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Theorem blssioo 18301
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 18297 . . . 4  |-  D  e.  ( * Met `  RR )
3 blrn 17962 . . . 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 10458 . . . . . 6  |-  ( r  e.  RR*  <->  ( r  e.  RR  \/  r  = 
+oo  \/  r  =  -oo ) )
61bl2ioo 18298 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y ( ball `  D ) r )  =  ( ( y  -  r ) (,) ( y  +  r ) ) )
7 resubcl 9111 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y  -  r
)  e.  RR )
8 readdcl 8820 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y  +  r )  e.  RR )
9 rexr 8877 . . . . . . . . . 10  |-  ( ( y  -  r )  e.  RR  ->  (
y  -  r )  e.  RR* )
10 rexr 8877 . . . . . . . . . 10  |-  ( ( y  +  r )  e.  RR  ->  (
y  +  r )  e.  RR* )
11 ioof 10741 . . . . . . . . . . . 12  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
12 ffn 5389 . . . . . . . . . . . 12  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
1311, 12ax-mp 8 . . . . . . . . . . 11  |-  (,)  Fn  ( RR*  X.  RR* )
14 fnovrn 5995 . . . . . . . . . . 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 2357 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y ( ball `  D ) r )  e.  ran  (,) )
19 oveq2 5866 . . . . . . . . 9  |-  ( r  =  +oo  ->  (
y ( ball `  D
) r )  =  ( y ( ball `  D )  +oo )
)
201remet 18296 . . . . . . . . . 10  |-  D  e.  ( Met `  RR )
21 blpnf 17954 . . . . . . . . . 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 2337 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  =  +oo )  -> 
( y ( ball `  D ) r )  =  RR )
24 ioomax 10724 . . . . . . . . 9  |-  (  -oo (,) 
+oo )  =  RR
25 ioorebas 10745 . . . . . . . . 9  |-  (  -oo (,) 
+oo )  e.  ran  (,)
2624, 25eqeltrri 2354 . . . . . . . 8  |-  RR  e.  ran  (,)
2723, 26syl6eqel 2371 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  =  +oo )  -> 
( y ( ball `  D ) r )  e.  ran  (,) )
28 oveq2 5866 . . . . . . . . 9  |-  ( r  =  -oo  ->  (
y ( ball `  D
) r )  =  ( y ( ball `  D )  -oo )
)
29 0xr 8878 . . . . . . . . . . 11  |-  0  e.  RR*
30 nltmnf 10468 . . . . . . . . . . 11  |-  ( 0  e.  RR*  ->  -.  0  <  -oo )
3129, 30ax-mp 8 . . . . . . . . . 10  |-  -.  0  <  -oo
32 mnfxr 10456 . . . . . . . . . . . 12  |-  -oo  e.  RR*
33 xbln0 17965 . . . . . . . . . . . 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 2500 . . . . . . . . . 10  |-  ( y  e.  RR  ->  ( -.  0  <  -oo  <->  ( y
( ball `  D )  -oo )  =  (/) ) )
3631, 35mpbii 202 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y ( ball `  D
)  -oo )  =  (/) )
3728, 36sylan9eqr 2337 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  =  -oo )  -> 
( y ( ball `  D ) r )  =  (/) )
38 iooid 10684 . . . . . . . . 9  |-  ( 0 (,) 0 )  =  (/)
39 ioorebas 10745 . . . . . . . . 9  |-  ( 0 (,) 0 )  e. 
ran  (,)
4038, 39eqeltrri 2354 . . . . . . . 8  |-  (/)  e.  ran  (,)
4137, 40syl6eqel 2371 . . . . . . 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 2343 . . . . 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 2672 . . 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 3184 1  |-  ran  ( ball `  D )  C_  ran  (,)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   class class class wbr 4023    X. cxp 4687   ran crn 4690    |` cres 4691    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737    + caddc 8740    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    - cmin 9037   (,)cioo 10656   abscabs 11719   * Metcxmt 16369   Metcme 16370   ballcbl 16371
This theorem is referenced by:  tgioo  18302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-xmet 16373  df-met 16374  df-bl 16375
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