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Theorem bl2ioo 12522
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 12519 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( A D x )  =  ( abs `  ( A  -  x
) ) )
3 recn 7671 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 7671 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  CC )
5 abssub 10759 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( abs `  ( A  -  x )
)  =  ( abs `  ( x  -  A
) ) )
63, 4, 5syl2an 285 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( abs `  ( A  -  x )
)  =  ( abs `  ( x  -  A
) ) )
72, 6eqtrd 2145 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( A D x )  =  ( abs `  ( x  -  A
) ) )
87breq1d 3903 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( A D x )  <  B  <->  ( abs `  ( x  -  A ) )  <  B ) )
98adantlr 466 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  ->  ( ( A D x )  < 
B  <->  ( abs `  (
x  -  A ) )  <  B ) )
10 absdiflt 10750 . . . . . . . 8  |-  ( ( x  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( abs `  (
x  -  A ) )  <  B  <->  ( ( A  -  B )  <  x  /\  x  < 
( A  +  B
) ) ) )
11103expb 1163 . . . . . . 7  |-  ( ( x  e.  RR  /\  ( A  e.  RR  /\  B  e.  RR ) )  ->  ( ( abs `  ( x  -  A ) )  < 
B  <->  ( ( A  -  B )  < 
x  /\  x  <  ( A  +  B ) ) ) )
1211ancoms 266 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  ->  ( ( abs `  ( x  -  A
) )  <  B  <->  ( ( A  -  B
)  <  x  /\  x  <  ( A  +  B ) ) ) )
139, 12bitrd 187 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  ->  ( ( A D x )  < 
B  <->  ( ( A  -  B )  < 
x  /\  x  <  ( A  +  B ) ) ) )
1413pm5.32da 445 . . . 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 947 . . . 4  |-  ( ( x  e.  RR  /\  ( A  -  B
)  <  x  /\  x  <  ( A  +  B ) )  <->  ( x  e.  RR  /\  ( ( A  -  B )  <  x  /\  x  <  ( A  +  B
) ) ) )
1614, 15syl6bbr 197 . . 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 7729 . . . 4  |-  ( B  e.  RR  ->  B  e.  RR* )
181rexmet 12521 . . . . 5  |-  D  e.  ( *Met `  RR )
19 elbl 12374 . . . . 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 1283 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( x  e.  ( A ( ball `  D
) B )  <->  ( x  e.  RR  /\  ( A D x )  < 
B ) ) )
2117, 20sylan2 282 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A ( ball `  D
) B )  <->  ( x  e.  RR  /\  ( A D x )  < 
B ) ) )
22 resubcl 7943 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
23 readdcl 7664 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
24 rexr 7729 . . . . 5  |-  ( ( A  -  B )  e.  RR  ->  ( A  -  B )  e.  RR* )
25 rexr 7729 . . . . 5  |-  ( ( A  +  B )  e.  RR  ->  ( A  +  B )  e.  RR* )
26 elioo2 9591 . . . . 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 285 . . . 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 406 . . 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 219 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A ( ball `  D
) B )  <->  x  e.  ( ( A  -  B ) (,) ( A  +  B )
) ) )
3029eqrdv 2111 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    /\ wa 103    <-> wb 104    /\ w3a 943    = wceq 1312    e. wcel 1461   class class class wbr 3893    X. cxp 4495    |` cres 4499    o. ccom 4501   ` cfv 5079  (class class class)co 5726   CCcc 7539   RRcr 7540    + caddc 7544   RR*cxr 7717    < clt 7718    - cmin 7850   (,)cioo 9558   abscabs 10655   *Metcxmet 11986   ballcbl 11988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655  ax-pre-mulgt0 7656  ax-pre-mulext 7657  ax-arch 7658  ax-caucvg 7659
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-if 3439  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-ilim 4249  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-frec 6240  df-map 6496  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-reap 8249  df-ap 8256  df-div 8340  df-inn 8625  df-2 8683  df-3 8684  df-4 8685  df-n0 8876  df-z 8953  df-uz 9223  df-rp 9338  df-xadd 9447  df-ioo 9562  df-seqfrec 10106  df-exp 10180  df-cj 10501  df-re 10502  df-im 10503  df-rsqrt 10656  df-abs 10657  df-psmet 11993  df-xmet 11994  df-met 11995  df-bl 11996
This theorem is referenced by:  ioo2bl  12523  blssioo  12525  tgioo  12526
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