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Theorem cnblcld 18497
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 12028 . . . . 5  |-  abs : CC
--> RR
2 ffn 5495 . . . . 5  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
3 elpreima 5752 . . . . 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 11970 . . . . . . . . . . 11  |-  ( x  e.  CC  ->  ( abs `  x )  e.  RR )
65rexrd 9028 . . . . . . . . . 10  |-  ( x  e.  CC  ->  ( abs `  x )  e. 
RR* )
7 absge0 11979 . . . . . . . . . 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 937 . . . . . . 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 9025 . . . . . . 7  |-  0  e.  RR*
14 simpl 443 . . . . . . 7  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  R  e.  RR* )
15 elicc1 10853 . . . . . . 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 8978 . . . . . . . . . 10  |-  0  e.  CC
18 cnblcld.1 . . . . . . . . . . . 12  |-  D  =  ( abs  o.  -  )
1918cnmetdval 18493 . . . . . . . . . . 11  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( 0  -  x
) ) )
20 abssub 12017 . . . . . . . . . . 11  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( abs `  (
0  -  x ) )  =  ( abs `  ( x  -  0 ) ) )
2119, 20eqtrd 2398 . . . . . . . . . 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 9215 . . . . . . . . . 10  |-  ( x  e.  CC  ->  (
x  -  0 )  =  x )
2423fveq2d 5636 . . . . . . . . 9  |-  ( x  e.  CC  ->  ( abs `  ( x  - 
0 ) )  =  ( abs `  x
) )
2522, 24eqtrd 2398 . . . . . . . 8  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  x
) )
2625adantl 452 . . . . . . 7  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
0 D x )  =  ( abs `  x
) )
2726breq1d 4135 . . . . . 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 2481 . 2  |-  ( R  e.  RR*  ->  ( `' abs " ( 0 [,] R ) )  =  { x  |  ( x  e.  CC  /\  ( 0 D x )  <_  R ) } )
32 df-rab 2637 . 2  |-  { x  e.  CC  |  ( 0 D x )  <_  R }  =  {
x  |  ( x  e.  CC  /\  (
0 D x )  <_  R ) }
3331, 32syl6eqr 2416 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 935    = wceq 1647    e. wcel 1715   {cab 2352   {crab 2632   class class class wbr 4125   `'ccnv 4791   "cima 4795    o. ccom 4796    Fn wfn 5353   -->wf 5354   ` cfv 5358  (class class class)co 5981   CCcc 8882   RRcr 8883   0cc0 8884   RR*cxr 9013    <_ cle 9015    - cmin 9184   [,]cicc 10812   abscabs 11926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-n0 10115  df-z 10176  df-uz 10382  df-rp 10506  df-icc 10816  df-seq 11211  df-exp 11270  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928
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