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Theorem ellogdm 22202
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 2529 . 2  |-  ( A  e.  D  <->  A  e.  ( CC  \  ( -oo (,] 0 ) ) )
3 eldif 3438 . 2  |-  ( A  e.  ( CC  \ 
( -oo (,] 0 ) )  <->  ( A  e.  CC  /\  -.  A  e.  ( -oo (,] 0
) ) )
4 mnfxr 11197 . . . . . . 7  |- -oo  e.  RR*
5 0re 9489 . . . . . . 7  |-  0  e.  RR
6 elioc2 11461 . . . . . . 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 967 . . . . . 6  |-  ( ( A  e.  RR  /\ -oo 
<  A  /\  A  <_ 
0 )  <->  ( ( A  e.  RR  /\ -oo  <  A )  /\  A  <_  0 ) )
9 mnflt 11207 . . . . . . . . 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 9556 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A  <_  0  <->  -.  0  <  A ) )
135, 12mpan2 671 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  <_  0  <->  -.  0  <  A ) )
14 elrp 11096 . . . . . . . . . . 11  |-  ( A  e.  RR+  <->  ( A  e.  RR  /\  0  < 
A ) )
1514baib 896 . . . . . . . . . 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 965    = wceq 1370    e. wcel 1758    \ cdif 3425   class class class wbr 4392  (class class class)co 6192   CCcc 9383   RRcr 9384   0cc0 9385   -oocmnf 9519   RR*cxr 9520    < clt 9521    <_ cle 9522   RR+crp 11094   (,]cioc 11404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-i2m1 9453  ax-1ne0 9454  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-rp 11095  df-ioc 11408
This theorem is referenced by:  logdmn0  22203  logdmnrp  22204  logdmss  22205  logcnlem2  22206  logcnlem3  22207  logcnlem4  22208  logcnlem5  22209  logcn  22210  dvloglem  22211  logf1o2  22213  cxpcn  22301  cxpcn2  22302  dmlogdmgm  27146  rpdmgm  27147  lgamgulmlem2  27152  lgamcvg2  27177
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