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Theorem logdmnrp 20489
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 3434 . . 3  |-  ( A  e.  ( CC  \ 
(  -oo (,] 0 ) )  ->  -.  A  e.  (  -oo (,] 0
) )
2 logcn.d . . 3  |-  D  =  ( CC  \  (  -oo (,] 0 ) )
31, 2eleq2s 2500 . 2  |-  ( A  e.  D  ->  -.  A  e.  (  -oo (,] 0 ) )
4 rpre 10578 . . . . 5  |-  ( -u A  e.  RR+  ->  -u A  e.  RR )
52ellogdm 20487 . . . . . . 7  |-  ( A  e.  D  <->  ( A  e.  CC  /\  ( A  e.  RR  ->  A  e.  RR+ ) ) )
65simplbi 447 . . . . . 6  |-  ( A  e.  D  ->  A  e.  CC )
7 negreb 9326 . . . . . 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 211 . . . 4  |-  ( A  e.  D  ->  ( -u A  e.  RR+  ->  A  e.  RR ) )
109imp 419 . . 3  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  A  e.  RR )
11 mnflt 10682 . . . 4  |-  ( A  e.  RR  ->  -oo  <  A )
1210, 11syl 16 . . 3  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  -oo  <  A )
13 rpgt0 10583 . . . . . 6  |-  ( -u A  e.  RR+  ->  0  <  -u A )
1413adantl 453 . . . . 5  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  0  <  -u A
)
1510lt0neg1d 9556 . . . . 5  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  ( A  <  0  <->  0  <  -u A ) )
1614, 15mpbird 224 . . . 4  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  A  <  0 )
17 0re 9051 . . . . 5  |-  0  e.  RR
18 ltle 9123 . . . . 5  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A  <  0  ->  A  <_  0 ) )
1910, 17, 18sylancl 644 . . . 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 10674 . . . 4  |-  -oo  e.  RR*
22 elioc2 10933 . . . 4  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR )  ->  ( A  e.  (  -oo (,] 0 )  <->  ( A  e.  RR  /\  -oo  <  A  /\  A  <_  0
) ) )
2321, 17, 22mp2an 654 . . 3  |-  ( A  e.  (  -oo (,] 0 )  <->  ( A  e.  RR  /\  -oo  <  A  /\  A  <_  0
) )
2410, 12, 20, 23syl3anbrc 1138 . 2  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  A  e.  (  -oo (,] 0 ) )
253, 24mtand 641 1  |-  ( A  e.  D  ->  -.  -u A  e.  RR+ )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    \ cdif 3281   class class class wbr 4176  (class class class)co 6044   CCcc 8948   RRcr 8949   0cc0 8950    -oocmnf 9078   RR*cxr 9079    < clt 9080    <_ cle 9081   -ucneg 9252   RR+crp 10572   (,]cioc 10877
This theorem is referenced by:  dvloglem  20496  logf1o2  20498
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-po 4467  df-so 4468  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-riota 6512  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-rp 10573  df-ioc 10881
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