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Theorem cnblcld 12741
Description: Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
Hypothesis
Ref Expression
cnblcld.1  |-  D  =  ( abs  o.  -  )
Assertion
Ref Expression
cnblcld  |-  ( R  e.  RR*  ->  ( `' abs " ( 0 [,] R ) )  =  { x  e.  CC  |  ( 0 D x )  <_  R } )
Distinct variable groups:    x, D    x, R

Proof of Theorem cnblcld
StepHypRef Expression
1 absf 10913 . . . . 5  |-  abs : CC
--> RR
2 ffn 5279 . . . . 5  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
3 elpreima 5546 . . . . 5  |-  ( abs 
Fn  CC  ->  ( x  e.  ( `' abs " ( 0 [,] R
) )  <->  ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,] R ) ) ) )
41, 2, 3mp2b 8 . . . 4  |-  ( x  e.  ( `' abs " ( 0 [,] R
) )  <->  ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,] R ) ) )
5 abscl 10854 . . . . . . . . . . 11  |-  ( x  e.  CC  ->  ( abs `  x )  e.  RR )
65rexrd 7838 . . . . . . . . . 10  |-  ( x  e.  CC  ->  ( abs `  x )  e. 
RR* )
7 absge0 10863 . . . . . . . . . 10  |-  ( x  e.  CC  ->  0  <_  ( abs `  x
) )
86, 7jca 304 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
( abs `  x
)  e.  RR*  /\  0  <_  ( abs `  x
) ) )
98adantl 275 . . . . . . . 8  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  RR*  /\  0  <_  ( abs `  x
) ) )
109biantrurd 303 . . . . . . 7  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  <_  R  <->  ( (
( abs `  x
)  e.  RR*  /\  0  <_  ( abs `  x
) )  /\  ( abs `  x )  <_  R ) ) )
11 df-3an 965 . . . . . . 7  |-  ( ( ( abs `  x
)  e.  RR*  /\  0  <_  ( abs `  x
)  /\  ( abs `  x )  <_  R
)  <->  ( ( ( abs `  x )  e.  RR*  /\  0  <_  ( abs `  x
) )  /\  ( abs `  x )  <_  R ) )
1210, 11syl6rbbr 198 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( ( abs `  x
)  e.  RR*  /\  0  <_  ( abs `  x
)  /\  ( abs `  x )  <_  R
)  <->  ( abs `  x
)  <_  R )
)
13 0xr 7835 . . . . . . 7  |-  0  e.  RR*
14 simpl 108 . . . . . . 7  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  R  e.  RR* )
15 elicc1 9736 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  R  e.  RR* )  ->  (
( abs `  x
)  e.  ( 0 [,] R )  <->  ( ( abs `  x )  e. 
RR*  /\  0  <_  ( abs `  x )  /\  ( abs `  x
)  <_  R )
) )
1613, 14, 15sylancr 411 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  ( 0 [,] R )  <->  ( ( abs `  x )  e. 
RR*  /\  0  <_  ( abs `  x )  /\  ( abs `  x
)  <_  R )
) )
17 0cn 7781 . . . . . . . . . 10  |-  0  e.  CC
18 cnblcld.1 . . . . . . . . . . . 12  |-  D  =  ( abs  o.  -  )
1918cnmetdval 12735 . . . . . . . . . . 11  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( 0  -  x
) ) )
20 abssub 10904 . . . . . . . . . . 11  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( abs `  (
0  -  x ) )  =  ( abs `  ( x  -  0 ) ) )
2119, 20eqtrd 2173 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( x  -  0 ) ) )
2217, 21mpan 421 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  (
x  -  0 ) ) )
23 subid1 8005 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
x  -  0 )  =  x )
2423fveq2d 5432 . . . . . . . . 9  |-  ( x  e.  CC  ->  ( abs `  ( x  - 
0 ) )  =  ( abs `  x
) )
2522, 24eqtrd 2173 . . . . . . . 8  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  x
) )
2625adantl 275 . . . . . . 7  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
0 D x )  =  ( abs `  x
) )
2726breq1d 3946 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( 0 D x )  <_  R  <->  ( abs `  x )  <_  R
) )
2812, 16, 273bitr4d 219 . . . . 5  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  ( 0 [,] R )  <->  ( 0 D x )  <_  R ) )
2928pm5.32da 448 . . . 4  |-  ( R  e.  RR*  ->  ( ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,] R ) )  <->  ( x  e.  CC  /\  ( 0 D x )  <_  R ) ) )
304, 29syl5bb 191 . . 3  |-  ( R  e.  RR*  ->  ( x  e.  ( `' abs " ( 0 [,] R
) )  <->  ( x  e.  CC  /\  ( 0 D x )  <_  R ) ) )
3130abbi2dv 2259 . 2  |-  ( R  e.  RR*  ->  ( `' abs " ( 0 [,] R ) )  =  { x  |  ( x  e.  CC  /\  ( 0 D x )  <_  R ) } )
32 df-rab 2426 . 2  |-  { x  e.  CC  |  ( 0 D x )  <_  R }  =  {
x  |  ( x  e.  CC  /\  (
0 D x )  <_  R ) }
3331, 32eqtr4di 2191 1  |-  ( R  e.  RR*  ->  ( `' abs " ( 0 [,] R ) )  =  { x  e.  CC  |  ( 0 D x )  <_  R } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1332    e. wcel 1481   {cab 2126   {crab 2421   class class class wbr 3936   `'ccnv 4545   "cima 4549    o. ccom 4550    Fn wfn 5125   -->wf 5126   ` cfv 5130  (class class class)co 5781   CCcc 7641   RRcr 7642   0cc0 7643   RR*cxr 7822    <_ cle 7824    - cmin 7956   [,]cicc 9703   abscabs 10800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761  ax-arch 7762  ax-caucvg 7763
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-2 8802  df-3 8803  df-4 8804  df-n0 9001  df-z 9078  df-uz 9350  df-rp 9470  df-icc 9707  df-seqfrec 10249  df-exp 10323  df-cj 10645  df-re 10646  df-im 10647  df-rsqrt 10801  df-abs 10802
This theorem is referenced by: (None)
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