r = +oo ) -> ( y ( ball ` D ) r ) e. ran (,) )
27		oveq2 6066	. . . . . . . . 9 |- ( r = -oo -> ( y ( ball ` D ) r ) = ( y ( ball ` D ) -oo ) )
28		0xr 8336	. . . . . . . . . . . 12 |- 0 e. RR*
29		nltmnf 10140	. . . . . . . . . . . 12 |- ( 0 e. RR* -> -. 0 < -oo )
30	28, 29	ax-mp 5	. . . . . . . . . . 11 |- -. 0 < -oo
31		xblm 15408	. . . . . . . . . . . 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 1365	. . . . . . . . . . 11 |- ( y e. RR -> ( E. w w e. ( y ( ball ` D ) -oo ) <-> 0 < -oo ) )
33	30, 32	mtbiri 682	. . . . . . . . . 10 |- ( y e. RR -> -. E. w w e. ( y ( ball ` D ) -oo ) )
34		notm0 3533	. . . . . . . . . 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 2289	. . . . . . . 8 |- ( ( y e. RR /\ r = -oo ) -> ( y ( ball ` D ) r ) = (/) )
37		iooidg 10261	. . . . . . . . . 10 |- ( 0 e. RR* -> ( 0 (,) 0 ) = (/) )
38	28, 37	ax-mp 5	. . . . . . . . 9 |- ( 0 (,) 0 ) = (/)
39		ioorebasg 10327	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ 0 e. RR* ) -> ( 0 (,) 0 ) e. ran (,) )
40	28, 28, 39	mp2an 426	. . . . . . . . 9 |- ( 0 (,) 0 ) e. ran (,)
41	38, 40	eqeltrri 2308	. . . . . . . 8 |- (/) e. ran (,)
42	36, 41	eqeltrdi 2325	. . . . . . 7 |- ( ( y e. RR /\ r = -oo ) -> ( y ( ball ` D ) r ) e. ran (,) )
43	14, 26, 42	3jaodan 1343	. . . . . 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 2297	. . . . 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 2668	. . 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 3246	1 |- ran ( ball ` D ) C_ ran (,)

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   \/ w3o 1004   = wceq 1398   E.wex 1541   e. wcel 2205   E.wrex 2523   C_ wss 3214   (/)c0 3512   class class class wbr 4114   X. cxp 4752   ran crn 4755   |` cres 4756   o. ccom 4758   ` cfv 5357  (class class class)co 6058   RRcr 8142   0cc0 8143   + caddc 8146   +oocpnf 8321   -oocmnf 8322   RR*cxr 8323   < clt 8324   - cmin 8460   (,)cioo 10240   abscabs 11707   *Metcxmet 14810   Metcmet 14811   ballcbl 14812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-map 6897  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-rp 10005  df-xadd 10125  df-ioo 10244  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-psmet 14817  df-xmet 14818  df-met 14819  df-bl 14820

This theorem is referenced by:  tgioo  15545