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Theorem dvrelog 21967
Description: The derivative of the real logarithm function. (Contributed by Mario Carneiro, 24-Feb-2015.)
Assertion
Ref Expression
dvrelog  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )

Proof of Theorem dvrelog
StepHypRef Expression
1 dfrelog 21902 . . 3  |-  ( log  |`  RR+ )  =  `' ( exp  |`  RR )
21oveq2i 6091 . 2  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( RR 
_D  `' ( exp  |`  RR ) )
3 reeff1o 21797 . . . . . . . . 9  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+
4 f1of 5629 . . . . . . . . 9  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
53, 4ax-mp 5 . . . . . . . 8  |-  ( exp  |`  RR ) : RR --> RR+
6 rpssre 10989 . . . . . . . 8  |-  RR+  C_  RR
7 fss 5555 . . . . . . . 8  |-  ( ( ( exp  |`  RR ) : RR --> RR+  /\  RR+  C_  RR )  ->  ( exp  |`  RR ) : RR --> RR )
85, 6, 7mp2an 665 . . . . . . 7  |-  ( exp  |`  RR ) : RR --> RR
9 ax-resscn 9327 . . . . . . . 8  |-  RR  C_  CC
10 efcn 21793 . . . . . . . . 9  |-  exp  e.  ( CC -cn-> CC )
11 rescncf 20315 . . . . . . . . 9  |-  ( RR  C_  CC  ->  ( exp  e.  ( CC -cn-> CC )  ->  ( exp  |`  RR )  e.  ( RR -cn-> CC ) ) )
129, 10, 11mp2 9 . . . . . . . 8  |-  ( exp  |`  RR )  e.  ( RR -cn-> CC )
13 cncffvrn 20316 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  ( exp  |`  RR )  e.  ( RR -cn-> CC ) )  ->  ( ( exp  |`  RR )  e.  ( RR -cn-> RR )  <-> 
( exp  |`  RR ) : RR --> RR ) )
149, 12, 13mp2an 665 . . . . . . 7  |-  ( ( exp  |`  RR )  e.  ( RR -cn-> RR )  <-> 
( exp  |`  RR ) : RR --> RR )
158, 14mpbir 209 . . . . . 6  |-  ( exp  |`  RR )  e.  ( RR -cn-> RR )
1615a1i 11 . . . . 5  |-  ( T. 
->  ( exp  |`  RR )  e.  ( RR -cn-> RR ) )
17 reelprrecn 9362 . . . . . . . . . 10  |-  RR  e.  { RR ,  CC }
18 eff 13350 . . . . . . . . . 10  |-  exp : CC
--> CC
19 ssid 3363 . . . . . . . . . 10  |-  CC  C_  CC
20 dvef 21294 . . . . . . . . . . . . 13  |-  ( CC 
_D  exp )  =  exp
2120dmeqi 5028 . . . . . . . . . . . 12  |-  dom  ( CC  _D  exp )  =  dom  exp
2218fdmi 5552 . . . . . . . . . . . 12  |-  dom  exp  =  CC
2321, 22eqtri 2453 . . . . . . . . . . 11  |-  dom  ( CC  _D  exp )  =  CC
249, 23sseqtr4i 3377 . . . . . . . . . 10  |-  RR  C_  dom  ( CC  _D  exp )
25 dvres3 21230 . . . . . . . . . 10  |-  ( ( ( RR  e.  { RR ,  CC }  /\  exp : CC --> CC )  /\  ( CC  C_  CC  /\  RR  C_  dom  ( CC  _D  exp )
) )  ->  ( RR  _D  ( exp  |`  RR ) )  =  ( ( CC  _D  exp )  |`  RR ) )
2617, 18, 19, 24, 25mp4an 666 . . . . . . . . 9  |-  ( RR 
_D  ( exp  |`  RR ) )  =  ( ( CC  _D  exp )  |`  RR )
2720reseq1i 5093 . . . . . . . . 9  |-  ( ( CC  _D  exp )  |`  RR )  =  ( exp  |`  RR )
2826, 27eqtri 2453 . . . . . . . 8  |-  ( RR 
_D  ( exp  |`  RR ) )  =  ( exp  |`  RR )
2928dmeqi 5028 . . . . . . 7  |-  dom  ( RR  _D  ( exp  |`  RR ) )  =  dom  ( exp  |`  RR )
305fdmi 5552 . . . . . . 7  |-  dom  ( exp  |`  RR )  =  RR
3129, 30eqtri 2453 . . . . . 6  |-  dom  ( RR  _D  ( exp  |`  RR ) )  =  RR
3231a1i 11 . . . . 5  |-  ( T. 
->  dom  ( RR  _D  ( exp  |`  RR )
)  =  RR )
33 0nrp 11009 . . . . . . 7  |-  -.  0  e.  RR+
3428rneqi 5053 . . . . . . . . 9  |-  ran  ( RR  _D  ( exp  |`  RR ) )  =  ran  ( exp  |`  RR )
35 f1ofo 5636 . . . . . . . . . 10  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  ->  ( exp  |`  RR ) : RR -onto-> RR+ )
36 forn 5611 . . . . . . . . . 10  |-  ( ( exp  |`  RR ) : RR -onto-> RR+  ->  ran  ( exp  |`  RR )  =  RR+ )
373, 35, 36mp2b 10 . . . . . . . . 9  |-  ran  ( exp  |`  RR )  = 
RR+
3834, 37eqtri 2453 . . . . . . . 8  |-  ran  ( RR  _D  ( exp  |`  RR ) )  =  RR+
3938eleq2i 2497 . . . . . . 7  |-  ( 0  e.  ran  ( RR 
_D  ( exp  |`  RR ) )  <->  0  e.  RR+ )
4033, 39mtbir 299 . . . . . 6  |-  -.  0  e.  ran  ( RR  _D  ( exp  |`  RR )
)
4140a1i 11 . . . . 5  |-  ( T. 
->  -.  0  e.  ran  ( RR  _D  ( exp  |`  RR ) ) )
423a1i 11 . . . . 5  |-  ( T. 
->  ( exp  |`  RR ) : RR -1-1-onto-> RR+ )
4316, 32, 41, 42dvcnvre 21333 . . . 4  |-  ( T. 
->  ( RR  _D  `' ( exp  |`  RR )
)  =  ( x  e.  RR+  |->  ( 1  /  ( ( RR 
_D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) ) ) ) )
4443trud 1371 . . 3  |-  ( RR 
_D  `' ( exp  |`  RR ) )  =  ( x  e.  RR+  |->  ( 1  /  (
( RR  _D  ( exp  |`  RR ) ) `
 ( `' ( exp  |`  RR ) `  x ) ) ) )
4528fveq1i 5680 . . . . . 6  |-  ( ( RR  _D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) )  =  ( ( exp  |`  RR ) `  ( `' ( exp  |`  RR ) `
 x ) )
46 f1ocnvfv2 5971 . . . . . . 7  |-  ( ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  /\  x  e.  RR+ )  ->  ( ( exp  |`  RR ) `  ( `' ( exp  |`  RR ) `  x
) )  =  x )
473, 46mpan 663 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( exp  |`  RR ) `  ( `' ( exp  |`  RR ) `  x
) )  =  x )
4845, 47syl5eq 2477 . . . . 5  |-  ( x  e.  RR+  ->  ( ( RR  _D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) )  =  x )
4948oveq2d 6096 . . . 4  |-  ( x  e.  RR+  ->  ( 1  /  ( ( RR 
_D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) ) )  =  ( 1  /  x ) )
5049mpteq2ia 4362 . . 3  |-  ( x  e.  RR+  |->  ( 1  /  ( ( RR 
_D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) ) ) )  =  ( x  e.  RR+  |->  ( 1  /  x ) )
5144, 50eqtri 2453 . 2  |-  ( RR 
_D  `' ( exp  |`  RR ) )  =  ( x  e.  RR+  |->  ( 1  /  x
) )
522, 51eqtri 2453 1  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1362   T. wtru 1363    e. wcel 1755    C_ wss 3316   {cpr 3867    e. cmpt 4338   `'ccnv 4826   dom cdm 4827   ran crn 4828    |` cres 4829   -->wf 5402   -onto->wfo 5404   -1-1-onto->wf1o 5405   ` cfv 5406  (class class class)co 6080   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271    / cdiv 9981   RR+crp 10979   expce 13330   -cn->ccncf 20294    _D cdv 21180   logclog 21891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-shft 12540  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-limsup 12933  df-clim 12950  df-rlim 12951  df-sum 13148  df-ef 13336  df-sin 13338  df-cos 13339  df-pi 13341  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-cmp 18832  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-log 21893
This theorem is referenced by:  relogcn  21968  advlog  21984  advlogexp  21985  logccv  21993  dvcxp1  22065  loglesqr  22081  logdivsum  22667  log2sumbnd  22678
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