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Theorem blssioo 21750
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 21746 . . . 4  |-  D  e.  ( *Met `  RR )
3 blrn 21361 . . . 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 11362 . . . . . 6  |-  ( r  e.  RR*  <->  ( r  e.  RR  \/  r  = +oo  \/  r  = -oo ) )
61bl2ioo 21747 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y ( ball `  D ) r )  =  ( ( y  -  r ) (,) ( y  +  r ) ) )
7 resubcl 9884 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y  -  r
)  e.  RR )
8 readdcl 9568 . . . . . . . . 9  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y  +  r )  e.  RR )
9 rexr 9632 . . . . . . . . . 10  |-  ( ( y  -  r )  e.  RR  ->  (
y  -  r )  e.  RR* )
10 rexr 9632 . . . . . . . . . 10  |-  ( ( y  +  r )  e.  RR  ->  (
y  +  r )  e.  RR* )
11 ioof 11678 . . . . . . . . . . . 12  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
12 ffn 5684 . . . . . . . . . . . 12  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
1311, 12ax-mp 5 . . . . . . . . . . 11  |-  (,)  Fn  ( RR*  X.  RR* )
14 fnovrn 6397 . . . . . . . . . . 11  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  (
y  -  r )  e.  RR*  /\  (
y  +  r )  e.  RR* )  ->  (
( y  -  r
) (,) ( y  +  r ) )  e.  ran  (,) )
1513, 14mp3an1 1347 . . . . . . . . . 10  |-  ( ( ( y  -  r
)  e.  RR*  /\  (
y  +  r )  e.  RR* )  ->  (
( y  -  r
) (,) ( y  +  r ) )  e.  ran  (,) )
169, 10, 15syl2an 479 . . . . . . . . 9  |-  ( ( ( y  -  r
)  e.  RR  /\  ( y  +  r )  e.  RR )  ->  ( ( y  -  r ) (,) ( y  +  r ) )  e.  ran  (,) )
177, 8, 16syl2anc 665 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( ( y  -  r ) (,) (
y  +  r ) )  e.  ran  (,) )
186, 17eqeltrd 2501 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  e.  RR )  ->  ( y ( ball `  D ) r )  e.  ran  (,) )
19 oveq2 6252 . . . . . . . . 9  |-  ( r  = +oo  ->  (
y ( ball `  D
) r )  =  ( y ( ball `  D ) +oo )
)
201remet 21745 . . . . . . . . . 10  |-  D  e.  ( Met `  RR )
21 blpnf 21349 . . . . . . . . . 10  |-  ( ( D  e.  ( Met `  RR )  /\  y  e.  RR )  ->  (
y ( ball `  D
) +oo )  =  RR )
2220, 21mpan 674 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y ( ball `  D
) +oo )  =  RR )
2319, 22sylan9eqr 2479 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  = +oo )  ->  ( y ( ball `  D ) r )  =  RR )
24 ioomax 11655 . . . . . . . . 9  |-  ( -oo (,) +oo )  =  RR
25 ioorebas 11682 . . . . . . . . 9  |-  ( -oo (,) +oo )  e.  ran  (,)
2624, 25eqeltrri 2498 . . . . . . . 8  |-  RR  e.  ran  (,)
2723, 26syl6eqel 2509 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  = +oo )  ->  ( y ( ball `  D ) r )  e.  ran  (,) )
28 oveq2 6252 . . . . . . . . 9  |-  ( r  = -oo  ->  (
y ( ball `  D
) r )  =  ( y ( ball `  D ) -oo )
)
29 0xr 9633 . . . . . . . . . . 11  |-  0  e.  RR*
30 nltmnf 11377 . . . . . . . . . . 11  |-  ( 0  e.  RR*  ->  -.  0  < -oo )
3129, 30ax-mp 5 . . . . . . . . . 10  |-  -.  0  < -oo
32 mnfxr 11360 . . . . . . . . . . . 12  |- -oo  e.  RR*
33 xbln0 21366 . . . . . . . . . . . 12  |-  ( ( D  e.  ( *Met `  RR )  /\  y  e.  RR  /\ -oo  e.  RR* )  ->  (
( y ( ball `  D ) -oo )  =/=  (/)  <->  0  < -oo ) )
342, 32, 33mp3an13 1351 . . . . . . . . . . 11  |-  ( y  e.  RR  ->  (
( y ( ball `  D ) -oo )  =/=  (/)  <->  0  < -oo ) )
3534necon1bbid 2635 . . . . . . . . . 10  |-  ( y  e.  RR  ->  ( -.  0  < -oo  <->  ( y
( ball `  D ) -oo )  =  (/) ) )
3631, 35mpbii 214 . . . . . . . . 9  |-  ( y  e.  RR  ->  (
y ( ball `  D
) -oo )  =  (/) )
3728, 36sylan9eqr 2479 . . . . . . . 8  |-  ( ( y  e.  RR  /\  r  = -oo )  ->  ( y ( ball `  D ) r )  =  (/) )
38 iooid 11610 . . . . . . . . 9  |-  ( 0 (,) 0 )  =  (/)
39 ioorebas 11682 . . . . . . . . 9  |-  ( 0 (,) 0 )  e. 
ran  (,)
4038, 39eqeltrri 2498 . . . . . . . 8  |-  (/)  e.  ran  (,)
4137, 40syl6eqel 2509 . . . . . . 7  |-  ( ( y  e.  RR  /\  r  = -oo )  ->  ( y ( ball `  D ) r )  e.  ran  (,) )
4218, 27, 413jaodan 1330 . . . . . 6  |-  ( ( y  e.  RR  /\  ( r  e.  RR  \/  r  = +oo  \/  r  = -oo ) )  ->  (
y ( ball `  D
) r )  e. 
ran  (,) )
435, 42sylan2b 477 . . . . 5  |-  ( ( y  e.  RR  /\  r  e.  RR* )  -> 
( y ( ball `  D ) r )  e.  ran  (,) )
44 eleq1 2489 . . . . 5  |-  ( z  =  ( y (
ball `  D )
r )  ->  (
z  e.  ran  (,)  <->  (
y ( ball `  D
) r )  e. 
ran  (,) ) )
4543, 44syl5ibrcom 225 . . . 4  |-  ( ( y  e.  RR  /\  r  e.  RR* )  -> 
( z  =  ( y ( ball `  D
) r )  -> 
z  e.  ran  (,) ) )
4645rexlimivv 2856 . . 3  |-  ( E. y  e.  RR  E. r  e.  RR*  z  =  ( y ( ball `  D ) r )  ->  z  e.  ran  (,) )
474, 46sylbi 198 . 2  |-  ( z  e.  ran  ( ball `  D )  ->  z  e.  ran  (,) )
4847ssriv 3406 1  |-  ran  ( ball `  D )  C_  ran  (,)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    /\ wa 370    \/ w3o 981    = wceq 1437    e. wcel 1872    =/= wne 2594   E.wrex 2710    C_ wss 3374   (/)c0 3699   ~Pcpw 3919   class class class wbr 4361    X. cxp 4789   ran crn 4792    |` cres 4793    o. ccom 4795    Fn wfn 5534   -->wf 5535   ` cfv 5539  (class class class)co 6244   RRcr 9484   0cc0 9485    + caddc 9488   +oocpnf 9618   -oocmnf 9619   RR*cxr 9620    < clt 9621    - cmin 9806   (,)cioo 11581   abscabs 13236   *Metcxmt 18893   Metcme 18894   ballcbl 18895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562  ax-pre-sup 9563
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-om 6646  df-1st 6746  df-2nd 6747  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-er 7313  df-map 7424  df-en 7520  df-dom 7521  df-sdom 7522  df-sup 7904  df-inf 7905  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-div 10216  df-nn 10556  df-2 10614  df-3 10615  df-n0 10816  df-z 10884  df-uz 11106  df-q 11211  df-rp 11249  df-xneg 11355  df-xadd 11356  df-xmul 11357  df-ioo 11585  df-seq 12159  df-exp 12218  df-cj 13101  df-re 13102  df-im 13103  df-sqrt 13237  df-abs 13238  df-psmet 18900  df-xmet 18901  df-met 18902  df-bl 18903
This theorem is referenced by:  tgioo  21751
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