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Theorem logdmnrp 20537
Description: A number in the continuous domain of  log is not a strictly negative number. (Contributed by Mario Carneiro, 18-Feb-2015.)
Hypothesis
Ref Expression
logcn.d  |-  D  =  ( CC  \  (  -oo (,] 0 ) )
Assertion
Ref Expression
logdmnrp  |-  ( A  e.  D  ->  -.  -u A  e.  RR+ )

Proof of Theorem logdmnrp
StepHypRef Expression
1 eldifn 3472 . . 3  |-  ( A  e.  ( CC  \ 
(  -oo (,] 0 ) )  ->  -.  A  e.  (  -oo (,] 0
) )
2 logcn.d . . 3  |-  D  =  ( CC  \  (  -oo (,] 0 ) )
31, 2eleq2s 2530 . 2  |-  ( A  e.  D  ->  -.  A  e.  (  -oo (,] 0 ) )
4 rpre 10623 . . . . 5  |-  ( -u A  e.  RR+  ->  -u A  e.  RR )
52ellogdm 20535 . . . . . . 7  |-  ( A  e.  D  <->  ( A  e.  CC  /\  ( A  e.  RR  ->  A  e.  RR+ ) ) )
65simplbi 448 . . . . . 6  |-  ( A  e.  D  ->  A  e.  CC )
7 negreb 9371 . . . . . 6  |-  ( A  e.  CC  ->  ( -u A  e.  RR  <->  A  e.  RR ) )
86, 7syl 16 . . . . 5  |-  ( A  e.  D  ->  ( -u A  e.  RR  <->  A  e.  RR ) )
94, 8syl5ib 212 . . . 4  |-  ( A  e.  D  ->  ( -u A  e.  RR+  ->  A  e.  RR ) )
109imp 420 . . 3  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  A  e.  RR )
11 mnflt 10727 . . . 4  |-  ( A  e.  RR  ->  -oo  <  A )
1210, 11syl 16 . . 3  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  -oo  <  A )
13 rpgt0 10628 . . . . . 6  |-  ( -u A  e.  RR+  ->  0  <  -u A )
1413adantl 454 . . . . 5  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  0  <  -u A
)
1510lt0neg1d 9601 . . . . 5  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  ( A  <  0  <->  0  <  -u A ) )
1614, 15mpbird 225 . . . 4  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  A  <  0 )
17 0re 9096 . . . . 5  |-  0  e.  RR
18 ltle 9168 . . . . 5  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A  <  0  ->  A  <_  0 ) )
1910, 17, 18sylancl 645 . . . 4  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  ( A  <  0  ->  A  <_  0 ) )
2016, 19mpd 15 . . 3  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  A  <_  0 )
21 mnfxr 10719 . . . 4  |-  -oo  e.  RR*
22 elioc2 10978 . . . 4  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR )  ->  ( A  e.  (  -oo (,] 0 )  <->  ( A  e.  RR  /\  -oo  <  A  /\  A  <_  0
) ) )
2321, 17, 22mp2an 655 . . 3  |-  ( A  e.  (  -oo (,] 0 )  <->  ( A  e.  RR  /\  -oo  <  A  /\  A  <_  0
) )
2410, 12, 20, 23syl3anbrc 1139 . 2  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  A  e.  (  -oo (,] 0 ) )
253, 24mtand 642 1  |-  ( A  e.  D  ->  -.  -u A  e.  RR+ )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    \ cdif 3319   class class class wbr 4215  (class class class)co 6084   CCcc 8993   RRcr 8994   0cc0 8995    -oocmnf 9123   RR*cxr 9124    < clt 9125    <_ cle 9126   -ucneg 9297   RR+crp 10617   (,]cioc 10922
This theorem is referenced by:  dvloglem  20544  logf1o2  20546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-rp 10618  df-ioc 10926
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