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Theorem dvrelog 20000
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 19939 . . 3  |-  ( log  |`  RR+ )  =  `' ( exp  |`  RR )
21oveq2i 5885 . 2  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( RR 
_D  `' ( exp  |`  RR ) )
3 reeff1o 19839 . . . . . . . . 9  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+
4 f1of 5488 . . . . . . . . 9  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
53, 4ax-mp 8 . . . . . . . 8  |-  ( exp  |`  RR ) : RR --> RR+
6 rpssre 10380 . . . . . . . 8  |-  RR+  C_  RR
7 fss 5413 . . . . . . . 8  |-  ( ( ( exp  |`  RR ) : RR --> RR+  /\  RR+  C_  RR )  ->  ( exp  |`  RR ) : RR --> RR )
85, 6, 7mp2an 653 . . . . . . 7  |-  ( exp  |`  RR ) : RR --> RR
9 ax-resscn 8810 . . . . . . . 8  |-  RR  C_  CC
10 efcn 19835 . . . . . . . . 9  |-  exp  e.  ( CC -cn-> CC )
11 rescncf 18417 . . . . . . . . 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 18418 . . . . . . . 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 8844 . . . . . . . . . . 11  |-  RR  e.  _V
1817prid1 3747 . . . . . . . . . 10  |-  RR  e.  { RR ,  CC }
19 eff 12379 . . . . . . . . . 10  |-  exp : CC
--> CC
20 ssid 3210 . . . . . . . . . 10  |-  CC  C_  CC
21 dvef 19343 . . . . . . . . . . . . 13  |-  ( CC 
_D  exp )  =  exp
2221dmeqi 4896 . . . . . . . . . . . 12  |-  dom  ( CC  _D  exp )  =  dom  exp
2319fdmi 5410 . . . . . . . . . . . 12  |-  dom  exp  =  CC
2422, 23eqtri 2316 . . . . . . . . . . 11  |-  dom  ( CC  _D  exp )  =  CC
259, 24sseqtr4i 3224 . . . . . . . . . 10  |-  RR  C_  dom  ( CC  _D  exp )
26 dvres3 19279 . . . . . . . . . 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 4967 . . . . . . . . 9  |-  ( ( CC  _D  exp )  |`  RR )  =  ( exp  |`  RR )
2927, 28eqtri 2316 . . . . . . . 8  |-  ( RR 
_D  ( exp  |`  RR ) )  =  ( exp  |`  RR )
3029dmeqi 4896 . . . . . . 7  |-  dom  ( RR  _D  ( exp  |`  RR ) )  =  dom  ( exp  |`  RR )
315fdmi 5410 . . . . . . 7  |-  dom  ( exp  |`  RR )  =  RR
3230, 31eqtri 2316 . . . . . 6  |-  dom  ( RR  _D  ( exp  |`  RR ) )  =  RR
3332a1i 10 . . . . 5  |-  (  T. 
->  dom  ( RR  _D  ( exp  |`  RR )
)  =  RR )
34 0nrp 10400 . . . . . . 7  |-  -.  0  e.  RR+
3529rneqi 4921 . . . . . . . . 9  |-  ran  ( RR  _D  ( exp  |`  RR ) )  =  ran  ( exp  |`  RR )
36 f1ofo 5495 . . . . . . . . . 10  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  ->  ( exp  |`  RR ) : RR -onto-> RR+ )
37 forn 5470 . . . . . . . . . 10  |-  ( ( exp  |`  RR ) : RR -onto-> RR+  ->  ran  ( exp  |`  RR )  =  RR+ )
383, 36, 37mp2b 9 . . . . . . . . 9  |-  ran  ( exp  |`  RR )  = 
RR+
3935, 38eqtri 2316 . . . . . . . 8  |-  ran  ( RR  _D  ( exp  |`  RR ) )  =  RR+
4039eleq2i 2360 . . . . . . 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 19382 . . . 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 5542 . . . . . 6  |-  ( ( RR  _D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) )  =  ( ( exp  |`  RR ) `  ( `' ( exp  |`  RR ) `
 x ) )
47 f1ocnvfv2 5809 . . . . . . 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 2340 . . . . 5  |-  ( x  e.  RR+  ->  ( ( RR  _D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) )  =  x )
5049oveq2d 5890 . . . 4  |-  ( x  e.  RR+  ->  ( 1  /  ( ( RR 
_D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) ) )  =  ( 1  /  x ) )
5150mpteq2ia 4118 . . 3  |-  ( x  e.  RR+  |->  ( 1  /  ( ( RR 
_D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) ) ) )  =  ( x  e.  RR+  |->  ( 1  /  x ) )
5245, 51eqtri 2316 . 2  |-  ( RR 
_D  `' ( exp  |`  RR ) )  =  ( x  e.  RR+  |->  ( 1  /  x
) )
532, 52eqtri 2316 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 1632    e. wcel 1696    C_ wss 3165   {cpr 3654    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   -->wf 5267   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    / cdiv 9439   RR+crp 10370   expce 12359   -cn->ccncf 18396    _D cdv 19229   logclog 19928
This theorem is referenced by:  relogcn  20001  advlog  20017  advlogexp  20018  logccv  20026  dvcxp1  20098  loglesqr  20114  logdivsum  20698  log2sumbnd  20709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930
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