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Theorem ellogdm 20091
Description: Elementhood in the "continuous domain" of the complex logarithm. (Contributed by Mario Carneiro, 18-Feb-2015.)
Hypothesis
Ref Expression
logcn.d  |-  D  =  ( CC  \  (  -oo (,] 0 ) )
Assertion
Ref Expression
ellogdm  |-  ( A  e.  D  <->  ( A  e.  CC  /\  ( A  e.  RR  ->  A  e.  RR+ ) ) )

Proof of Theorem ellogdm
StepHypRef Expression
1 logcn.d . . 3  |-  D  =  ( CC  \  (  -oo (,] 0 ) )
21eleq2i 2422 . 2  |-  ( A  e.  D  <->  A  e.  ( CC  \  (  -oo (,] 0 ) ) )
3 eldif 3238 . 2  |-  ( A  e.  ( CC  \ 
(  -oo (,] 0 ) )  <->  ( A  e.  CC  /\  -.  A  e.  (  -oo (,] 0
) ) )
4 mnfxr 10545 . . . . . . 7  |-  -oo  e.  RR*
5 0re 8925 . . . . . . 7  |-  0  e.  RR
6 elioc2 10802 . . . . . . 7  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR )  ->  ( A  e.  (  -oo (,] 0 )  <->  ( A  e.  RR  /\  -oo  <  A  /\  A  <_  0
) ) )
74, 5, 6mp2an 653 . . . . . 6  |-  ( A  e.  (  -oo (,] 0 )  <->  ( A  e.  RR  /\  -oo  <  A  /\  A  <_  0
) )
8 df-3an 936 . . . . . 6  |-  ( ( A  e.  RR  /\  -oo 
<  A  /\  A  <_ 
0 )  <->  ( ( A  e.  RR  /\  -oo  <  A )  /\  A  <_  0 ) )
9 mnflt 10553 . . . . . . . . 9  |-  ( A  e.  RR  ->  -oo  <  A )
109pm4.71i 613 . . . . . . . 8  |-  ( A  e.  RR  <->  ( A  e.  RR  /\  -oo  <  A ) )
1110anbi1i 676 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  <_  0 )  <->  ( ( A  e.  RR  /\  -oo  <  A )  /\  A  <_  0 ) )
12 lenlt 8988 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A  <_  0  <->  -.  0  <  A ) )
135, 12mpan2 652 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  <_  0  <->  -.  0  <  A ) )
14 elrp 10445 . . . . . . . . . . 11  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
1514baib 871 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  e.  RR+  <->  0  <  A ) )
1615notbid 285 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( -.  A  e.  RR+  <->  -.  0  <  A ) )
1713, 16bitr4d 247 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  <_  0  <->  -.  A  e.  RR+ ) )
1817pm5.32i 618 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  <_  0 )  <->  ( A  e.  RR  /\  -.  A  e.  RR+ ) )
1911, 18bitr3i 242 . . . . . 6  |-  ( ( ( A  e.  RR  /\ 
-oo  <  A )  /\  A  <_  0 )  <->  ( A  e.  RR  /\  -.  A  e.  RR+ ) )
207, 8, 193bitri 262 . . . . 5  |-  ( A  e.  (  -oo (,] 0 )  <->  ( A  e.  RR  /\  -.  A  e.  RR+ ) )
2120notbii 287 . . . 4  |-  ( -.  A  e.  (  -oo (,] 0 )  <->  -.  ( A  e.  RR  /\  -.  A  e.  RR+ ) )
22 iman 413 . . . 4  |-  ( ( A  e.  RR  ->  A  e.  RR+ )  <->  -.  ( A  e.  RR  /\  -.  A  e.  RR+ ) )
2321, 22bitr4i 243 . . 3  |-  ( -.  A  e.  (  -oo (,] 0 )  <->  ( A  e.  RR  ->  A  e.  RR+ ) )
2423anbi2i 675 . 2  |-  ( ( A  e.  CC  /\  -.  A  e.  (  -oo (,] 0 ) )  <-> 
( A  e.  CC  /\  ( A  e.  RR  ->  A  e.  RR+ )
) )
252, 3, 243bitri 262 1  |-  ( A  e.  D  <->  ( A  e.  CC  /\  ( A  e.  RR  ->  A  e.  RR+ ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    \ cdif 3225   class class class wbr 4102  (class class class)co 5942   CCcc 8822   RRcr 8823   0cc0 8824    -oocmnf 8952   RR*cxr 8953    < clt 8954    <_ cle 8955   RR+crp 10443   (,]cioc 10746
This theorem is referenced by:  logdmn0  20092  logdmnrp  20093  logdmss  20094  logcnlem2  20095  logcnlem3  20096  logcnlem4  20097  logcnlem5  20098  logcn  20099  dvloglem  20100  logf1o2  20102  cxpcn  20190  cxpcn2  20191  dmlogdmgm  24057  rpdmgm  24058  lgamgulmlem2  24063  lgamcvg2  24088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-i2m1 8892  ax-1ne0 8893  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-po 4393  df-so 4394  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-rp 10444  df-ioc 10750
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