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Theorem ellogdm 22886
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 2545 . 2  |-  ( A  e.  D  <->  A  e.  ( CC  \  ( -oo (,] 0 ) ) )
3 eldif 3491 . 2  |-  ( A  e.  ( CC  \ 
( -oo (,] 0 ) )  <->  ( A  e.  CC  /\  -.  A  e.  ( -oo (,] 0
) ) )
4 mnfxr 11335 . . . . . . 7  |- -oo  e.  RR*
5 0re 9608 . . . . . . 7  |-  0  e.  RR
6 elioc2 11599 . . . . . . 7  |-  ( ( -oo  e.  RR*  /\  0  e.  RR )  ->  ( A  e.  ( -oo (,] 0 )  <->  ( A  e.  RR  /\ -oo  <  A  /\  A  <_  0
) ) )
74, 5, 6mp2an 672 . . . . . 6  |-  ( A  e.  ( -oo (,] 0 )  <->  ( A  e.  RR  /\ -oo  <  A  /\  A  <_  0
) )
8 df-3an 975 . . . . . 6  |-  ( ( A  e.  RR  /\ -oo 
<  A  /\  A  <_ 
0 )  <->  ( ( A  e.  RR  /\ -oo  <  A )  /\  A  <_  0 ) )
9 mnflt 11345 . . . . . . . . 9  |-  ( A  e.  RR  -> -oo  <  A )
109pm4.71i 632 . . . . . . . 8  |-  ( A  e.  RR  <->  ( A  e.  RR  /\ -oo  <  A ) )
1110anbi1i 695 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  <_  0 )  <->  ( ( A  e.  RR  /\ -oo  <  A )  /\  A  <_  0 ) )
12 lenlt 9675 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A  <_  0  <->  -.  0  <  A ) )
135, 12mpan2 671 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  <_  0  <->  -.  0  <  A ) )
14 elrp 11234 . . . . . . . . . . 11  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
1514baib 901 . . . . . . . . . 10  |-  ( A  e.  RR  ->  ( A  e.  RR+  <->  0  <  A ) )
1615notbid 294 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( -.  A  e.  RR+  <->  -.  0  <  A ) )
1713, 16bitr4d 256 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  <_  0  <->  -.  A  e.  RR+ ) )
1817pm5.32i 637 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  <_  0 )  <->  ( A  e.  RR  /\  -.  A  e.  RR+ ) )
1911, 18bitr3i 251 . . . . . 6  |-  ( ( ( A  e.  RR  /\ -oo  <  A )  /\  A  <_  0 )  <->  ( A  e.  RR  /\  -.  A  e.  RR+ ) )
207, 8, 193bitri 271 . . . . 5  |-  ( A  e.  ( -oo (,] 0 )  <->  ( A  e.  RR  /\  -.  A  e.  RR+ ) )
2120notbii 296 . . . 4  |-  ( -.  A  e.  ( -oo (,] 0 )  <->  -.  ( A  e.  RR  /\  -.  A  e.  RR+ ) )
22 iman 424 . . . 4  |-  ( ( A  e.  RR  ->  A  e.  RR+ )  <->  -.  ( A  e.  RR  /\  -.  A  e.  RR+ ) )
2321, 22bitr4i 252 . . 3  |-  ( -.  A  e.  ( -oo (,] 0 )  <->  ( A  e.  RR  ->  A  e.  RR+ ) )
2423anbi2i 694 . 2  |-  ( ( A  e.  CC  /\  -.  A  e.  ( -oo (,] 0 ) )  <-> 
( A  e.  CC  /\  ( A  e.  RR  ->  A  e.  RR+ )
) )
252, 3, 243bitri 271 1  |-  ( A  e.  D  <->  ( A  e.  CC  /\  ( A  e.  RR  ->  A  e.  RR+ ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    \ cdif 3478   class class class wbr 4453  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504   -oocmnf 9638   RR*cxr 9639    < clt 9640    <_ cle 9641   RR+crp 11232   (,]cioc 11542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-i2m1 9572  ax-1ne0 9573  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-rp 11233  df-ioc 11546
This theorem is referenced by:  logdmn0  22887  logdmnrp  22888  logdmss  22889  logcnlem2  22890  logcnlem3  22891  logcnlem4  22892  logcnlem5  22893  logcn  22894  dvloglem  22895  logf1o2  22897  cxpcn  22985  cxpcn2  22986  dmlogdmgm  28391  rpdmgm  28392  lgamgulmlem2  28397  lgamcvg2  28422
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