Theorem blssioo 13596

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 13592	. . . 4 |- D e. ( *Met ` RR )
3		blrn 13463	. . . 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 9745	. . . . . 6 |- ( r e. RR* <-> ( r e. RR \/ r = +oo \/ r = -oo ) )
6	1	bl2ioo 13593	. . . . . . . 8 |- ( ( y e. RR /\ r e. RR ) -> ( y ( ball ` D ) r ) = ( ( y - r ) (,) ( y + r ) ) )
7		resubcl 8195	. . . . . . . . 9 |- ( ( y e. RR /\ r e. RR ) -> ( y - r ) e. RR )
8		readdcl 7912	. . . . . . . . 9 |- ( ( y e. RR /\ r e. RR ) -> ( y + r ) e. RR )
9		rexr 7977	. . . . . . . . . 10 |- ( ( y - r ) e. RR -> ( y - r ) e. RR* )
10		rexr 7977	. . . . . . . . . 10 |- ( ( y + r ) e. RR -> ( y + r ) e. RR* )
11		ioorebasg 9944	. . . . . . . . . 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 2252	. . . . . . 7 |- ( ( y e. RR /\ r e. RR ) -> ( y ( ball ` D ) r ) e. ran (,) )
15		oveq2 5873	. . . . . . . . 9 |- ( r = +oo -> ( y ( ball ` D ) r ) = ( y ( ball ` D ) +oo ) )
16	1	remet 13591	. . . . . . . . . 10 |- D e. ( Met ` RR )
17		blpnf 13451	. . . . . . . . . 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 2230	. . . . . . . 8 |- ( ( y e. RR /\ r = +oo ) -> ( y ( ball ` D ) r ) = RR )
20		ioomax 9917	. . . . . . . . 9 |- ( -oo (,) +oo ) = RR
21		mnfxr 7988	. . . . . . . . . 10 |- -oo e. RR*
22		pnfxr 7984	. . . . . . . . . 10 |- +oo e. RR*
23		ioorebasg 9944	. . . . . . . . . 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 2249	. . . . . . . 8 |- RR e. ran (,)
26	19, 25	eqeltrdi 2266	. . . . . . 7 |- ( ( y e. RR /\ r = +oo ) -> ( y ( ball ` D ) r ) e. ran (,) )
27		oveq2 5873	. . . . . . . . 9 |- ( r = -oo -> ( y ( ball ` D ) r ) = ( y ( ball ` D ) -oo ) )
28		0xr 7978	. . . . . . . . . . . 12 |- 0 e. RR*
29		nltmnf 9757	. . . . . . . . . . . 12 |- ( 0 e. RR* -> -. 0 < -oo )
30	28, 29	ax-mp 5	. . . . . . . . . . 11 |- -. 0 < -oo
31		xblm 13468	. . . . . . . . . . . 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 1328	. . . . . . . . . . 11 |- ( y e. RR -> ( E. w w e. ( y ( ball ` D ) -oo ) <-> 0 < -oo ) )
33	30, 32	mtbiri 675	. . . . . . . . . 10 |- ( y e. RR -> -. E. w w e. ( y ( ball ` D ) -oo ) )
34		notm0 3441	. . . . . . . . . 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 2230	. . . . . . . 8 |- ( ( y e. RR /\ r = -oo ) -> ( y ( ball ` D ) r ) = (/) )
37		iooidg 9878	. . . . . . . . . 10 |- ( 0 e. RR* -> ( 0 (,) 0 ) = (/) )
38	28, 37	ax-mp 5	. . . . . . . . 9 |- ( 0 (,) 0 ) = (/)
39		ioorebasg 9944	. . . . . . . . . 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 2249	. . . . . . . 8 |- (/) e. ran (,)
42	36, 41	eqeltrdi 2266	. . . . . . 7 |- ( ( y e. RR /\ r = -oo ) -> ( y ( ball ` D ) r ) e. ran (,) )
43	14, 26, 42	3jaodan 1306	. . . . . 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 2238	. . . . 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 2598	. . 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 3157	1 |- ran ( ball ` D ) C_ ran (,)

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   \/ w3o 977   = wceq 1353   E.wex 1490   e. wcel 2146   E.wrex 2454   C_ wss 3127   (/)c0 3420   class class class wbr 3998   X. cxp 4618   ran crn 4621   |` cres 4622   o. ccom 4624   ` cfv 5208  (class class class)co 5865   RRcr 7785   0cc0 7786   + caddc 7789   +oocpnf 7963   -oocmnf 7964   RR*cxr 7965   < clt 7966   - cmin 8102   (,)cioo 9857   abscabs 10972   *Metcxmet 13031   Metcmet 13032   ballcbl 13033

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-map 6640  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-rp 9623  df-xadd 9742  df-ioo 9861  df-seqfrec 10414  df-exp 10488  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-psmet 13038  df-xmet 13039  df-met 13040  df-bl 13041

This theorem is referenced by:  tgioo  13597