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Theorem blssioo 18317
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 18313 . . . 4  |-  D  e.  ( * Met `  RR )
3 blrn 17978 . . . 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 8 . . 3  |-  ( z  e.  ran  ( ball `  D )  <->  E. y  e.  RR  E. r  e. 
RR*  z  =  ( y ( ball `  D
) r ) )
5 elxr 10474 . . . . . 6  |-  ( r  e.  RR*  <->  ( r  e.  RR  \/  r  = 
+oo  \/  r  =  -oo ) )
61bl2ioo 18314 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y ( ball `  D ) r )  =  ( ( y  -  r ) (,) ( y  +  r ) ) )
7 resubcl 9127 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y  -  r
)  e.  RR )
8 readdcl 8836 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y  +  r )  e.  RR )
9 rexr 8893 . . . . . . . . . 10  |-  ( ( y  -  r )  e.  RR  ->  (
y  -  r )  e.  RR* )
10 rexr 8893 . . . . . . . . . 10  |-  ( ( y  +  r )  e.  RR  ->  (
y  +  r )  e.  RR* )
11 ioof 10757 . . . . . . . . . . . 12  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
12 ffn 5405 . . . . . . . . . . . 12  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
1311, 12ax-mp 8 . . . . . . . . . . 11  |-  (,)  Fn  ( RR*  X.  RR* )
14 fnovrn 6011 . . . . . . . . . . 11  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  (
y  -  r )  e.  RR*  /\  (
y  +  r )  e.  RR* )  ->  (
( y  -  r
) (,) ( y  +  r ) )  e.  ran  (,) )
1513, 14mp3an1 1264 . . . . . . . . . 10  |-  ( ( ( y  -  r
)  e.  RR*  /\  (
y  +  r )  e.  RR* )  ->  (
( y  -  r
) (,) ( y  +  r ) )  e.  ran  (,) )
169, 10, 15syl2an 463 . . . . . . . . 9  |-  ( ( ( y  -  r
)  e.  RR  /\  ( y  +  r )  e.  RR )  ->  ( ( y  -  r ) (,) ( y  +  r ) )  e.  ran  (,) )
177, 8, 16syl2anc 642 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( ( y  -  r ) (,) (
y  +  r ) )  e.  ran  (,) )
186, 17eqeltrd 2370 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y ( ball `  D ) r )  e.  ran  (,) )
19 oveq2 5882 . . . . . . . . 9  |-  ( r  =  +oo  ->  (
y ( ball `  D
) r )  =  ( y ( ball `  D )  +oo )
)
201remet 18312 . . . . . . . . . 10  |-  D  e.  ( Met `  RR )
21 blpnf 17970 . . . . . . . . . 10  |-  ( ( D  e.  ( Met `  RR )  /\  y  e.  RR )  ->  (
y ( ball `  D
)  +oo )  =  RR )
2220, 21mpan 651 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y ( ball `  D
)  +oo )  =  RR )
2319, 22sylan9eqr 2350 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  =  +oo )  -> 
( y ( ball `  D ) r )  =  RR )
24 ioomax 10740 . . . . . . . . 9  |-  (  -oo (,) 
+oo )  =  RR
25 ioorebas 10761 . . . . . . . . 9  |-  (  -oo (,) 
+oo )  e.  ran  (,)
2624, 25eqeltrri 2367 . . . . . . . 8  |-  RR  e.  ran  (,)
2723, 26syl6eqel 2384 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  =  +oo )  -> 
( y ( ball `  D ) r )  e.  ran  (,) )
28 oveq2 5882 . . . . . . . . 9  |-  ( r  =  -oo  ->  (
y ( ball `  D
) r )  =  ( y ( ball `  D )  -oo )
)
29 0xr 8894 . . . . . . . . . . 11  |-  0  e.  RR*
30 nltmnf 10484 . . . . . . . . . . 11  |-  ( 0  e.  RR*  ->  -.  0  <  -oo )
3129, 30ax-mp 8 . . . . . . . . . 10  |-  -.  0  <  -oo
32 mnfxr 10472 . . . . . . . . . . . 12  |-  -oo  e.  RR*
33 xbln0 17981 . . . . . . . . . . . 12  |-  ( ( D  e.  ( * Met `  RR )  /\  y  e.  RR  /\ 
-oo  e.  RR* )  -> 
( ( y (
ball `  D )  -oo )  =/=  (/)  <->  0  <  -oo ) )
342, 32, 33mp3an13 1268 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  (
( y ( ball `  D )  -oo )  =/=  (/)  <->  0  <  -oo ) )
3534necon1bbid 2513 . . . . . . . . . 10  |-  ( y  e.  RR  ->  ( -.  0  <  -oo  <->  ( y
( ball `  D )  -oo )  =  (/) ) )
3631, 35mpbii 202 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y ( ball `  D
)  -oo )  =  (/) )
3728, 36sylan9eqr 2350 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  =  -oo )  -> 
( y ( ball `  D ) r )  =  (/) )
38 iooid 10700 . . . . . . . . 9  |-  ( 0 (,) 0 )  =  (/)
39 ioorebas 10761 . . . . . . . . 9  |-  ( 0 (,) 0 )  e. 
ran  (,)
4038, 39eqeltrri 2367 . . . . . . . 8  |-  (/)  e.  ran  (,)
4137, 40syl6eqel 2384 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  =  -oo )  -> 
( y ( ball `  D ) r )  e.  ran  (,) )
4218, 27, 413jaodan 1248 . . . . . 6  |-  ( ( y  e.  RR  /\  ( r  e.  RR  \/  r  =  +oo  \/  r  =  -oo )
)  ->  ( y
( ball `  D )
r )  e.  ran  (,) )
435, 42sylan2b 461 . . . . 5  |-  ( ( y  e.  RR  /\  r  e.  RR* )  -> 
( y ( ball `  D ) r )  e.  ran  (,) )
44 eleq1 2356 . . . . 5  |-  ( z  =  ( y (
ball `  D )
r )  ->  (
z  e.  ran  (,)  <->  (
y ( ball `  D
) r )  e. 
ran  (,) ) )
4543, 44syl5ibrcom 213 . . . 4  |-  ( ( y  e.  RR  /\  r  e.  RR* )  -> 
( z  =  ( y ( ball `  D
) r )  -> 
z  e.  ran  (,) ) )
4645rexlimivv 2685 . . 3  |-  ( E. y  e.  RR  E. r  e.  RR*  z  =  ( y ( ball `  D ) r )  ->  z  e.  ran  (,) )
474, 46sylbi 187 . 2  |-  ( z  e.  ran  ( ball `  D )  ->  z  e.  ran  (,) )
4847ssriv 3197 1  |-  ran  ( ball `  D )  C_  ran  (,)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    \/ w3o 933    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   class class class wbr 4039    X. cxp 4703   ran crn 4706    |` cres 4707    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753    + caddc 8756    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882    < clt 8883    - cmin 9053   (,)cioo 10672   abscabs 11735   * Metcxmt 16385   Metcme 16386   ballcbl 16387
This theorem is referenced by:  tgioo  18318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-xmet 16389  df-met 16390  df-bl 16391
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