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Theorem blssioo 20507
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 20503 . . . 4  |-  D  e.  ( *Met `  RR )
3 blrn 20119 . . . 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 11210 . . . . . 6  |-  ( r  e.  RR*  <->  ( r  e.  RR  \/  r  = +oo  \/  r  = -oo ) )
61bl2ioo 20504 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y ( ball `  D ) r )  =  ( ( y  -  r ) (,) ( y  +  r ) ) )
7 resubcl 9787 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y  -  r
)  e.  RR )
8 readdcl 9479 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y  +  r )  e.  RR )
9 rexr 9543 . . . . . . . . . 10  |-  ( ( y  -  r )  e.  RR  ->  (
y  -  r )  e.  RR* )
10 rexr 9543 . . . . . . . . . 10  |-  ( ( y  +  r )  e.  RR  ->  (
y  +  r )  e.  RR* )
11 ioof 11507 . . . . . . . . . . . 12  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
12 ffn 5670 . . . . . . . . . . . 12  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
1311, 12ax-mp 5 . . . . . . . . . . 11  |-  (,)  Fn  ( RR*  X.  RR* )
14 fnovrn 6351 . . . . . . . . . . 11  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  (
y  -  r )  e.  RR*  /\  (
y  +  r )  e.  RR* )  ->  (
( y  -  r
) (,) ( y  +  r ) )  e.  ran  (,) )
1513, 14mp3an1 1302 . . . . . . . . . 10  |-  ( ( ( y  -  r
)  e.  RR*  /\  (
y  +  r )  e.  RR* )  ->  (
( y  -  r
) (,) ( y  +  r ) )  e.  ran  (,) )
169, 10, 15syl2an 477 . . . . . . . . 9  |-  ( ( ( y  -  r
)  e.  RR  /\  ( y  +  r )  e.  RR )  ->  ( ( y  -  r ) (,) ( y  +  r ) )  e.  ran  (,) )
177, 8, 16syl2anc 661 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( ( y  -  r ) (,) (
y  +  r ) )  e.  ran  (,) )
186, 17eqeltrd 2542 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y ( ball `  D ) r )  e.  ran  (,) )
19 oveq2 6211 . . . . . . . . 9  |-  ( r  = +oo  ->  (
y ( ball `  D
) r )  =  ( y ( ball `  D ) +oo )
)
201remet 20502 . . . . . . . . . 10  |-  D  e.  ( Met `  RR )
21 blpnf 20107 . . . . . . . . . 10  |-  ( ( D  e.  ( Met `  RR )  /\  y  e.  RR )  ->  (
y ( ball `  D
) +oo )  =  RR )
2220, 21mpan 670 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y ( ball `  D
) +oo )  =  RR )
2319, 22sylan9eqr 2517 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  = +oo )  ->  ( y ( ball `  D ) r )  =  RR )
24 ioomax 11484 . . . . . . . . 9  |-  ( -oo (,) +oo )  =  RR
25 ioorebas 11511 . . . . . . . . 9  |-  ( -oo (,) +oo )  e.  ran  (,)
2624, 25eqeltrri 2539 . . . . . . . 8  |-  RR  e.  ran  (,)
2723, 26syl6eqel 2550 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  = +oo )  ->  ( y ( ball `  D ) r )  e.  ran  (,) )
28 oveq2 6211 . . . . . . . . 9  |-  ( r  = -oo  ->  (
y ( ball `  D
) r )  =  ( y ( ball `  D ) -oo )
)
29 0xr 9544 . . . . . . . . . . 11  |-  0  e.  RR*
30 nltmnf 11223 . . . . . . . . . . 11  |-  ( 0  e.  RR*  ->  -.  0  < -oo )
3129, 30ax-mp 5 . . . . . . . . . 10  |-  -.  0  < -oo
32 mnfxr 11208 . . . . . . . . . . . 12  |- -oo  e.  RR*
33 xbln0 20124 . . . . . . . . . . . 12  |-  ( ( D  e.  ( *Met `  RR )  /\  y  e.  RR  /\ -oo  e.  RR* )  ->  (
( y ( ball `  D ) -oo )  =/=  (/)  <->  0  < -oo ) )
342, 32, 33mp3an13 1306 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  (
( y ( ball `  D ) -oo )  =/=  (/)  <->  0  < -oo ) )
3534necon1bbid 2702 . . . . . . . . . 10  |-  ( y  e.  RR  ->  ( -.  0  < -oo  <->  ( y
( ball `  D ) -oo )  =  (/) ) )
3631, 35mpbii 211 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y ( ball `  D
) -oo )  =  (/) )
3728, 36sylan9eqr 2517 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  = -oo )  ->  ( y ( ball `  D ) r )  =  (/) )
38 iooid 11442 . . . . . . . . 9  |-  ( 0 (,) 0 )  =  (/)
39 ioorebas 11511 . . . . . . . . 9  |-  ( 0 (,) 0 )  e. 
ran  (,)
4038, 39eqeltrri 2539 . . . . . . . 8  |-  (/)  e.  ran  (,)
4137, 40syl6eqel 2550 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  = -oo )  ->  ( y ( ball `  D ) r )  e.  ran  (,) )
4218, 27, 413jaodan 1285 . . . . . 6  |-  ( ( y  e.  RR  /\  ( r  e.  RR  \/  r  = +oo  \/  r  = -oo ) )  ->  (
y ( ball `  D
) r )  e. 
ran  (,) )
435, 42sylan2b 475 . . . . 5  |-  ( ( y  e.  RR  /\  r  e.  RR* )  -> 
( y ( ball `  D ) r )  e.  ran  (,) )
44 eleq1 2526 . . . . 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 2952 . . 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 3471 1  |-  ran  ( ball `  D )  C_  ran  (,)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    \/ w3o 964    = wceq 1370    e. wcel 1758    =/= wne 2648   E.wrex 2800    C_ wss 3439   (/)c0 3748   ~Pcpw 3971   class class class wbr 4403    X. cxp 4949   ran crn 4952    |` cres 4953    o. ccom 4955    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203   RRcr 9395   0cc0 9396    + caddc 9399   +oocpnf 9529   -oocmnf 9530   RR*cxr 9531    < clt 9532    - cmin 9709   (,)cioo 11414   abscabs 12844   *Metcxmt 17929   Metcme 17930   ballcbl 17931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-q 11068  df-rp 11106  df-xneg 11203  df-xadd 11204  df-xmul 11205  df-ioo 11418  df-seq 11927  df-exp 11986  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-psmet 17937  df-xmet 17938  df-met 17939  df-bl 17940
This theorem is referenced by:  tgioo  20508
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