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Theorem cnblcld 18284
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 11821 . . . . 5  |-  abs : CC
--> RR
2 ffn 5389 . . . . 5  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
3 elpreima 5645 . . . . 5  |-  ( abs 
Fn  CC  ->  ( x  e.  ( `' abs " ( 0 [,] R
) )  <->  ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,] R ) ) ) )
41, 2, 3mp2b 9 . . . 4  |-  ( x  e.  ( `' abs " ( 0 [,] R
) )  <->  ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,] R ) ) )
5 abscl 11763 . . . . . . . . . . 11  |-  ( x  e.  CC  ->  ( abs `  x )  e.  RR )
65rexrd 8881 . . . . . . . . . 10  |-  ( x  e.  CC  ->  ( abs `  x )  e. 
RR* )
7 absge0 11772 . . . . . . . . . 10  |-  ( x  e.  CC  ->  0  <_  ( abs `  x
) )
86, 7jca 518 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
( abs `  x
)  e.  RR*  /\  0  <_  ( abs `  x
) ) )
98adantl 452 . . . . . . . 8  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  RR*  /\  0  <_  ( abs `  x
) ) )
109biantrurd 494 . . . . . . 7  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  <_  R  <->  ( (
( abs `  x
)  e.  RR*  /\  0  <_  ( abs `  x
) )  /\  ( abs `  x )  <_  R ) ) )
11 df-3an 936 . . . . . . 7  |-  ( ( ( abs `  x
)  e.  RR*  /\  0  <_  ( abs `  x
)  /\  ( abs `  x )  <_  R
)  <->  ( ( ( abs `  x )  e.  RR*  /\  0  <_  ( abs `  x
) )  /\  ( abs `  x )  <_  R ) )
1210, 11syl6rbbr 255 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( ( abs `  x
)  e.  RR*  /\  0  <_  ( abs `  x
)  /\  ( abs `  x )  <_  R
)  <->  ( abs `  x
)  <_  R )
)
13 0xr 8878 . . . . . . 7  |-  0  e.  RR*
14 simpl 443 . . . . . . 7  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  R  e.  RR* )
15 elicc1 10700 . . . . . . 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 644 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  ( 0 [,] R )  <->  ( ( abs `  x )  e. 
RR*  /\  0  <_  ( abs `  x )  /\  ( abs `  x
)  <_  R )
) )
17 0cn 8831 . . . . . . . . . 10  |-  0  e.  CC
18 cnblcld.1 . . . . . . . . . . . 12  |-  D  =  ( abs  o.  -  )
1918cnmetdval 18280 . . . . . . . . . . 11  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( 0  -  x
) ) )
20 abssub 11810 . . . . . . . . . . 11  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( abs `  (
0  -  x ) )  =  ( abs `  ( x  -  0 ) ) )
2119, 20eqtrd 2315 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( x  -  0 ) ) )
2217, 21mpan 651 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  (
x  -  0 ) ) )
23 subid1 9068 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
x  -  0 )  =  x )
2423fveq2d 5529 . . . . . . . . 9  |-  ( x  e.  CC  ->  ( abs `  ( x  - 
0 ) )  =  ( abs `  x
) )
2522, 24eqtrd 2315 . . . . . . . 8  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  x
) )
2625adantl 452 . . . . . . 7  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
0 D x )  =  ( abs `  x
) )
2726breq1d 4033 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( 0 D x )  <_  R  <->  ( abs `  x )  <_  R
) )
2812, 16, 273bitr4d 276 . . . . 5  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  ( 0 [,] R )  <->  ( 0 D x )  <_  R ) )
2928pm5.32da 622 . . . 4  |-  ( R  e.  RR*  ->  ( ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,] R ) )  <->  ( x  e.  CC  /\  ( 0 D x )  <_  R ) ) )
304, 29syl5bb 248 . . 3  |-  ( R  e.  RR*  ->  ( x  e.  ( `' abs " ( 0 [,] R
) )  <->  ( x  e.  CC  /\  ( 0 D x )  <_  R ) ) )
3130abbi2dv 2398 . 2  |-  ( R  e.  RR*  ->  ( `' abs " ( 0 [,] R ) )  =  { x  |  ( x  e.  CC  /\  ( 0 D x )  <_  R ) } )
32 df-rab 2552 . 2  |-  { x  e.  CC  |  ( 0 D x )  <_  R }  =  {
x  |  ( x  e.  CC  /\  (
0 D x )  <_  R ) }
3331, 32syl6eqr 2333 1  |-  ( R  e.  RR*  ->  ( `' abs " ( 0 [,] R ) )  =  { x  e.  CC  |  ( 0 D x )  <_  R } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547   class class class wbr 4023   `'ccnv 4688   "cima 4692    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   RR*cxr 8866    <_ cle 8868    - cmin 9037   [,]cicc 10659   abscabs 11719
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-icc 10663  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
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