MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  blssioo Structured version   Unicode version

Theorem blssioo 18827
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 18823 . . . 4  |-  D  e.  ( * Met `  RR )
3 blrn 18440 . . . 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 10717 . . . . . 6  |-  ( r  e.  RR*  <->  ( r  e.  RR  \/  r  = 
+oo  \/  r  =  -oo ) )
61bl2ioo 18824 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y ( ball `  D ) r )  =  ( ( y  -  r ) (,) ( y  +  r ) ) )
7 resubcl 9366 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y  -  r
)  e.  RR )
8 readdcl 9074 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y  +  r )  e.  RR )
9 rexr 9131 . . . . . . . . . 10  |-  ( ( y  -  r )  e.  RR  ->  (
y  -  r )  e.  RR* )
10 rexr 9131 . . . . . . . . . 10  |-  ( ( y  +  r )  e.  RR  ->  (
y  +  r )  e.  RR* )
11 ioof 11003 . . . . . . . . . . . 12  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
12 ffn 5592 . . . . . . . . . . . 12  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
1311, 12ax-mp 8 . . . . . . . . . . 11  |-  (,)  Fn  ( RR*  X.  RR* )
14 fnovrn 6222 . . . . . . . . . . 11  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  (
y  -  r )  e.  RR*  /\  (
y  +  r )  e.  RR* )  ->  (
( y  -  r
) (,) ( y  +  r ) )  e.  ran  (,) )
1513, 14mp3an1 1267 . . . . . . . . . 10  |-  ( ( ( y  -  r
)  e.  RR*  /\  (
y  +  r )  e.  RR* )  ->  (
( y  -  r
) (,) ( y  +  r ) )  e.  ran  (,) )
169, 10, 15syl2an 465 . . . . . . . . 9  |-  ( ( ( y  -  r
)  e.  RR  /\  ( y  +  r )  e.  RR )  ->  ( ( y  -  r ) (,) ( y  +  r ) )  e.  ran  (,) )
177, 8, 16syl2anc 644 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( ( y  -  r ) (,) (
y  +  r ) )  e.  ran  (,) )
186, 17eqeltrd 2511 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y ( ball `  D ) r )  e.  ran  (,) )
19 oveq2 6090 . . . . . . . . 9  |-  ( r  =  +oo  ->  (
y ( ball `  D
) r )  =  ( y ( ball `  D )  +oo )
)
201remet 18822 . . . . . . . . . 10  |-  D  e.  ( Met `  RR )
21 blpnf 18428 . . . . . . . . . 10  |-  ( ( D  e.  ( Met `  RR )  /\  y  e.  RR )  ->  (
y ( ball `  D
)  +oo )  =  RR )
2220, 21mpan 653 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y ( ball `  D
)  +oo )  =  RR )
2319, 22sylan9eqr 2491 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  =  +oo )  -> 
( y ( ball `  D ) r )  =  RR )
24 ioomax 10986 . . . . . . . . 9  |-  (  -oo (,) 
+oo )  =  RR
25 ioorebas 11007 . . . . . . . . 9  |-  (  -oo (,) 
+oo )  e.  ran  (,)
2624, 25eqeltrri 2508 . . . . . . . 8  |-  RR  e.  ran  (,)
2723, 26syl6eqel 2525 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  =  +oo )  -> 
( y ( ball `  D ) r )  e.  ran  (,) )
28 oveq2 6090 . . . . . . . . 9  |-  ( r  =  -oo  ->  (
y ( ball `  D
) r )  =  ( y ( ball `  D )  -oo )
)
29 0xr 9132 . . . . . . . . . . 11  |-  0  e.  RR*
30 nltmnf 10727 . . . . . . . . . . 11  |-  ( 0  e.  RR*  ->  -.  0  <  -oo )
3129, 30ax-mp 8 . . . . . . . . . 10  |-  -.  0  <  -oo
32 mnfxr 10715 . . . . . . . . . . . 12  |-  -oo  e.  RR*
33 xbln0 18445 . . . . . . . . . . . 12  |-  ( ( D  e.  ( * Met `  RR )  /\  y  e.  RR  /\ 
-oo  e.  RR* )  -> 
( ( y (
ball `  D )  -oo )  =/=  (/)  <->  0  <  -oo ) )
342, 32, 33mp3an13 1271 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  (
( y ( ball `  D )  -oo )  =/=  (/)  <->  0  <  -oo ) )
3534necon1bbid 2659 . . . . . . . . . 10  |-  ( y  e.  RR  ->  ( -.  0  <  -oo  <->  ( y
( ball `  D )  -oo )  =  (/) ) )
3631, 35mpbii 204 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y ( ball `  D
)  -oo )  =  (/) )
3728, 36sylan9eqr 2491 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  =  -oo )  -> 
( y ( ball `  D ) r )  =  (/) )
38 iooid 10945 . . . . . . . . 9  |-  ( 0 (,) 0 )  =  (/)
39 ioorebas 11007 . . . . . . . . 9  |-  ( 0 (,) 0 )  e. 
ran  (,)
4038, 39eqeltrri 2508 . . . . . . . 8  |-  (/)  e.  ran  (,)
4137, 40syl6eqel 2525 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  =  -oo )  -> 
( y ( ball `  D ) r )  e.  ran  (,) )
4218, 27, 413jaodan 1251 . . . . . 6  |-  ( ( y  e.  RR  /\  ( r  e.  RR  \/  r  =  +oo  \/  r  =  -oo )
)  ->  ( y
( ball `  D )
r )  e.  ran  (,) )
435, 42sylan2b 463 . . . . 5  |-  ( ( y  e.  RR  /\  r  e.  RR* )  -> 
( y ( ball `  D ) r )  e.  ran  (,) )
44 eleq1 2497 . . . . 5  |-  ( z  =  ( y (
ball `  D )
r )  ->  (
z  e.  ran  (,)  <->  (
y ( ball `  D
) r )  e. 
ran  (,) ) )
4543, 44syl5ibrcom 215 . . . 4  |-  ( ( y  e.  RR  /\  r  e.  RR* )  -> 
( z  =  ( y ( ball `  D
) r )  -> 
z  e.  ran  (,) ) )
4645rexlimivv 2836 . . 3  |-  ( E. y  e.  RR  E. r  e.  RR*  z  =  ( y ( ball `  D ) r )  ->  z  e.  ran  (,) )
474, 46sylbi 189 . 2  |-  ( z  e.  ran  ( ball `  D )  ->  z  e.  ran  (,) )
4847ssriv 3353 1  |-  ran  ( ball `  D )  C_  ran  (,)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    /\ wa 360    \/ w3o 936    = wceq 1653    e. wcel 1726    =/= wne 2600   E.wrex 2707    C_ wss 3321   (/)c0 3629   ~Pcpw 3800   class class class wbr 4213    X. cxp 4877   ran crn 4880    |` cres 4881    o. ccom 4883    Fn wfn 5450   -->wf 5451   ` cfv 5455  (class class class)co 6082   RRcr 8990   0cc0 8991    + caddc 8994    +oocpnf 9118    -oocmnf 9119   RR*cxr 9120    < clt 9121    - cmin 9292   (,)cioo 10917   abscabs 12040   * Metcxmt 16687   Metcme 16688   ballcbl 16689
This theorem is referenced by:  tgioo  18828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-q 10576  df-rp 10614  df-xneg 10711  df-xadd 10712  df-xmul 10713  df-ioo 10921  df-seq 11325  df-exp 11384  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-psmet 16695  df-xmet 16696  df-met 16697  df-bl 16698
  Copyright terms: Public domain W3C validator