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Theorem ellogdm 20522
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 2499 . 2  |-  ( A  e.  D  <->  A  e.  ( CC  \  (  -oo (,] 0 ) ) )
3 eldif 3322 . 2  |-  ( A  e.  ( CC  \ 
(  -oo (,] 0 ) )  <->  ( A  e.  CC  /\  -.  A  e.  (  -oo (,] 0
) ) )
4 mnfxr 10706 . . . . . . 7  |-  -oo  e.  RR*
5 0re 9083 . . . . . . 7  |-  0  e.  RR
6 elioc2 10965 . . . . . . 7  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR )  ->  ( A  e.  (  -oo (,] 0 )  <->  ( A  e.  RR  /\  -oo  <  A  /\  A  <_  0
) ) )
74, 5, 6mp2an 654 . . . . . 6  |-  ( A  e.  (  -oo (,] 0 )  <->  ( A  e.  RR  /\  -oo  <  A  /\  A  <_  0
) )
8 df-3an 938 . . . . . 6  |-  ( ( A  e.  RR  /\  -oo 
<  A  /\  A  <_ 
0 )  <->  ( ( A  e.  RR  /\  -oo  <  A )  /\  A  <_  0 ) )
9 mnflt 10714 . . . . . . . . 9  |-  ( A  e.  RR  ->  -oo  <  A )
109pm4.71i 614 . . . . . . . 8  |-  ( A  e.  RR  <->  ( A  e.  RR  /\  -oo  <  A ) )
1110anbi1i 677 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  <_  0 )  <->  ( ( A  e.  RR  /\  -oo  <  A )  /\  A  <_  0 ) )
12 lenlt 9146 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A  <_  0  <->  -.  0  <  A ) )
135, 12mpan2 653 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  <_  0  <->  -.  0  <  A ) )
14 elrp 10606 . . . . . . . . . . 11  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
1514baib 872 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  e.  RR+  <->  0  <  A ) )
1615notbid 286 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( -.  A  e.  RR+  <->  -.  0  <  A ) )
1713, 16bitr4d 248 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  <_  0  <->  -.  A  e.  RR+ ) )
1817pm5.32i 619 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  <_  0 )  <->  ( A  e.  RR  /\  -.  A  e.  RR+ ) )
1911, 18bitr3i 243 . . . . . 6  |-  ( ( ( A  e.  RR  /\ 
-oo  <  A )  /\  A  <_  0 )  <->  ( A  e.  RR  /\  -.  A  e.  RR+ ) )
207, 8, 193bitri 263 . . . . 5  |-  ( A  e.  (  -oo (,] 0 )  <->  ( A  e.  RR  /\  -.  A  e.  RR+ ) )
2120notbii 288 . . . 4  |-  ( -.  A  e.  (  -oo (,] 0 )  <->  -.  ( A  e.  RR  /\  -.  A  e.  RR+ ) )
22 iman 414 . . . 4  |-  ( ( A  e.  RR  ->  A  e.  RR+ )  <->  -.  ( A  e.  RR  /\  -.  A  e.  RR+ ) )
2321, 22bitr4i 244 . . 3  |-  ( -.  A  e.  (  -oo (,] 0 )  <->  ( A  e.  RR  ->  A  e.  RR+ ) )
2423anbi2i 676 . 2  |-  ( ( A  e.  CC  /\  -.  A  e.  (  -oo (,] 0 ) )  <-> 
( A  e.  CC  /\  ( A  e.  RR  ->  A  e.  RR+ )
) )
252, 3, 243bitri 263 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    \ cdif 3309   class class class wbr 4204  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982    -oocmnf 9110   RR*cxr 9111    < clt 9112    <_ cle 9113   RR+crp 10604   (,]cioc 10909
This theorem is referenced by:  logdmn0  20523  logdmnrp  20524  logdmss  20525  logcnlem2  20526  logcnlem3  20527  logcnlem4  20528  logcnlem5  20529  logcn  20530  dvloglem  20531  logf1o2  20533  cxpcn  20621  cxpcn2  20622  dmlogdmgm  24800  rpdmgm  24801  lgamgulmlem2  24806  lgamcvg2  24831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-i2m1 9050  ax-1ne0 9051  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-rp 10605  df-ioc 10913
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