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Theorem cnbl0 12456
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 abscl 10663 . . . . . . . . 9  |-  ( x  e.  CC  ->  ( abs `  x )  e.  RR )
2 absge0 10672 . . . . . . . . 9  |-  ( x  e.  CC  ->  0  <_  ( abs `  x
) )
31, 2jca 302 . . . . . . . 8  |-  ( x  e.  CC  ->  (
( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
) ) )
43adantl 273 . . . . . . 7  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
) ) )
54biantrurd 301 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  <  R  <->  ( (
( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
) )  /\  ( abs `  x )  < 
R ) ) )
6 df-3an 932 . . . . . 6  |-  ( ( ( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
)  /\  ( abs `  x )  <  R
)  <->  ( ( ( abs `  x )  e.  RR  /\  0  <_  ( abs `  x
) )  /\  ( abs `  x )  < 
R ) )
75, 6syl6rbbr 198 . . . . 5  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( ( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
)  /\  ( abs `  x )  <  R
)  <->  ( abs `  x
)  <  R )
)
8 0re 7638 . . . . . 6  |-  0  e.  RR
9 simpl 108 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  R  e.  RR* )
10 elico2 9561 . . . . . 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 408 . . . . 5  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  ( 0 [,) R )  <->  ( ( abs `  x )  e.  RR  /\  0  <_ 
( abs `  x
)  /\  ( abs `  x )  <  R
) ) )
12 0cn 7630 . . . . . . . . 9  |-  0  e.  CC
13 cnblcld.1 . . . . . . . . . . 11  |-  D  =  ( abs  o.  -  )
1413cnmetdval 12451 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( 0  -  x
) ) )
15 abssub 10713 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( abs `  (
0  -  x ) )  =  ( abs `  ( x  -  0 ) ) )
1614, 15eqtrd 2132 . . . . . . . . 9  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( x  -  0 ) ) )
1712, 16mpan 418 . . . . . . . 8  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  (
x  -  0 ) ) )
18 subid1 7853 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
x  -  0 )  =  x )
1918fveq2d 5357 . . . . . . . 8  |-  ( x  e.  CC  ->  ( abs `  ( x  - 
0 ) )  =  ( abs `  x
) )
2017, 19eqtrd 2132 . . . . . . 7  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  x
) )
2120adantl 273 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
0 D x )  =  ( abs `  x
) )
2221breq1d 3885 . . . . 5  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( 0 D x )  <  R  <->  ( abs `  x )  <  R
) )
237, 11, 223bitr4d 219 . . . 4  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  ( 0 [,) R )  <->  ( 0 D x )  < 
R ) )
2423pm5.32da 443 . . 3  |-  ( R  e.  RR*  ->  ( ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,) R ) )  <->  ( x  e.  CC  /\  ( 0 D x )  < 
R ) ) )
25 absf 10722 . . . . 5  |-  abs : CC
--> RR
26 ffn 5208 . . . . 5  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
2725, 26ax-mp 7 . . . 4  |-  abs  Fn  CC
28 elpreima 5471 . . . 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 12453 . . . . 5  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
3113, 30eqeltri 2172 . . . 4  |-  D  e.  ( *Met `  CC )
32 elbl 12319 . . . 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 1273 . . 3  |-  ( R  e.  RR*  ->  ( x  e.  ( 0 (
ball `  D ) R )  <->  ( x  e.  CC  /\  ( 0 D x )  < 
R ) ) )
3424, 29, 333bitr4d 219 . 2  |-  ( R  e.  RR*  ->  ( x  e.  ( `' abs " ( 0 [,) R
) )  <->  x  e.  ( 0 ( ball `  D ) R ) ) )
3534eqrdv 2098 1  |-  ( R  e.  RR*  ->  ( `' abs " ( 0 [,) R ) )  =  ( 0 (
ball `  D ) R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 930    = wceq 1299    e. wcel 1448   class class class wbr 3875   `'ccnv 4476   "cima 4480    o. ccom 4481    Fn wfn 5054   -->wf 5055   ` cfv 5059  (class class class)co 5706   CCcc 7498   RRcr 7499   0cc0 7500   RR*cxr 7671    < clt 7672    <_ cle 7673    - cmin 7804   [,)cico 9514   abscabs 10609   *Metcxmet 11931   ballcbl 11933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-map 6474  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-rp 9292  df-xadd 9401  df-ico 9518  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-psmet 11938  df-xmet 11939  df-met 11940  df-bl 11941
This theorem is referenced by: (None)
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