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Theorem dvrelog 19984
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 19923 . . 3  |-  ( log  |`  RR+ )  =  `' ( exp  |`  RR )
21oveq2i 5869 . 2  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( RR 
_D  `' ( exp  |`  RR ) )
3 reeff1o 19823 . . . . . . . . 9  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+
4 f1of 5472 . . . . . . . . 9  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
53, 4ax-mp 8 . . . . . . . 8  |-  ( exp  |`  RR ) : RR --> RR+
6 rpssre 10364 . . . . . . . 8  |-  RR+  C_  RR
7 fss 5397 . . . . . . . 8  |-  ( ( ( exp  |`  RR ) : RR --> RR+  /\  RR+  C_  RR )  ->  ( exp  |`  RR ) : RR --> RR )
85, 6, 7mp2an 653 . . . . . . 7  |-  ( exp  |`  RR ) : RR --> RR
9 ax-resscn 8794 . . . . . . . 8  |-  RR  C_  CC
10 efcn 19819 . . . . . . . . 9  |-  exp  e.  ( CC -cn-> CC )
11 rescncf 18401 . . . . . . . . 9  |-  ( RR  C_  CC  ->  ( exp  e.  ( CC -cn-> CC )  ->  ( exp  |`  RR )  e.  ( RR -cn-> CC ) ) )
129, 10, 11mp2 17 . . . . . . . 8  |-  ( exp  |`  RR )  e.  ( RR -cn-> CC )
13 cncffvrn 18402 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  ( exp  |`  RR )  e.  ( RR -cn-> CC ) )  ->  ( ( exp  |`  RR )  e.  ( RR -cn-> RR )  <-> 
( exp  |`  RR ) : RR --> RR ) )
149, 12, 13mp2an 653 . . . . . . 7  |-  ( ( exp  |`  RR )  e.  ( RR -cn-> RR )  <-> 
( exp  |`  RR ) : RR --> RR )
158, 14mpbir 200 . . . . . 6  |-  ( exp  |`  RR )  e.  ( RR -cn-> RR )
1615a1i 10 . . . . 5  |-  (  T. 
->  ( exp  |`  RR )  e.  ( RR -cn-> RR ) )
17 reex 8828 . . . . . . . . . . 11  |-  RR  e.  _V
1817prid1 3734 . . . . . . . . . 10  |-  RR  e.  { RR ,  CC }
19 eff 12363 . . . . . . . . . 10  |-  exp : CC
--> CC
20 ssid 3197 . . . . . . . . . 10  |-  CC  C_  CC
21 dvef 19327 . . . . . . . . . . . . 13  |-  ( CC 
_D  exp )  =  exp
2221dmeqi 4880 . . . . . . . . . . . 12  |-  dom  ( CC  _D  exp )  =  dom  exp
2319fdmi 5394 . . . . . . . . . . . 12  |-  dom  exp  =  CC
2422, 23eqtri 2303 . . . . . . . . . . 11  |-  dom  ( CC  _D  exp )  =  CC
259, 24sseqtr4i 3211 . . . . . . . . . 10  |-  RR  C_  dom  ( CC  _D  exp )
26 dvres3 19263 . . . . . . . . . 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 ) )
2718, 19, 20, 25, 26mp4an 654 . . . . . . . . 9  |-  ( RR 
_D  ( exp  |`  RR ) )  =  ( ( CC  _D  exp )  |`  RR )
2821reseq1i 4951 . . . . . . . . 9  |-  ( ( CC  _D  exp )  |`  RR )  =  ( exp  |`  RR )
2927, 28eqtri 2303 . . . . . . . 8  |-  ( RR 
_D  ( exp  |`  RR ) )  =  ( exp  |`  RR )
3029dmeqi 4880 . . . . . . 7  |-  dom  ( RR  _D  ( exp  |`  RR ) )  =  dom  ( exp  |`  RR )
315fdmi 5394 . . . . . . 7  |-  dom  ( exp  |`  RR )  =  RR
3230, 31eqtri 2303 . . . . . 6  |-  dom  ( RR  _D  ( exp  |`  RR ) )  =  RR
3332a1i 10 . . . . 5  |-  (  T. 
->  dom  ( RR  _D  ( exp  |`  RR )
)  =  RR )
34 0nrp 10384 . . . . . . 7  |-  -.  0  e.  RR+
3529rneqi 4905 . . . . . . . . 9  |-  ran  ( RR  _D  ( exp  |`  RR ) )  =  ran  ( exp  |`  RR )
36 f1ofo 5479 . . . . . . . . . 10  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  ->  ( exp  |`  RR ) : RR -onto-> RR+ )
37 forn 5454 . . . . . . . . . 10  |-  ( ( exp  |`  RR ) : RR -onto-> RR+  ->  ran  ( exp  |`  RR )  =  RR+ )
383, 36, 37mp2b 9 . . . . . . . . 9  |-  ran  ( exp  |`  RR )  = 
RR+
3935, 38eqtri 2303 . . . . . . . 8  |-  ran  ( RR  _D  ( exp  |`  RR ) )  =  RR+
4039eleq2i 2347 . . . . . . 7  |-  ( 0  e.  ran  ( RR 
_D  ( exp  |`  RR ) )  <->  0  e.  RR+ )
4134, 40mtbir 290 . . . . . 6  |-  -.  0  e.  ran  ( RR  _D  ( exp  |`  RR )
)
4241a1i 10 . . . . 5  |-  (  T. 
->  -.  0  e.  ran  ( RR  _D  ( exp  |`  RR ) ) )
433a1i 10 . . . . 5  |-  (  T. 
->  ( exp  |`  RR ) : RR -1-1-onto-> RR+ )
4416, 33, 42, 43dvcnvre 19366 . . . 4  |-  (  T. 
->  ( RR  _D  `' ( exp  |`  RR )
)  =  ( x  e.  RR+  |->  ( 1  /  ( ( RR 
_D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) ) ) ) )
4544trud 1314 . . 3  |-  ( RR 
_D  `' ( exp  |`  RR ) )  =  ( x  e.  RR+  |->  ( 1  /  (
( RR  _D  ( exp  |`  RR ) ) `
 ( `' ( exp  |`  RR ) `  x ) ) ) )
4629fveq1i 5526 . . . . . 6  |-  ( ( RR  _D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) )  =  ( ( exp  |`  RR ) `  ( `' ( exp  |`  RR ) `
 x ) )
47 f1ocnvfv2 5793 . . . . . . 7  |-  ( ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  /\  x  e.  RR+ )  ->  ( ( exp  |`  RR ) `  ( `' ( exp  |`  RR ) `  x
) )  =  x )
483, 47mpan 651 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( exp  |`  RR ) `  ( `' ( exp  |`  RR ) `  x
) )  =  x )
4946, 48syl5eq 2327 . . . . 5  |-  ( x  e.  RR+  ->  ( ( RR  _D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) )  =  x )
5049oveq2d 5874 . . . 4  |-  ( x  e.  RR+  ->  ( 1  /  ( ( RR 
_D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) ) )  =  ( 1  /  x ) )
5150mpteq2ia 4102 . . 3  |-  ( x  e.  RR+  |->  ( 1  /  ( ( RR 
_D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) ) ) )  =  ( x  e.  RR+  |->  ( 1  /  x ) )
5245, 51eqtri 2303 . 2  |-  ( RR 
_D  `' ( exp  |`  RR ) )  =  ( x  e.  RR+  |->  ( 1  /  x
) )
532, 52eqtri 2303 1  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    T. wtru 1307    = wceq 1623    e. wcel 1684    C_ wss 3152   {cpr 3641    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   -->wf 5251   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    / cdiv 9423   RR+crp 10354   expce 12343   -cn->ccncf 18380    _D cdv 19213   logclog 19912
This theorem is referenced by:  relogcn  19985  advlog  20001  advlogexp  20002  logccv  20010  dvcxp1  20082  loglesqr  20098  logdivsum  20682  log2sumbnd  20693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914
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