Theorem blssioo 15347

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 15343	. . . 4 |- D e. ( *Met ` RR )
3		blrn 15206	. . . 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 10055	. . . . . 6 |- ( r e. RR* <-> ( r e. RR \/ r = +oo \/ r = -oo ) )
6	1	bl2ioo 15344	. . . . . . . 8 |- ( ( y e. RR /\ r e. RR ) -> ( y ( ball ` D ) r ) = ( ( y - r ) (,) ( y + r ) ) )
7		resubcl 8485	. . . . . . . . 9 |- ( ( y e. RR /\ r e. RR ) -> ( y - r ) e. RR )
8		readdcl 8201	. . . . . . . . 9 |- ( ( y e. RR /\ r e. RR ) -> ( y + r ) e. RR )
9		rexr 8267	. . . . . . . . . 10 |- ( ( y - r ) e. RR -> ( y - r ) e. RR* )
10		rexr 8267	. . . . . . . . . 10 |- ( ( y + r ) e. RR -> ( y + r ) e. RR* )
11		ioorebasg 10254	. . . . . . . . . 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 2308	. . . . . . 7 |- ( ( y e. RR /\ r e. RR ) -> ( y ( ball ` D ) r ) e. ran (,) )
15		oveq2 6036	. . . . . . . . 9 |- ( r = +oo -> ( y ( ball ` D ) r ) = ( y ( ball ` D ) +oo ) )
16	1	remet 15342	. . . . . . . . . 10 |- D e. ( Met ` RR )
17		blpnf 15194	. . . . . . . . . 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 2286	. . . . . . . 8 |- ( ( y e. RR /\ r = +oo ) -> ( y ( ball ` D ) r ) = RR )
20		ioomax 10227	. . . . . . . . 9 |- ( -oo (,) +oo ) = RR
21		mnfxr 8278	. . . . . . . . . 10 |- -oo e. RR*
22		pnfxr 8274	. . . . . . . . . 10 |- +oo e. RR*
23		ioorebasg 10254	. . . . . . . . . 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 2305	. . . . . . . 8 |- RR e. ran (,)
26	19, 25	eqeltrdi 2322	. . . . . . 7 |- ( ( y e. RR /\ r = +oo ) -> ( y ( ball ` D ) r ) e. ran (,) )
27		oveq2 6036	. . . . . . . . 9 |- ( r = -oo -> ( y ( ball ` D ) r ) = ( y ( ball ` D ) -oo ) )
28		0xr 8268	. . . . . . . . . . . 12 |- 0 e. RR*
29		nltmnf 10067	. . . . . . . . . . . 12 |- ( 0 e. RR* -> -. 0 < -oo )
30	28, 29	ax-mp 5	. . . . . . . . . . 11 |- -. 0 < -oo
31		xblm 15211	. . . . . . . . . . . 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 3517	. . . . . . . . . 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 2286	. . . . . . . 8 |- ( ( y e. RR /\ r = -oo ) -> ( y ( ball ` D ) r ) = (/) )
37		iooidg 10188	. . . . . . . . . 10 |- ( 0 e. RR* -> ( 0 (,) 0 ) = (/) )
38	28, 37	ax-mp 5	. . . . . . . . 9 |- ( 0 (,) 0 ) = (/)
39		ioorebasg 10254	. . . . . . . . . 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 2305	. . . . . . . 8 |- (/) e. ran (,)
42	36, 41	eqeltrdi 2322	. . . . . . 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 2294	. . . . 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 2657	. . 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 3232	1 |- ran ( ball ` D ) C_ ran (,)

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   \/ w3o 1004   = wceq 1398   E.wex 1541   e. wcel 2202   E.wrex 2512   C_ wss 3201   (/)c0 3496   class class class wbr 4093   X. cxp 4729   ran crn 4732   |` cres 4733   o. ccom 4735   ` cfv 5333  (class class class)co 6028   RRcr 8074   0cc0 8075   + caddc 8078   +oocpnf 8253   -oocmnf 8254   RR*cxr 8255   < clt 8256   - cmin 8392   (,)cioo 10167   abscabs 11620   *Metcxmet 14615   Metcmet 14616   ballcbl 14617

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-map 6862  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-rp 9933  df-xadd 10052  df-ioo 10171  df-seqfrec 10756  df-exp 10847  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-psmet 14622  df-xmet 14623  df-met 14624  df-bl 14625

This theorem is referenced by:  tgioo  15348