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Theorem dvrelog 22746
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 22681 . . 3  |-  ( log  |`  RR+ )  =  `' ( exp  |`  RR )
21oveq2i 6293 . 2  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( RR 
_D  `' ( exp  |`  RR ) )
3 reeff1o 22576 . . . . . . . . 9  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+
4 f1of 5814 . . . . . . . . 9  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
53, 4ax-mp 5 . . . . . . . 8  |-  ( exp  |`  RR ) : RR --> RR+
6 rpssre 11226 . . . . . . . 8  |-  RR+  C_  RR
7 fss 5737 . . . . . . . 8  |-  ( ( ( exp  |`  RR ) : RR --> RR+  /\  RR+  C_  RR )  ->  ( exp  |`  RR ) : RR --> RR )
85, 6, 7mp2an 672 . . . . . . 7  |-  ( exp  |`  RR ) : RR --> RR
9 ax-resscn 9545 . . . . . . . 8  |-  RR  C_  CC
10 efcn 22572 . . . . . . . . 9  |-  exp  e.  ( CC -cn-> CC )
11 rescncf 21136 . . . . . . . . 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 21137 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  ( exp  |`  RR )  e.  ( RR -cn-> CC ) )  ->  ( ( exp  |`  RR )  e.  ( RR -cn-> RR )  <-> 
( exp  |`  RR ) : RR --> RR ) )
149, 12, 13mp2an 672 . . . . . . 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 9580 . . . . . . . . . 10  |-  RR  e.  { RR ,  CC }
18 eff 13675 . . . . . . . . . 10  |-  exp : CC
--> CC
19 ssid 3523 . . . . . . . . . 10  |-  CC  C_  CC
20 dvef 22116 . . . . . . . . . . . . 13  |-  ( CC 
_D  exp )  =  exp
2120dmeqi 5202 . . . . . . . . . . . 12  |-  dom  ( CC  _D  exp )  =  dom  exp
2218fdmi 5734 . . . . . . . . . . . 12  |-  dom  exp  =  CC
2321, 22eqtri 2496 . . . . . . . . . . 11  |-  dom  ( CC  _D  exp )  =  CC
249, 23sseqtr4i 3537 . . . . . . . . . 10  |-  RR  C_  dom  ( CC  _D  exp )
25 dvres3 22052 . . . . . . . . . 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 673 . . . . . . . . 9  |-  ( RR 
_D  ( exp  |`  RR ) )  =  ( ( CC  _D  exp )  |`  RR )
2720reseq1i 5267 . . . . . . . . 9  |-  ( ( CC  _D  exp )  |`  RR )  =  ( exp  |`  RR )
2826, 27eqtri 2496 . . . . . . . 8  |-  ( RR 
_D  ( exp  |`  RR ) )  =  ( exp  |`  RR )
2928dmeqi 5202 . . . . . . 7  |-  dom  ( RR  _D  ( exp  |`  RR ) )  =  dom  ( exp  |`  RR )
305fdmi 5734 . . . . . . 7  |-  dom  ( exp  |`  RR )  =  RR
3129, 30eqtri 2496 . . . . . 6  |-  dom  ( RR  _D  ( exp  |`  RR ) )  =  RR
3231a1i 11 . . . . 5  |-  ( T. 
->  dom  ( RR  _D  ( exp  |`  RR )
)  =  RR )
33 0nrp 11246 . . . . . . 7  |-  -.  0  e.  RR+
3428rneqi 5227 . . . . . . . . 9  |-  ran  ( RR  _D  ( exp  |`  RR ) )  =  ran  ( exp  |`  RR )
35 f1ofo 5821 . . . . . . . . . 10  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  ->  ( exp  |`  RR ) : RR -onto-> RR+ )
36 forn 5796 . . . . . . . . . 10  |-  ( ( exp  |`  RR ) : RR -onto-> RR+  ->  ran  ( exp  |`  RR )  =  RR+ )
373, 35, 36mp2b 10 . . . . . . . . 9  |-  ran  ( exp  |`  RR )  = 
RR+
3834, 37eqtri 2496 . . . . . . . 8  |-  ran  ( RR  _D  ( exp  |`  RR ) )  =  RR+
3938eleq2i 2545 . . . . . . 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 22155 . . . 4  |-  ( T. 
->  ( RR  _D  `' ( exp  |`  RR )
)  =  ( x  e.  RR+  |->  ( 1  /  ( ( RR 
_D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) ) ) ) )
4443trud 1388 . . 3  |-  ( RR 
_D  `' ( exp  |`  RR ) )  =  ( x  e.  RR+  |->  ( 1  /  (
( RR  _D  ( exp  |`  RR ) ) `
 ( `' ( exp  |`  RR ) `  x ) ) ) )
4528fveq1i 5865 . . . . . 6  |-  ( ( RR  _D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) )  =  ( ( exp  |`  RR ) `  ( `' ( exp  |`  RR ) `
 x ) )
46 f1ocnvfv2 6169 . . . . . . 7  |-  ( ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  /\  x  e.  RR+ )  ->  ( ( exp  |`  RR ) `  ( `' ( exp  |`  RR ) `  x
) )  =  x )
473, 46mpan 670 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( exp  |`  RR ) `  ( `' ( exp  |`  RR ) `  x
) )  =  x )
4845, 47syl5eq 2520 . . . . 5  |-  ( x  e.  RR+  ->  ( ( RR  _D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) )  =  x )
4948oveq2d 6298 . . . 4  |-  ( x  e.  RR+  ->  ( 1  /  ( ( RR 
_D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) ) )  =  ( 1  /  x ) )
5049mpteq2ia 4529 . . 3  |-  ( x  e.  RR+  |->  ( 1  /  ( ( RR 
_D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) ) ) )  =  ( x  e.  RR+  |->  ( 1  /  x ) )
5144, 50eqtri 2496 . 2  |-  ( RR 
_D  `' ( exp  |`  RR ) )  =  ( x  e.  RR+  |->  ( 1  /  x
) )
522, 51eqtri 2496 1  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1379   T. wtru 1380    e. wcel 1767    C_ wss 3476   {cpr 4029    |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001   -->wf 5582   -onto->wfo 5584   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    / cdiv 10202   RR+crp 11216   expce 13655   -cn->ccncf 21115    _D cdv 22002   logclog 22670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-shft 12859  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-limsup 13253  df-clim 13270  df-rlim 13271  df-sum 13468  df-ef 13661  df-sin 13663  df-cos 13664  df-pi 13666  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-lp 19403  df-perf 19404  df-cn 19494  df-cnp 19495  df-haus 19582  df-cmp 19653  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cncf 21117  df-limc 22005  df-dv 22006  df-log 22672
This theorem is referenced by:  relogcn  22747  advlog  22763  advlogexp  22764  logccv  22772  dvcxp1  22844  loglesqrt  22860  logdivsum  23446  log2sumbnd  23457
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