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Theorem cnbl0 18299
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 abscl 11779 . . . . . . . . 9  |-  ( x  e.  CC  ->  ( abs `  x )  e.  RR )
2 absge0 11788 . . . . . . . . 9  |-  ( x  e.  CC  ->  0  <_  ( abs `  x
) )
31, 2jca 518 . . . . . . . 8  |-  ( x  e.  CC  ->  (
( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
) ) )
43adantl 452 . . . . . . 7  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
) ) )
54biantrurd 494 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  <  R  <->  ( (
( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
) )  /\  ( abs `  x )  < 
R ) ) )
6 df-3an 936 . . . . . 6  |-  ( ( ( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
)  /\  ( abs `  x )  <  R
)  <->  ( ( ( abs `  x )  e.  RR  /\  0  <_  ( abs `  x
) )  /\  ( abs `  x )  < 
R ) )
75, 6syl6rbbr 255 . . . . 5  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( ( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
)  /\  ( abs `  x )  <  R
)  <->  ( abs `  x
)  <  R )
)
8 0re 8854 . . . . . 6  |-  0  e.  RR
9 simpl 443 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  R  e.  RR* )
10 elico2 10730 . . . . . 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 644 . . . . 5  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  ( 0 [,) R )  <->  ( ( abs `  x )  e.  RR  /\  0  <_ 
( abs `  x
)  /\  ( abs `  x )  <  R
) ) )
12 0cn 8847 . . . . . . . . 9  |-  0  e.  CC
13 cnblcld.1 . . . . . . . . . . 11  |-  D  =  ( abs  o.  -  )
1413cnmetdval 18296 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( 0  -  x
) ) )
15 abssub 11826 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( abs `  (
0  -  x ) )  =  ( abs `  ( x  -  0 ) ) )
1614, 15eqtrd 2328 . . . . . . . . 9  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( x  -  0 ) ) )
1712, 16mpan 651 . . . . . . . 8  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  (
x  -  0 ) ) )
18 subid1 9084 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
x  -  0 )  =  x )
1918fveq2d 5545 . . . . . . . 8  |-  ( x  e.  CC  ->  ( abs `  ( x  - 
0 ) )  =  ( abs `  x
) )
2017, 19eqtrd 2328 . . . . . . 7  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  x
) )
2120adantl 452 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
0 D x )  =  ( abs `  x
) )
2221breq1d 4049 . . . . 5  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( 0 D x )  <  R  <->  ( abs `  x )  <  R
) )
237, 11, 223bitr4d 276 . . . 4  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  ( 0 [,) R )  <->  ( 0 D x )  < 
R ) )
2423pm5.32da 622 . . 3  |-  ( R  e.  RR*  ->  ( ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,) R ) )  <->  ( x  e.  CC  /\  ( 0 D x )  < 
R ) ) )
25 absf 11837 . . . . 5  |-  abs : CC
--> RR
26 ffn 5405 . . . . 5  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
2725, 26ax-mp 8 . . . 4  |-  abs  Fn  CC
28 elpreima 5661 . . . 4  |-  ( abs 
Fn  CC  ->  ( x  e.  ( `' abs " ( 0 [,) R
) )  <->  ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,) R ) ) ) )
2927, 28mp1i 11 . . 3  |-  ( R  e.  RR*  ->  ( x  e.  ( `' abs " ( 0 [,) R
) )  <->  ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,) R ) ) ) )
30 cnxmet 18298 . . . . 5  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
3113, 30eqeltri 2366 . . . 4  |-  D  e.  ( * Met `  CC )
32 elbl 17965 . . . 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 1267 . . 3  |-  ( R  e.  RR*  ->  ( x  e.  ( 0 (
ball `  D ) R )  <->  ( x  e.  CC  /\  ( 0 D x )  < 
R ) ) )
3424, 29, 333bitr4d 276 . 2  |-  ( R  e.  RR*  ->  ( x  e.  ( `' abs " ( 0 [,) R
) )  <->  x  e.  ( 0 ( ball `  D ) R ) ) )
3534eqrdv 2294 1  |-  ( R  e.  RR*  ->  ( `' abs " ( 0 [,) R ) )  =  ( 0 (
ball `  D ) R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   `'ccnv 4704   "cima 4708    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053   [,)cico 10674   abscabs 11735   * Metcxmt 16385   ballcbl 16387
This theorem is referenced by:  psercnlem2  19816  efopnlem1  20019
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-xadd 10469  df-ico 10678  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-xmet 16389  df-met 16390  df-bl 16391
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