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Theorem blssioo 21592
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.
StepHypRef Expression
1 remet.1 . . . . 5  |-  D  =  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )
21rexmet 21588 . . . 4  |-  D  e.  ( *Met `  RR )
3 blrn 21204 . . . 4  |-  ( D  e.  ( *Met `  RR )  ->  (
z  e.  ran  ( ball `  D )  <->  E. y  e.  RR  E. r  e. 
RR*  z  =  ( y ( ball `  D
) r ) ) )
42, 3ax-mp 5 . . 3  |-  ( z  e.  ran  ( ball `  D )  <->  E. y  e.  RR  E. r  e. 
RR*  z  =  ( y ( ball `  D
) r ) )
5 elxr 11378 . . . . . 6  |-  ( r  e.  RR*  <->  ( r  e.  RR  \/  r  = +oo  \/  r  = -oo ) )
61bl2ioo 21589 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y ( ball `  D ) r )  =  ( ( y  -  r ) (,) ( y  +  r ) ) )
7 resubcl 9919 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y  -  r
)  e.  RR )
8 readdcl 9605 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y  +  r )  e.  RR )
9 rexr 9669 . . . . . . . . . 10  |-  ( ( y  -  r )  e.  RR  ->  (
y  -  r )  e.  RR* )
10 rexr 9669 . . . . . . . . . 10  |-  ( ( y  +  r )  e.  RR  ->  (
y  +  r )  e.  RR* )
11 ioof 11676 . . . . . . . . . . . 12  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
12 ffn 5714 . . . . . . . . . . . 12  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
1311, 12ax-mp 5 . . . . . . . . . . 11  |-  (,)  Fn  ( RR*  X.  RR* )
14 fnovrn 6431 . . . . . . . . . . 11  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  (
y  -  r )  e.  RR*  /\  (
y  +  r )  e.  RR* )  ->  (
( y  -  r
) (,) ( y  +  r ) )  e.  ran  (,) )
1513, 14mp3an1 1313 . . . . . . . . . 10  |-  ( ( ( y  -  r
)  e.  RR*  /\  (
y  +  r )  e.  RR* )  ->  (
( y  -  r
) (,) ( y  +  r ) )  e.  ran  (,) )
169, 10, 15syl2an 475 . . . . . . . . 9  |-  ( ( ( y  -  r
)  e.  RR  /\  ( y  +  r )  e.  RR )  ->  ( ( y  -  r ) (,) ( y  +  r ) )  e.  ran  (,) )
177, 8, 16syl2anc 659 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( ( y  -  r ) (,) (
y  +  r ) )  e.  ran  (,) )
186, 17eqeltrd 2490 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y ( ball `  D ) r )  e.  ran  (,) )
19 oveq2 6286 . . . . . . . . 9  |-  ( r  = +oo  ->  (
y ( ball `  D
) r )  =  ( y ( ball `  D ) +oo )
)
201remet 21587 . . . . . . . . . 10  |-  D  e.  ( Met `  RR )
21 blpnf 21192 . . . . . . . . . 10  |-  ( ( D  e.  ( Met `  RR )  /\  y  e.  RR )  ->  (
y ( ball `  D
) +oo )  =  RR )
2220, 21mpan 668 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y ( ball `  D
) +oo )  =  RR )
2319, 22sylan9eqr 2465 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  = +oo )  ->  ( y ( ball `  D ) r )  =  RR )
24 ioomax 11653 . . . . . . . . 9  |-  ( -oo (,) +oo )  =  RR
25 ioorebas 11680 . . . . . . . . 9  |-  ( -oo (,) +oo )  e.  ran  (,)
2624, 25eqeltrri 2487 . . . . . . . 8  |-  RR  e.  ran  (,)
2723, 26syl6eqel 2498 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  = +oo )  ->  ( y ( ball `  D ) r )  e.  ran  (,) )
28 oveq2 6286 . . . . . . . . 9  |-  ( r  = -oo  ->  (
y ( ball `  D
) r )  =  ( y ( ball `  D ) -oo )
)
29 0xr 9670 . . . . . . . . . . 11  |-  0  e.  RR*
30 nltmnf 11391 . . . . . . . . . . 11  |-  ( 0  e.  RR*  ->  -.  0  < -oo )
3129, 30ax-mp 5 . . . . . . . . . 10  |-  -.  0  < -oo
32 mnfxr 11376 . . . . . . . . . . . 12  |- -oo  e.  RR*
33 xbln0 21209 . . . . . . . . . . . 12  |-  ( ( D  e.  ( *Met `  RR )  /\  y  e.  RR  /\ -oo  e.  RR* )  ->  (
( y ( ball `  D ) -oo )  =/=  (/)  <->  0  < -oo ) )
342, 32, 33mp3an13 1317 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  (
( y ( ball `  D ) -oo )  =/=  (/)  <->  0  < -oo ) )
3534necon1bbid 2653 . . . . . . . . . 10  |-  ( y  e.  RR  ->  ( -.  0  < -oo  <->  ( y
( ball `  D ) -oo )  =  (/) ) )
3631, 35mpbii 211 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y ( ball `  D
) -oo )  =  (/) )
3728, 36sylan9eqr 2465 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  = -oo )  ->  ( y ( ball `  D ) r )  =  (/) )
38 iooid 11610 . . . . . . . . 9  |-  ( 0 (,) 0 )  =  (/)
39 ioorebas 11680 . . . . . . . . 9  |-  ( 0 (,) 0 )  e. 
ran  (,)
4038, 39eqeltrri 2487 . . . . . . . 8  |-  (/)  e.  ran  (,)
4137, 40syl6eqel 2498 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  = -oo )  ->  ( y ( ball `  D ) r )  e.  ran  (,) )
4218, 27, 413jaodan 1296 . . . . . 6  |-  ( ( y  e.  RR  /\  ( r  e.  RR  \/  r  = +oo  \/  r  = -oo ) )  ->  (
y ( ball `  D
) r )  e. 
ran  (,) )
435, 42sylan2b 473 . . . . 5  |-  ( ( y  e.  RR  /\  r  e.  RR* )  -> 
( y ( ball `  D ) r )  e.  ran  (,) )
44 eleq1 2474 . . . . 5  |-  ( z  =  ( y (
ball `  D )
r )  ->  (
z  e.  ran  (,)  <->  (
y ( ball `  D
) r )  e. 
ran  (,) ) )
4543, 44syl5ibrcom 222 . . . 4  |-  ( ( y  e.  RR  /\  r  e.  RR* )  -> 
( z  =  ( y ( ball `  D
) r )  -> 
z  e.  ran  (,) ) )
4645rexlimivv 2901 . . 3  |-  ( E. y  e.  RR  E. r  e.  RR*  z  =  ( y ( ball `  D ) r )  ->  z  e.  ran  (,) )
474, 46sylbi 195 . 2  |-  ( z  e.  ran  ( ball `  D )  ->  z  e.  ran  (,) )
4847ssriv 3446 1  |-  ran  ( ball `  D )  C_  ran  (,)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 367    \/ w3o 973    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2755    C_ wss 3414   (/)c0 3738   ~Pcpw 3955   class class class wbr 4395    X. cxp 4821   ran crn 4824    |` cres 4825    o. ccom 4827    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   RRcr 9521   0cc0 9522    + caddc 9525   +oocpnf 9655   -oocmnf 9656   RR*cxr 9657    < clt 9658    - cmin 9841   (,)cioo 11582   abscabs 13216   *Metcxmt 18723   Metcme 18724   ballcbl 18725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-seq 12152  df-exp 12211  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734
This theorem is referenced by:  tgioo  21593
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