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Theorem bl2ioo 18314
Description: A ball in terms of an open interval of reals. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
Hypothesis
Ref Expression
remet.1  |-  D  =  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )
Assertion
Ref Expression
bl2ioo  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A ( ball `  D ) B )  =  ( ( A  -  B ) (,) ( A  +  B
) ) )

Proof of Theorem bl2ioo
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 remet.1 . . . . . . . . . 10  |-  D  =  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )
21remetdval 18311 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( A D x )  =  ( abs `  ( A  -  x
) ) )
3 recn 8843 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 8843 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  CC )
5 abssub 11826 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( abs `  ( A  -  x )
)  =  ( abs `  ( x  -  A
) ) )
63, 4, 5syl2an 463 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( abs `  ( A  -  x )
)  =  ( abs `  ( x  -  A
) ) )
72, 6eqtrd 2328 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( A D x )  =  ( abs `  ( x  -  A
) ) )
87breq1d 4049 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( A D x )  <  B  <->  ( abs `  ( x  -  A ) )  <  B ) )
98adantlr 695 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  ->  ( ( A D x )  < 
B  <->  ( abs `  (
x  -  A ) )  <  B ) )
10 absdiflt 11817 . . . . . . . 8  |-  ( ( x  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( abs `  (
x  -  A ) )  <  B  <->  ( ( A  -  B )  <  x  /\  x  < 
( A  +  B
) ) ) )
11103expb 1152 . . . . . . 7  |-  ( ( x  e.  RR  /\  ( A  e.  RR  /\  B  e.  RR ) )  ->  ( ( abs `  ( x  -  A ) )  < 
B  <->  ( ( A  -  B )  < 
x  /\  x  <  ( A  +  B ) ) ) )
1211ancoms 439 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  ->  ( ( abs `  ( x  -  A
) )  <  B  <->  ( ( A  -  B
)  <  x  /\  x  <  ( A  +  B ) ) ) )
139, 12bitrd 244 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  ->  ( ( A D x )  < 
B  <->  ( ( A  -  B )  < 
x  /\  x  <  ( A  +  B ) ) ) )
1413pm5.32da 622 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( x  e.  RR  /\  ( A D x )  < 
B )  <->  ( x  e.  RR  /\  ( ( A  -  B )  <  x  /\  x  <  ( A  +  B
) ) ) ) )
15 3anass 938 . . . 4  |-  ( ( x  e.  RR  /\  ( A  -  B
)  <  x  /\  x  <  ( A  +  B ) )  <->  ( x  e.  RR  /\  ( ( A  -  B )  <  x  /\  x  <  ( A  +  B
) ) ) )
1614, 15syl6bbr 254 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( x  e.  RR  /\  ( A D x )  < 
B )  <->  ( x  e.  RR  /\  ( A  -  B )  < 
x  /\  x  <  ( A  +  B ) ) ) )
17 rexr 8893 . . . 4  |-  ( B  e.  RR  ->  B  e.  RR* )
181rexmet 18313 . . . . 5  |-  D  e.  ( * Met `  RR )
19 elbl 17965 . . . . 5  |-  ( ( D  e.  ( * Met `  RR )  /\  A  e.  RR  /\  B  e.  RR* )  ->  ( x  e.  ( A ( ball `  D
) B )  <->  ( x  e.  RR  /\  ( A D x )  < 
B ) ) )
2018, 19mp3an1 1264 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( x  e.  ( A ( ball `  D
) B )  <->  ( x  e.  RR  /\  ( A D x )  < 
B ) ) )
2117, 20sylan2 460 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A ( ball `  D
) B )  <->  ( x  e.  RR  /\  ( A D x )  < 
B ) ) )
22 resubcl 9127 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
23 readdcl 8836 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
24 rexr 8893 . . . . 5  |-  ( ( A  -  B )  e.  RR  ->  ( A  -  B )  e.  RR* )
25 rexr 8893 . . . . 5  |-  ( ( A  +  B )  e.  RR  ->  ( A  +  B )  e.  RR* )
26 elioo2 10713 . . . . 5  |-  ( ( ( A  -  B
)  e.  RR*  /\  ( A  +  B )  e.  RR* )  ->  (
x  e.  ( ( A  -  B ) (,) ( A  +  B ) )  <->  ( x  e.  RR  /\  ( A  -  B )  < 
x  /\  x  <  ( A  +  B ) ) ) )
2724, 25, 26syl2an 463 . . . 4  |-  ( ( ( A  -  B
)  e.  RR  /\  ( A  +  B
)  e.  RR )  ->  ( x  e.  ( ( A  -  B ) (,) ( A  +  B )
)  <->  ( x  e.  RR  /\  ( A  -  B )  < 
x  /\  x  <  ( A  +  B ) ) ) )
2822, 23, 27syl2anc 642 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( ( A  -  B
) (,) ( A  +  B ) )  <-> 
( x  e.  RR  /\  ( A  -  B
)  <  x  /\  x  <  ( A  +  B ) ) ) )
2916, 21, 283bitr4d 276 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A ( ball `  D
) B )  <->  x  e.  ( ( A  -  B ) (,) ( A  +  B )
) ) )
3029eqrdv 2294 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A ( ball `  D ) B )  =  ( ( A  -  B ) (,) ( A  +  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039    X. cxp 4703    |` cres 4707    o. ccom 4709   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752    + caddc 8756   RR*cxr 8882    < clt 8883    - cmin 9053   (,)cioo 10672   abscabs 11735   * Metcxmt 16385   ballcbl 16387
This theorem is referenced by:  ioo2bl  18315  blssioo  18317  tgioo  18318  iccntr  18342  icccmplem2  18344  reconnlem2  18348  opnreen  18352  lebnumii  18480  opnmbllem  18972  lhop  19379  dvcnvre  19382  altretop  25703  opnrebl  26338  opnrebl2  26339
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-rp 10371  df-xadd 10469  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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