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Theorem cnbl0 13585
Description: Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
Hypothesis
Ref Expression
cnblcld.1  |-  D  =  ( abs  o.  -  )
Assertion
Ref Expression
cnbl0  |-  ( R  e.  RR*  ->  ( `' abs " ( 0 [,) R ) )  =  ( 0 (
ball `  D ) R ) )

Proof of Theorem cnbl0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-3an 980 . . . . . 6  |-  ( ( ( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
)  /\  ( abs `  x )  <  R
)  <->  ( ( ( abs `  x )  e.  RR  /\  0  <_  ( abs `  x
) )  /\  ( abs `  x )  < 
R ) )
2 abscl 11026 . . . . . . . . 9  |-  ( x  e.  CC  ->  ( abs `  x )  e.  RR )
3 absge0 11035 . . . . . . . . 9  |-  ( x  e.  CC  ->  0  <_  ( abs `  x
) )
42, 3jca 306 . . . . . . . 8  |-  ( x  e.  CC  ->  (
( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
) ) )
54adantl 277 . . . . . . 7  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
) ) )
65biantrurd 305 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  <  R  <->  ( (
( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
) )  /\  ( abs `  x )  < 
R ) ) )
71, 6bitr4id 199 . . . . 5  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( ( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
)  /\  ( abs `  x )  <  R
)  <->  ( abs `  x
)  <  R )
)
8 0re 7932 . . . . . 6  |-  0  e.  RR
9 simpl 109 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  R  e.  RR* )
10 elico2 9906 . . . . . 6  |-  ( ( 0  e.  RR  /\  R  e.  RR* )  -> 
( ( abs `  x
)  e.  ( 0 [,) R )  <->  ( ( abs `  x )  e.  RR  /\  0  <_ 
( abs `  x
)  /\  ( abs `  x )  <  R
) ) )
118, 9, 10sylancr 414 . . . . 5  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  ( 0 [,) R )  <->  ( ( abs `  x )  e.  RR  /\  0  <_ 
( abs `  x
)  /\  ( abs `  x )  <  R
) ) )
12 0cn 7924 . . . . . . . . 9  |-  0  e.  CC
13 cnblcld.1 . . . . . . . . . . 11  |-  D  =  ( abs  o.  -  )
1413cnmetdval 13580 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( 0  -  x
) ) )
15 abssub 11076 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( abs `  (
0  -  x ) )  =  ( abs `  ( x  -  0 ) ) )
1614, 15eqtrd 2208 . . . . . . . . 9  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( x  -  0 ) ) )
1712, 16mpan 424 . . . . . . . 8  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  (
x  -  0 ) ) )
18 subid1 8151 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
x  -  0 )  =  x )
1918fveq2d 5511 . . . . . . . 8  |-  ( x  e.  CC  ->  ( abs `  ( x  - 
0 ) )  =  ( abs `  x
) )
2017, 19eqtrd 2208 . . . . . . 7  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  x
) )
2120adantl 277 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
0 D x )  =  ( abs `  x
) )
2221breq1d 4008 . . . . 5  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( 0 D x )  <  R  <->  ( abs `  x )  <  R
) )
237, 11, 223bitr4d 220 . . . 4  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  ( 0 [,) R )  <->  ( 0 D x )  < 
R ) )
2423pm5.32da 452 . . 3  |-  ( R  e.  RR*  ->  ( ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,) R ) )  <->  ( x  e.  CC  /\  ( 0 D x )  < 
R ) ) )
25 absf 11085 . . . . 5  |-  abs : CC
--> RR
26 ffn 5357 . . . . 5  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
2725, 26ax-mp 5 . . . 4  |-  abs  Fn  CC
28 elpreima 5627 . . . 4  |-  ( abs 
Fn  CC  ->  ( x  e.  ( `' abs " ( 0 [,) R
) )  <->  ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,) R ) ) ) )
2927, 28mp1i 10 . . 3  |-  ( R  e.  RR*  ->  ( x  e.  ( `' abs " ( 0 [,) R
) )  <->  ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,) R ) ) ) )
30 cnxmet 13582 . . . . 5  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
3113, 30eqeltri 2248 . . . 4  |-  D  e.  ( *Met `  CC )
32 elbl 13442 . . . 4  |-  ( ( D  e.  ( *Met `  CC )  /\  0  e.  CC  /\  R  e.  RR* )  ->  ( x  e.  ( 0 ( ball `  D
) R )  <->  ( x  e.  CC  /\  ( 0 D x )  < 
R ) ) )
3331, 12, 32mp3an12 1327 . . 3  |-  ( R  e.  RR*  ->  ( x  e.  ( 0 (
ball `  D ) R )  <->  ( x  e.  CC  /\  ( 0 D x )  < 
R ) ) )
3424, 29, 333bitr4d 220 . 2  |-  ( R  e.  RR*  ->  ( x  e.  ( `' abs " ( 0 [,) R
) )  <->  x  e.  ( 0 ( ball `  D ) R ) ) )
3534eqrdv 2173 1  |-  ( R  e.  RR*  ->  ( `' abs " ( 0 [,) R ) )  =  ( 0 (
ball `  D ) R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2146   class class class wbr 3998   `'ccnv 4619   "cima 4623    o. ccom 4624    Fn wfn 5203   -->wf 5204   ` cfv 5208  (class class class)co 5865   CCcc 7784   RRcr 7785   0cc0 7786   RR*cxr 7965    < clt 7966    <_ cle 7967    - cmin 8102   [,)cico 9859   abscabs 10972   *Metcxmet 13031   ballcbl 13033
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-map 6640  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-rp 9623  df-xadd 9742  df-ico 9863  df-seqfrec 10414  df-exp 10488  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-psmet 13038  df-xmet 13039  df-met 13040  df-bl 13041
This theorem is referenced by: (None)
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