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Theorem cnbl0 18810
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 12085 . . . . . . . . 9  |-  ( x  e.  CC  ->  ( abs `  x )  e.  RR )
2 absge0 12094 . . . . . . . . 9  |-  ( x  e.  CC  ->  0  <_  ( abs `  x
) )
31, 2jca 520 . . . . . . . 8  |-  ( x  e.  CC  ->  (
( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
) ) )
43adantl 454 . . . . . . 7  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
) ) )
54biantrurd 496 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  <  R  <->  ( (
( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
) )  /\  ( abs `  x )  < 
R ) ) )
6 df-3an 939 . . . . . 6  |-  ( ( ( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
)  /\  ( abs `  x )  <  R
)  <->  ( ( ( abs `  x )  e.  RR  /\  0  <_  ( abs `  x
) )  /\  ( abs `  x )  < 
R ) )
75, 6syl6rbbr 257 . . . . 5  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( ( abs `  x
)  e.  RR  /\  0  <_  ( abs `  x
)  /\  ( abs `  x )  <  R
)  <->  ( abs `  x
)  <  R )
)
8 0re 9093 . . . . . 6  |-  0  e.  RR
9 simpl 445 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  R  e.  RR* )
10 elico2 10976 . . . . . 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 646 . . . . 5  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  ( 0 [,) R )  <->  ( ( abs `  x )  e.  RR  /\  0  <_ 
( abs `  x
)  /\  ( abs `  x )  <  R
) ) )
12 0cn 9086 . . . . . . . . 9  |-  0  e.  CC
13 cnblcld.1 . . . . . . . . . . 11  |-  D  =  ( abs  o.  -  )
1413cnmetdval 18807 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( 0  -  x
) ) )
15 abssub 12132 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( abs `  (
0  -  x ) )  =  ( abs `  ( x  -  0 ) ) )
1614, 15eqtrd 2470 . . . . . . . . 9  |-  ( ( 0  e.  CC  /\  x  e.  CC )  ->  ( 0 D x )  =  ( abs `  ( x  -  0 ) ) )
1712, 16mpan 653 . . . . . . . 8  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  (
x  -  0 ) ) )
18 subid1 9324 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
x  -  0 )  =  x )
1918fveq2d 5734 . . . . . . . 8  |-  ( x  e.  CC  ->  ( abs `  ( x  - 
0 ) )  =  ( abs `  x
) )
2017, 19eqtrd 2470 . . . . . . 7  |-  ( x  e.  CC  ->  (
0 D x )  =  ( abs `  x
) )
2120adantl 454 . . . . . 6  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
0 D x )  =  ( abs `  x
) )
2221breq1d 4224 . . . . 5  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( 0 D x )  <  R  <->  ( abs `  x )  <  R
) )
237, 11, 223bitr4d 278 . . . 4  |-  ( ( R  e.  RR*  /\  x  e.  CC )  ->  (
( abs `  x
)  e.  ( 0 [,) R )  <->  ( 0 D x )  < 
R ) )
2423pm5.32da 624 . . 3  |-  ( R  e.  RR*  ->  ( ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,) R ) )  <->  ( x  e.  CC  /\  ( 0 D x )  < 
R ) ) )
25 absf 12143 . . . . 5  |-  abs : CC
--> RR
26 ffn 5593 . . . . 5  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
2725, 26ax-mp 8 . . . 4  |-  abs  Fn  CC
28 elpreima 5852 . . . 4  |-  ( abs 
Fn  CC  ->  ( x  e.  ( `' abs " ( 0 [,) R
) )  <->  ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,) R ) ) ) )
2927, 28mp1i 12 . . 3  |-  ( R  e.  RR*  ->  ( x  e.  ( `' abs " ( 0 [,) R
) )  <->  ( x  e.  CC  /\  ( abs `  x )  e.  ( 0 [,) R ) ) ) )
30 cnxmet 18809 . . . . 5  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
3113, 30eqeltri 2508 . . . 4  |-  D  e.  ( * Met `  CC )
32 elbl 18420 . . . 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 1270 . . 3  |-  ( R  e.  RR*  ->  ( x  e.  ( 0 (
ball `  D ) R )  <->  ( x  e.  CC  /\  ( 0 D x )  < 
R ) ) )
3424, 29, 333bitr4d 278 . 2  |-  ( R  e.  RR*  ->  ( x  e.  ( `' abs " ( 0 [,) R
) )  <->  x  e.  ( 0 ( ball `  D ) R ) ) )
3534eqrdv 2436 1  |-  ( R  e.  RR*  ->  ( `' abs " ( 0 [,) R ) )  =  ( 0 (
ball `  D ) R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214   `'ccnv 4879   "cima 4883    o. ccom 4884    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   RR*cxr 9121    < clt 9122    <_ cle 9123    - cmin 9293   [,)cico 10920   abscabs 12041   * Metcxmt 16688   ballcbl 16690
This theorem is referenced by:  psercnlem2  20342  efopnlem1  20549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-xadd 10713  df-ico 10924  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699
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