ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnbl0 Unicode version

Theorem cnbl0 14486
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 982 . . . . . 6  |-  ( ( ( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
)  /\  ( abs `  x )  <  R
)  <->  ( ( ( abs `  x )  e.  RR  /\  0  <_  ( abs `  x
) )  /\  ( abs `  x )  < 
R ) )
2 abscl 11091 . . . . . . . . 9  |-  ( x  e.  CC  ->  ( abs `  x )  e.  RR )
3 absge0 11100 . . . . . . . . 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 7986 . . . . . 6  |-  0  e.  RR
9 simpl 109 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  R  e.  RR* )
10 elico2 9966 . . . . . 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 7978 . . . . . . . . 9  |-  0  e.  CC
13 cnblcld.1 . . . . . . . . . . 11  |-  D  =  ( abs  o.  -  )
1413cnmetdval 14481 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( 0  -  x
) ) )
15 abssub 11141 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( abs `  (
0  -  x ) )  =  ( abs `  ( x  -  0 ) ) )
1614, 15eqtrd 2222 . . . . . . . . 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 8206 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
x  -  0 )  =  x )
1918fveq2d 5538 . . . . . . . 8  |-  ( x  e.  CC  ->  ( abs `  ( x  - 
0 ) )  =  ( abs `  x
) )
2017, 19eqtrd 2222 . . . . . . 7  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  x
) )
2120adantl 277 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
0 D x )  =  ( abs `  x
) )
2221breq1d 4028 . . . . 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 11150 . . . . 5  |-  abs : CC
--> RR
26 ffn 5384 . . . . 5  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
2725, 26ax-mp 5 . . . 4  |-  abs  Fn  CC
28 elpreima 5655 . . . 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 14483 . . . . 5  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
3113, 30eqeltri 2262 . . . 4  |-  D  e.  ( *Met `  CC )
32 elbl 14343 . . . 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 1338 . . 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 2187 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 980    = wceq 1364    e. wcel 2160   class class class wbr 4018   `'ccnv 4643   "cima 4647    o. ccom 4648    Fn wfn 5230   -->wf 5231   ` cfv 5235  (class class class)co 5895   CCcc 7838   RRcr 7839   0cc0 7840   RR*cxr 8020    < clt 8021    <_ cle 8022    - cmin 8157   [,)cico 9919   abscabs 11037   *Metcxmet 13846   ballcbl 13848
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 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959  ax-caucvg 7960
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 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-frec 6415  df-map 6675  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-2 9007  df-3 9008  df-4 9009  df-n0 9206  df-z 9283  df-uz 9558  df-rp 9683  df-xadd 9802  df-ico 9923  df-seqfrec 10476  df-exp 10550  df-cj 10882  df-re 10883  df-im 10884  df-rsqrt 11038  df-abs 11039  df-psmet 13853  df-xmet 13854  df-met 13855  df-bl 13856
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator