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Theorem bl2ioo 14527
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 14524 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( A D x )  =  ( abs `  ( A  -  x
) ) )
3 recn 7979 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
4 recn 7979 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  CC )
5 abssub 11151 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  CC )  ->  ( abs `  ( A  -  x )
)  =  ( abs `  ( x  -  A
) ) )
63, 4, 5syl2an 289 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( abs `  ( A  -  x )
)  =  ( abs `  ( x  -  A
) ) )
72, 6eqtrd 2222 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( A D x )  =  ( abs `  ( x  -  A
) ) )
87breq1d 4031 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  RR )  ->  ( ( A D x )  <  B  <->  ( abs `  ( x  -  A ) )  <  B ) )
98adantlr 477 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  ->  ( ( A D x )  < 
B  <->  ( abs `  (
x  -  A ) )  <  B ) )
10 absdiflt 11142 . . . . . . . 8  |-  ( ( x  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( abs `  (
x  -  A ) )  <  B  <->  ( ( A  -  B )  <  x  /\  x  < 
( A  +  B
) ) ) )
11103expb 1206 . . . . . . 7  |-  ( ( x  e.  RR  /\  ( A  e.  RR  /\  B  e.  RR ) )  ->  ( ( abs `  ( x  -  A ) )  < 
B  <->  ( ( A  -  B )  < 
x  /\  x  <  ( A  +  B ) ) ) )
1211ancoms 268 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  ->  ( ( abs `  ( x  -  A
) )  <  B  <->  ( ( A  -  B
)  <  x  /\  x  <  ( A  +  B ) ) ) )
139, 12bitrd 188 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  ->  ( ( A D x )  < 
B  <->  ( ( A  -  B )  < 
x  /\  x  <  ( A  +  B ) ) ) )
1413pm5.32da 452 . . . 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 984 . . . 4  |-  ( ( x  e.  RR  /\  ( A  -  B
)  <  x  /\  x  <  ( A  +  B ) )  <->  ( x  e.  RR  /\  ( ( A  -  B )  <  x  /\  x  <  ( A  +  B
) ) ) )
1614, 15bitr4di 198 . . 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 8038 . . . 4  |-  ( B  e.  RR  ->  B  e.  RR* )
181rexmet 14526 . . . . 5  |-  D  e.  ( *Met `  RR )
19 elbl 14376 . . . . 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 1335 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( x  e.  ( A ( ball `  D
) B )  <->  ( x  e.  RR  /\  ( A D x )  < 
B ) ) )
2117, 20sylan2 286 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A ( ball `  D
) B )  <->  ( x  e.  RR  /\  ( A D x )  < 
B ) ) )
22 resubcl 8256 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  RR )
23 readdcl 7972 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
24 rexr 8038 . . . . 5  |-  ( ( A  -  B )  e.  RR  ->  ( A  -  B )  e.  RR* )
25 rexr 8038 . . . . 5  |-  ( ( A  +  B )  e.  RR  ->  ( A  +  B )  e.  RR* )
26 elioo2 9957 . . . . 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 289 . . . 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 411 . . 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 220 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  ( A ( ball `  D
) B )  <->  x  e.  ( ( A  -  B ) (,) ( A  +  B )
) ) )
3029eqrdv 2187 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 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   class class class wbr 4021    X. cxp 4645    |` cres 4649    o. ccom 4651   ` cfv 5238  (class class class)co 5900   CCcc 7844   RRcr 7845    + caddc 7849   RR*cxr 8026    < clt 8027    - cmin 8163   (,)cioo 9924   abscabs 11047   *Metcxmet 13874   ballcbl 13876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-mulrcl 7945  ax-addcom 7946  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-1rid 7953  ax-0id 7954  ax-rnegex 7955  ax-precex 7956  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-apti 7961  ax-pre-ltadd 7962  ax-pre-mulgt0 7963  ax-pre-mulext 7964  ax-arch 7965  ax-caucvg 7966
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-po 4317  df-iso 4318  df-iord 4387  df-on 4389  df-ilim 4390  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-frec 6420  df-map 6680  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-reap 8567  df-ap 8574  df-div 8665  df-inn 8955  df-2 9013  df-3 9014  df-4 9015  df-n0 9212  df-z 9289  df-uz 9564  df-rp 9690  df-xadd 9809  df-ioo 9928  df-seqfrec 10485  df-exp 10560  df-cj 10892  df-re 10893  df-im 10894  df-rsqrt 11048  df-abs 11049  df-psmet 13881  df-xmet 13882  df-met 13883  df-bl 13884
This theorem is referenced by:  ioo2bl  14528  blssioo  14530  tgioo  14531
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