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Theorem ellogdm 20002
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 2360 . 2  |-  ( A  e.  D  <->  A  e.  ( CC  \  (  -oo (,] 0 ) ) )
3 eldif 3175 . 2  |-  ( A  e.  ( CC  \ 
(  -oo (,] 0 ) )  <->  ( A  e.  CC  /\  -.  A  e.  (  -oo (,] 0
) ) )
4 mnfxr 10472 . . . . . . 7  |-  -oo  e.  RR*
5 0re 8854 . . . . . . 7  |-  0  e.  RR
6 elioc2 10729 . . . . . . 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 10480 . . . . . . . . 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 8917 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A  <_  0  <->  -.  0  <  A ) )
135, 12mpan2 652 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  <_  0  <->  -.  0  <  A ) )
14 elrp 10372 . . . . . . . . . . 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 1632    e. wcel 1696    \ cdif 3162   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    -oocmnf 8881   RR*cxr 8882    < clt 8883    <_ cle 8884   RR+crp 10370   (,]cioc 10673
This theorem is referenced by:  logdmn0  20003  logdmnrp  20004  logdmss  20005  logcnlem2  20006  logcnlem3  20007  logcnlem4  20008  logcnlem5  20009  logcn  20010  dvloglem  20011  logf1o2  20013  cxpcn  20101  cxpcn2  20102
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-i2m1 8821  ax-1ne0 8822  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828
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-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  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-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-rp 10371  df-ioc 10677
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