MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bl2ioo Structured version   Unicode version

Theorem bl2ioo 20391
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 20388 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( A D x )  =  ( abs `  ( A  -  x
) ) )
3 recn 9393 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 9393 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  CC )
5 abssub 12835 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( abs `  ( A  -  x )
)  =  ( abs `  ( x  -  A
) ) )
63, 4, 5syl2an 477 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( abs `  ( A  -  x )
)  =  ( abs `  ( x  -  A
) ) )
72, 6eqtrd 2475 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( A D x )  =  ( abs `  ( x  -  A
) ) )
87breq1d 4323 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( A D x )  <  B  <->  ( abs `  ( x  -  A ) )  <  B ) )
98adantlr 714 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  ->  ( ( A D x )  < 
B  <->  ( abs `  (
x  -  A ) )  <  B ) )
10 absdiflt 12826 . . . . . . . 8  |-  ( ( x  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( abs `  (
x  -  A ) )  <  B  <->  ( ( A  -  B )  <  x  /\  x  < 
( A  +  B
) ) ) )
11103expb 1188 . . . . . . 7  |-  ( ( x  e.  RR  /\  ( A  e.  RR  /\  B  e.  RR ) )  ->  ( ( abs `  ( x  -  A ) )  < 
B  <->  ( ( A  -  B )  < 
x  /\  x  <  ( A  +  B ) ) ) )
1211ancoms 453 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  ->  ( ( abs `  ( x  -  A
) )  <  B  <->  ( ( A  -  B
)  <  x  /\  x  <  ( A  +  B ) ) ) )
139, 12bitrd 253 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  ->  ( ( A D x )  < 
B  <->  ( ( A  -  B )  < 
x  /\  x  <  ( A  +  B ) ) ) )
1413pm5.32da 641 . . . 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 969 . . . 4  |-  ( ( x  e.  RR  /\  ( A  -  B
)  <  x  /\  x  <  ( A  +  B ) )  <->  ( x  e.  RR  /\  ( ( A  -  B )  <  x  /\  x  <  ( A  +  B
) ) ) )
1614, 15syl6bbr 263 . . 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 9450 . . . 4  |-  ( B  e.  RR  ->  B  e.  RR* )
181rexmet 20390 . . . . 5  |-  D  e.  ( *Met `  RR )
19 elbl 19985 . . . . 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 1301 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( x  e.  ( A ( ball `  D
) B )  <->  ( x  e.  RR  /\  ( A D x )  < 
B ) ) )
2117, 20sylan2 474 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A ( ball `  D
) B )  <->  ( x  e.  RR  /\  ( A D x )  < 
B ) ) )
22 resubcl 9694 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
23 readdcl 9386 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
24 rexr 9450 . . . . 5  |-  ( ( A  -  B )  e.  RR  ->  ( A  -  B )  e.  RR* )
25 rexr 9450 . . . . 5  |-  ( ( A  +  B )  e.  RR  ->  ( A  +  B )  e.  RR* )
26 elioo2 11362 . . . . 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 477 . . . 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 661 . . 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 285 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A ( ball `  D
) B )  <->  x  e.  ( ( A  -  B ) (,) ( A  +  B )
) ) )
3029eqrdv 2441 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A ( ball `  D ) B )  =  ( ( A  -  B ) (,) ( A  +  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4313    X. cxp 4859    |` cres 4863    o. ccom 4865   ` cfv 5439  (class class class)co 6112   CCcc 9301   RRcr 9302    + caddc 9306   RR*cxr 9438    < clt 9439    - cmin 9616   (,)cioo 11321   abscabs 12744   *Metcxmt 17823   ballcbl 17825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-xadd 11111  df-ioo 11325  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-psmet 17831  df-xmet 17832  df-met 17833  df-bl 17834
This theorem is referenced by:  ioo2bl  20392  blssioo  20394  tgioo  20395  iccntr  20420  icccmplem2  20422  reconnlem2  20426  opnreen  20430  lebnumii  20560  opnmbllem  21103  lhop  21510  dvcnvre  21513  dya2icoseg2  26715  opnmbllem0  28453  opnrebl  28541  opnrebl2  28542
  Copyright terms: Public domain W3C validator