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Theorem logdmn0 22044
Description: A number in the continuous domain of  log is nonzero. (Contributed by Mario Carneiro, 18-Feb-2015.)
Hypothesis
Ref Expression
logcn.d  |-  D  =  ( CC  \  ( -oo (,] 0 ) )
Assertion
Ref Expression
logdmn0  |-  ( A  e.  D  ->  A  =/=  0 )

Proof of Theorem logdmn0
StepHypRef Expression
1 0nrp 11017 . . . 4  |-  -.  0  e.  RR+
2 0re 9382 . . . . 5  |-  0  e.  RR
3 logcn.d . . . . . . 7  |-  D  =  ( CC  \  ( -oo (,] 0 ) )
43ellogdm 22043 . . . . . 6  |-  ( 0  e.  D  <->  ( 0  e.  CC  /\  (
0  e.  RR  ->  0  e.  RR+ ) ) )
54simprbi 461 . . . . 5  |-  ( 0  e.  D  ->  (
0  e.  RR  ->  0  e.  RR+ ) )
62, 5mpi 17 . . . 4  |-  ( 0  e.  D  ->  0  e.  RR+ )
71, 6mto 176 . . 3  |-  -.  0  e.  D
8 eleq1 2501 . . 3  |-  ( A  =  0  ->  ( A  e.  D  <->  0  e.  D ) )
97, 8mtbiri 303 . 2  |-  ( A  =  0  ->  -.  A  e.  D )
109necon2ai 2654 1  |-  ( A  e.  D  ->  A  =/=  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761    =/= wne 2604    \ cdif 3322  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   -oocmnf 9412   RR+crp 10987   (,]cioc 11297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-i2m1 9346  ax-1ne0 9347  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-rp 10988  df-ioc 11301
This theorem is referenced by:  logdmss  22046  logcnlem2  22047  logcnlem3  22048  logcnlem4  22049  logcnlem5  22050  logcn  22051  dvloglem  22052  logf1o2  22054  logtayl  22064  logtayl2  22066  cxpcn  22142  atansssdm  22287  lgamgulmlem2  26946  dvcncxp1  28402  dvcnsqr  28403
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