Theorem blssioo 22598

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 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 22594	. . . 4 |- D e. ( *Met ` RR )
3		blrn 22214	. . . 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 11950	. . . . . 6 |- ( r e. RR* <-> ( r e. RR \/ r = +oo \/ r = -oo ) )
6	1	bl2ioo 22595	. . . . . . . 8 |- ( ( y e. RR /\ r e. RR ) -> ( y ( ball ` D ) r ) = ( ( y - r ) (,) ( y + r ) ) )
7		resubcl 10345	. . . . . . . . 9 |- ( ( y e. RR /\ r e. RR ) -> ( y - r ) e. RR )
8		readdcl 10019	. . . . . . . . 9 |- ( ( y e. RR /\ r e. RR ) -> ( y + r ) e. RR )
9		rexr 10085	. . . . . . . . . 10 |- ( ( y - r ) e. RR -> ( y - r ) e. RR* )
10		rexr 10085	. . . . . . . . . 10 |- ( ( y + r ) e. RR -> ( y + r ) e. RR* )
11		ioof 12271	. . . . . . . . . . . 12 |- (,) : ( RR* X. RR* ) --> ~P RR
12		ffn 6045	. . . . . . . . . . . 12 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) )
13	11, 12	ax-mp 5	. . . . . . . . . . 11 |- (,) Fn ( RR* X. RR* )
14		fnovrn 6809	. . . . . . . . . . 11 |- ( ( (,) Fn ( RR* X. RR* ) /\ ( y - r ) e. RR* /\ ( y + r ) e. RR* ) -> ( ( y - r ) (,) ( y + r ) ) e. ran (,) )
15	13, 14	mp3an1 1411	. . . . . . . . . 10 |- ( ( ( y - r ) e. RR* /\ ( y + r ) e. RR* ) -> ( ( y - r ) (,) ( y + r ) ) e. ran (,) )
16	9, 10, 15	syl2an 494	. . . . . . . . 9 |- ( ( ( y - r ) e. RR /\ ( y + r ) e. RR ) -> ( ( y - r ) (,) ( y + r ) ) e. ran (,) )
17	7, 8, 16	syl2anc 693	. . . . . . . 8 |- ( ( y e. RR /\ r e. RR ) -> ( ( y - r ) (,) ( y + r ) ) e. ran (,) )
18	6, 17	eqeltrd 2701	. . . . . . 7 |- ( ( y e. RR /\ r e. RR ) -> ( y ( ball ` D ) r ) e. ran (,) )
19		oveq2 6658	. . . . . . . . 9 |- ( r = +oo -> ( y ( ball ` D ) r ) = ( y ( ball ` D ) +oo ) )
20	1	remet 22593	. . . . . . . . . 10 |- D e. ( Met ` RR )
21		blpnf 22202	. . . . . . . . . 10 |- ( ( D e. ( Met ` RR ) /\ y e. RR ) -> ( y ( ball ` D ) +oo ) = RR )
22	20, 21	mpan 706	. . . . . . . . 9 |- ( y e. RR -> ( y ( ball ` D ) +oo ) = RR )
23	19, 22	sylan9eqr 2678	. . . . . . . 8 |- ( ( y e. RR /\ r = +oo ) -> ( y ( ball ` D ) r ) = RR )
24		ioomax 12248	. . . . . . . . 9 |- ( -oo (,) +oo ) = RR
25		ioorebas 12275	. . . . . . . . 9 |- ( -oo (,) +oo ) e. ran (,)
26	24, 25	eqeltrri 2698	. . . . . . . 8 |- RR e. ran (,)
27	23, 26	syl6eqel 2709	. . . . . . 7 |- ( ( y e. RR /\ r = +oo ) -> ( y ( ball ` D ) r ) e. ran (,) )
28		oveq2 6658	. . . . . . . . 9 |- ( r = -oo -> ( y ( ball ` D ) r ) = ( y ( ball ` D ) -oo ) )
29		0xr 10086	. . . . . . . . . . 11 |- 0 e. RR*
30		nltmnf 11963	. . . . . . . . . . 11 |- ( 0 e. RR* -> -. 0 < -oo )
31	29, 30	ax-mp 5	. . . . . . . . . 10 |- -. 0 < -oo
32		mnfxr 10096	. . . . . . . . . . . 12 |- -oo e. RR*
33		xbln0 22219	. . . . . . . . . . . 12 |- ( ( D e. ( *Met ` RR ) /\ y e. RR /\ -oo e. RR* ) -> ( ( y ( ball ` D ) -oo ) =/= (/) <-> 0 < -oo ) )
34	2, 32, 33	mp3an13 1415	. . . . . . . . . . 11 |- ( y e. RR -> ( ( y ( ball ` D ) -oo ) =/= (/) <-> 0 < -oo ) )
35	34	necon1bbid 2833	. . . . . . . . . 10 |- ( y e. RR -> ( -. 0 < -oo <-> ( y ( ball ` D ) -oo ) = (/) ) )
36	31, 35	mpbii 223	. . . . . . . . 9 |- ( y e. RR -> ( y ( ball ` D ) -oo ) = (/) )
37	28, 36	sylan9eqr 2678	. . . . . . . 8 |- ( ( y e. RR /\ r = -oo ) -> ( y ( ball ` D ) r ) = (/) )
38		iooid 12203	. . . . . . . . 9 |- ( 0 (,) 0 ) = (/)
39		ioorebas 12275	. . . . . . . . 9 |- ( 0 (,) 0 ) e. ran (,)
40	38, 39	eqeltrri 2698	. . . . . . . 8 |- (/) e. ran (,)
41	37, 40	syl6eqel 2709	. . . . . . 7 |- ( ( y e. RR /\ r = -oo ) -> ( y ( ball ` D ) r ) e. ran (,) )
42	18, 27, 41	3jaodan 1394	. . . . . 6 |- ( ( y e. RR /\ ( r e. RR \/ r = +oo \/ r = -oo ) ) -> ( y ( ball ` D ) r ) e. ran (,) )
43	5, 42	sylan2b 492	. . . . 5 |- ( ( y e. RR /\ r e. RR* ) -> ( y ( ball ` D ) r ) e. ran (,) )
44		eleq1 2689	. . . . 5 |- ( z = ( y ( ball ` D ) r ) -> ( z e. ran (,) <-> ( y ( ball ` D ) r ) e. ran (,) ) )
45	43, 44	syl5ibrcom 237	. . . 4 |- ( ( y e. RR /\ r e. RR* ) -> ( z = ( y ( ball ` D ) r ) -> z e. ran (,) ) )
46	45	rexlimivv 3036	. . 3 |- ( E. y e. RR E. r e. RR* z = ( y ( ball ` D ) r ) -> z e. ran (,) )
47	4, 46	sylbi 207	. 2 |- ( z e. ran ( ball ` D ) -> z e. ran (,) )
48	47	ssriv 3607	1 |- ran ( ball ` D ) C_ ran (,)

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   X. cxp 5112   ran crn 5115   |` cres 5116   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   - cmin 10266   (,)cioo 12175   abscabs 13974   *Metcxmt 19731   Metcme 19732   ballcbl 19733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741

This theorem is referenced by:  tgioo  22599