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Theorem ioo2bl 20483
Description: An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
Hypothesis
Ref Expression
remet.1  |-  D  =  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )
Assertion
Ref Expression
ioo2bl  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A (,) B
)  =  ( ( ( A  +  B
)  /  2 ) ( ball `  D
) ( ( B  -  A )  / 
2 ) ) )

Proof of Theorem ioo2bl
StepHypRef Expression
1 readdcl 9463 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  +  A
)  e.  RR )
21ancoms 453 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  +  A
)  e.  RR )
32rehalfcld 10669 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( B  +  A )  /  2
)  e.  RR )
4 resubcl 9771 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  -  A
)  e.  RR )
54ancoms 453 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  -  A
)  e.  RR )
65rehalfcld 10669 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( B  -  A )  /  2
)  e.  RR )
7 remet.1 . . . 4  |-  D  =  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )
87bl2ioo 20482 . . 3  |-  ( ( ( ( B  +  A )  /  2
)  e.  RR  /\  ( ( B  -  A )  /  2
)  e.  RR )  ->  ( ( ( B  +  A )  /  2 ) (
ball `  D )
( ( B  -  A )  /  2
) )  =  ( ( ( ( B  +  A )  / 
2 )  -  (
( B  -  A
)  /  2 ) ) (,) ( ( ( B  +  A
)  /  2 )  +  ( ( B  -  A )  / 
2 ) ) ) )
93, 6, 8syl2anc 661 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( B  +  A )  / 
2 ) ( ball `  D ) ( ( B  -  A )  /  2 ) )  =  ( ( ( ( B  +  A
)  /  2 )  -  ( ( B  -  A )  / 
2 ) ) (,) ( ( ( B  +  A )  / 
2 )  +  ( ( B  -  A
)  /  2 ) ) ) )
10 recn 9470 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
11 recn 9470 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
12 addcom 9653 . . . . 5  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B  +  A
)  =  ( A  +  B ) )
1310, 11, 12syl2anr 478 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  +  A
)  =  ( A  +  B ) )
1413oveq1d 6202 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( B  +  A )  /  2
)  =  ( ( A  +  B )  /  2 ) )
1514oveq1d 6202 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( B  +  A )  / 
2 ) ( ball `  D ) ( ( B  -  A )  /  2 ) )  =  ( ( ( A  +  B )  /  2 ) (
ball `  D )
( ( B  -  A )  /  2
) ) )
16 halfaddsub 10656 . . . . 5  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( ( ( ( B  +  A )  /  2 )  +  ( ( B  -  A )  /  2
) )  =  B  /\  ( ( ( B  +  A )  /  2 )  -  ( ( B  -  A )  /  2
) )  =  A ) )
1710, 11, 16syl2anr 478 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( B  +  A )  /  2 )  +  ( ( B  -  A )  /  2
) )  =  B  /\  ( ( ( B  +  A )  /  2 )  -  ( ( B  -  A )  /  2
) )  =  A ) )
1817simprd 463 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( B  +  A )  / 
2 )  -  (
( B  -  A
)  /  2 ) )  =  A )
1917simpld 459 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( B  +  A )  / 
2 )  +  ( ( B  -  A
)  /  2 ) )  =  B )
2018, 19oveq12d 6205 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( B  +  A )  /  2 )  -  ( ( B  -  A )  /  2
) ) (,) (
( ( B  +  A )  /  2
)  +  ( ( B  -  A )  /  2 ) ) )  =  ( A (,) B ) )
219, 15, 203eqtr3rd 2500 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A (,) B
)  =  ( ( ( A  +  B
)  /  2 ) ( ball `  D
) ( ( B  -  A )  / 
2 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    X. cxp 4933    |` cres 4937    o. ccom 4939   ` cfv 5513  (class class class)co 6187   CCcc 9378   RRcr 9379    + caddc 9383    - cmin 9693    / cdiv 10091   2c2 10469   (,)cioo 11398   abscabs 12822   ballcbl 17909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457  ax-pre-sup 9458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-er 7198  df-map 7313  df-en 7408  df-dom 7409  df-sdom 7410  df-sup 7789  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-div 10092  df-nn 10421  df-2 10478  df-3 10479  df-n0 10678  df-z 10745  df-uz 10960  df-rp 11090  df-xadd 11188  df-ioo 11402  df-seq 11905  df-exp 11964  df-cj 12687  df-re 12688  df-im 12689  df-sqr 12823  df-abs 12824  df-psmet 17915  df-xmet 17916  df-met 17917  df-bl 17918
This theorem is referenced by:  ioo2blex  20484
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