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Theorem cnblcld 21408
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 13182 . . . . 5  |-  abs : CC
--> RR
2 ffn 5737 . . . . 5  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
3 elpreima 6008 . . . . 5  |-  ( abs 
Fn  CC  ->  ( x  e.  ( `' abs " ( 0 [,] R
) )  <->  ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,] R ) ) ) )
41, 2, 3mp2b 10 . . . 4  |-  ( x  e.  ( `' abs " ( 0 [,] R
) )  <->  ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,] R ) ) )
5 abscl 13123 . . . . . . . . . . 11  |-  ( x  e.  CC  ->  ( abs `  x )  e.  RR )
65rexrd 9660 . . . . . . . . . 10  |-  ( x  e.  CC  ->  ( abs `  x )  e. 
RR* )
7 absge0 13132 . . . . . . . . . 10  |-  ( x  e.  CC  ->  0  <_  ( abs `  x
) )
86, 7jca 532 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
( abs `  x
)  e.  RR*  /\  0  <_  ( abs `  x
) ) )
98adantl 466 . . . . . . . 8  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  RR*  /\  0  <_  ( abs `  x
) ) )
109biantrurd 508 . . . . . . 7  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  <_  R  <->  ( (
( abs `  x
)  e.  RR*  /\  0  <_  ( abs `  x
) )  /\  ( abs `  x )  <_  R ) ) )
11 df-3an 975 . . . . . . 7  |-  ( ( ( abs `  x
)  e.  RR*  /\  0  <_  ( abs `  x
)  /\  ( abs `  x )  <_  R
)  <->  ( ( ( abs `  x )  e.  RR*  /\  0  <_  ( abs `  x
) )  /\  ( abs `  x )  <_  R ) )
1210, 11syl6rbbr 264 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( ( abs `  x
)  e.  RR*  /\  0  <_  ( abs `  x
)  /\  ( abs `  x )  <_  R
)  <->  ( abs `  x
)  <_  R )
)
13 0xr 9657 . . . . . . 7  |-  0  e.  RR*
14 simpl 457 . . . . . . 7  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  R  e.  RR* )
15 elicc1 11598 . . . . . . 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 663 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  ( 0 [,] R )  <->  ( ( abs `  x )  e. 
RR*  /\  0  <_  ( abs `  x )  /\  ( abs `  x
)  <_  R )
) )
17 0cn 9605 . . . . . . . . . 10  |-  0  e.  CC
18 cnblcld.1 . . . . . . . . . . . 12  |-  D  =  ( abs  o.  -  )
1918cnmetdval 21404 . . . . . . . . . . 11  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( 0  -  x
) ) )
20 abssub 13171 . . . . . . . . . . 11  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( abs `  (
0  -  x ) )  =  ( abs `  ( x  -  0 ) ) )
2119, 20eqtrd 2498 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( x  -  0 ) ) )
2217, 21mpan 670 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  (
x  -  0 ) ) )
23 subid1 9858 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
x  -  0 )  =  x )
2423fveq2d 5876 . . . . . . . . 9  |-  ( x  e.  CC  ->  ( abs `  ( x  - 
0 ) )  =  ( abs `  x
) )
2522, 24eqtrd 2498 . . . . . . . 8  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  x
) )
2625adantl 466 . . . . . . 7  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
0 D x )  =  ( abs `  x
) )
2726breq1d 4466 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( 0 D x )  <_  R  <->  ( abs `  x )  <_  R
) )
2812, 16, 273bitr4d 285 . . . . 5  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  ( 0 [,] R )  <->  ( 0 D x )  <_  R ) )
2928pm5.32da 641 . . . 4  |-  ( R  e.  RR*  ->  ( ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,] R ) )  <->  ( x  e.  CC  /\  ( 0 D x )  <_  R ) ) )
304, 29syl5bb 257 . . 3  |-  ( R  e.  RR*  ->  ( x  e.  ( `' abs " ( 0 [,] R
) )  <->  ( x  e.  CC  /\  ( 0 D x )  <_  R ) ) )
3130abbi2dv 2594 . 2  |-  ( R  e.  RR*  ->  ( `' abs " ( 0 [,] R ) )  =  { x  |  ( x  e.  CC  /\  ( 0 D x )  <_  R ) } )
32 df-rab 2816 . 2  |-  { x  e.  CC  |  ( 0 D x )  <_  R }  =  {
x  |  ( x  e.  CC  /\  (
0 D x )  <_  R ) }
3331, 32syl6eqr 2516 1  |-  ( R  e.  RR*  ->  ( `' abs " ( 0 [,] R ) )  =  { x  e.  CC  |  ( 0 D x )  <_  R } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {cab 2442   {crab 2811   class class class wbr 4456   `'ccnv 5007   "cima 5011    o. ccom 5012    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   RR*cxr 9644    <_ cle 9646    - cmin 9824   [,]cicc 11557   abscabs 13079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-icc 11561  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081
This theorem is referenced by: (None)
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