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Theorem ioo2bl 21467
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 9564 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  +  A
)  e.  RR )
21ancoms 451 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  +  A
)  e.  RR )
32rehalfcld 10781 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( B  +  A )  /  2
)  e.  RR )
4 resubcl 9874 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  -  A
)  e.  RR )
54ancoms 451 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  -  A
)  e.  RR )
65rehalfcld 10781 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( B  -  A )  /  2
)  e.  RR )
7 remet.1 . . . 4  |-  D  =  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )
87bl2ioo 21466 . . 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 659 . 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 9571 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
11 recn 9571 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
12 addcom 9755 . . . . 5  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( B  +  A
)  =  ( A  +  B ) )
1310, 11, 12syl2anr 476 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  +  A
)  =  ( A  +  B ) )
1413oveq1d 6285 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( B  +  A )  /  2
)  =  ( ( A  +  B )  /  2 ) )
1514oveq1d 6285 . 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 10768 . . . . 5  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( ( ( ( B  +  A )  /  2 )  +  ( ( B  -  A )  /  2
) )  =  B  /\  ( ( ( B  +  A )  /  2 )  -  ( ( B  -  A )  /  2
) )  =  A ) )
1710, 11, 16syl2anr 476 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( B  +  A )  /  2 )  +  ( ( B  -  A )  /  2
) )  =  B  /\  ( ( ( B  +  A )  /  2 )  -  ( ( B  -  A )  /  2
) )  =  A ) )
1817simprd 461 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( B  +  A )  / 
2 )  -  (
( B  -  A
)  /  2 ) )  =  A )
1917simpld 457 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( B  +  A )  / 
2 )  +  ( ( B  -  A
)  /  2 ) )  =  B )
2018, 19oveq12d 6288 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( B  +  A )  /  2 )  -  ( ( B  -  A )  /  2
) ) (,) (
( ( B  +  A )  /  2
)  +  ( ( B  -  A )  /  2 ) ) )  =  ( A (,) B ) )
219, 15, 203eqtr3rd 2504 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 367    = wceq 1398    e. wcel 1823    X. cxp 4986    |` cres 4990    o. ccom 4992   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480    + caddc 9484    - cmin 9796    / cdiv 10202   2c2 10581   (,)cioo 11532   abscabs 13152   ballcbl 18603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-xadd 11322  df-ioo 11536  df-seq 12093  df-exp 12152  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612
This theorem is referenced by:  ioo2blex  21468
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