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Theorem logdmnrp 22220
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 3588 . . 3  |-  ( A  e.  ( CC  \ 
( -oo (,] 0 ) )  ->  -.  A  e.  ( -oo (,] 0
) )
2 logcn.d . . 3  |-  D  =  ( CC  \  ( -oo (,] 0 ) )
31, 2eleq2s 2562 . 2  |-  ( A  e.  D  ->  -.  A  e.  ( -oo (,] 0 ) )
4 rpre 11109 . . . . 5  |-  ( -u A  e.  RR+  ->  -u A  e.  RR )
52ellogdm 22218 . . . . . . 7  |-  ( A  e.  D  <->  ( A  e.  CC  /\  ( A  e.  RR  ->  A  e.  RR+ ) ) )
65simplbi 460 . . . . . 6  |-  ( A  e.  D  ->  A  e.  CC )
7 negreb 9786 . . . . . 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 219 . . . 4  |-  ( A  e.  D  ->  ( -u A  e.  RR+  ->  A  e.  RR ) )
109imp 429 . . 3  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  A  e.  RR )
11 mnflt 11216 . . . 4  |-  ( A  e.  RR  -> -oo  <  A )
1210, 11syl 16 . . 3  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  -> -oo  <  A )
13 rpgt0 11114 . . . . . 6  |-  ( -u A  e.  RR+  ->  0  <  -u A )
1413adantl 466 . . . . 5  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  0  <  -u A
)
1510lt0neg1d 10021 . . . . 5  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  ( A  <  0  <->  0  <  -u A ) )
1614, 15mpbird 232 . . . 4  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  A  <  0 )
17 0re 9498 . . . . 5  |-  0  e.  RR
18 ltle 9575 . . . . 5  |-  ( ( A  e.  RR  /\  0  e.  RR )  ->  ( A  <  0  ->  A  <_  0 ) )
1910, 17, 18sylancl 662 . . . 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 11206 . . . 4  |- -oo  e.  RR*
22 elioc2 11470 . . . 4  |-  ( ( -oo  e.  RR*  /\  0  e.  RR )  ->  ( A  e.  ( -oo (,] 0 )  <->  ( A  e.  RR  /\ -oo  <  A  /\  A  <_  0
) ) )
2321, 17, 22mp2an 672 . . 3  |-  ( A  e.  ( -oo (,] 0 )  <->  ( A  e.  RR  /\ -oo  <  A  /\  A  <_  0
) )
2410, 12, 20, 23syl3anbrc 1172 . 2  |-  ( ( A  e.  D  /\  -u A  e.  RR+ )  ->  A  e.  ( -oo (,] 0 ) )
253, 24mtand 659 1  |-  ( A  e.  D  ->  -.  -u 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 3434   class class class wbr 4401  (class class class)co 6201   CCcc 9392   RRcr 9393   0cc0 9394   -oocmnf 9528   RR*cxr 9529    < clt 9530    <_ cle 9531   -ucneg 9708   RR+crp 11103   (,]cioc 11413
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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-rp 11104  df-ioc 11417
This theorem is referenced by:  dvloglem  22227  logf1o2  22229
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