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Theorem dvrelog 23574
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 23507 . . 3  |-  ( log  |`  RR+ )  =  `' ( exp  |`  RR )
21oveq2i 6314 . 2  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( RR 
_D  `' ( exp  |`  RR ) )
3 reeff1o 23394 . . . . . . . . 9  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+
4 f1of 5829 . . . . . . . . 9  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
53, 4ax-mp 5 . . . . . . . 8  |-  ( exp  |`  RR ) : RR --> RR+
6 rpssre 11314 . . . . . . . 8  |-  RR+  C_  RR
7 fss 5752 . . . . . . . 8  |-  ( ( ( exp  |`  RR ) : RR --> RR+  /\  RR+  C_  RR )  ->  ( exp  |`  RR ) : RR --> RR )
85, 6, 7mp2an 677 . . . . . . 7  |-  ( exp  |`  RR ) : RR --> RR
9 ax-resscn 9598 . . . . . . . 8  |-  RR  C_  CC
10 efcn 23390 . . . . . . . . 9  |-  exp  e.  ( CC -cn-> CC )
11 rescncf 21921 . . . . . . . . 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 21922 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  ( exp  |`  RR )  e.  ( RR -cn-> CC ) )  ->  ( ( exp  |`  RR )  e.  ( RR -cn-> RR )  <-> 
( exp  |`  RR ) : RR --> RR ) )
149, 12, 13mp2an 677 . . . . . . 7  |-  ( ( exp  |`  RR )  e.  ( RR -cn-> RR )  <-> 
( exp  |`  RR ) : RR --> RR )
158, 14mpbir 213 . . . . . 6  |-  ( exp  |`  RR )  e.  ( RR -cn-> RR )
1615a1i 11 . . . . 5  |-  ( T. 
->  ( exp  |`  RR )  e.  ( RR -cn-> RR ) )
17 reelprrecn 9633 . . . . . . . . . 10  |-  RR  e.  { RR ,  CC }
18 eff 14129 . . . . . . . . . 10  |-  exp : CC
--> CC
19 ssid 3484 . . . . . . . . . 10  |-  CC  C_  CC
20 dvef 22924 . . . . . . . . . . . . 13  |-  ( CC 
_D  exp )  =  exp
2120dmeqi 5053 . . . . . . . . . . . 12  |-  dom  ( CC  _D  exp )  =  dom  exp
2218fdmi 5749 . . . . . . . . . . . 12  |-  dom  exp  =  CC
2321, 22eqtri 2452 . . . . . . . . . . 11  |-  dom  ( CC  _D  exp )  =  CC
249, 23sseqtr4i 3498 . . . . . . . . . 10  |-  RR  C_  dom  ( CC  _D  exp )
25 dvres3 22860 . . . . . . . . . 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 678 . . . . . . . . 9  |-  ( RR 
_D  ( exp  |`  RR ) )  =  ( ( CC  _D  exp )  |`  RR )
2720reseq1i 5118 . . . . . . . . 9  |-  ( ( CC  _D  exp )  |`  RR )  =  ( exp  |`  RR )
2826, 27eqtri 2452 . . . . . . . 8  |-  ( RR 
_D  ( exp  |`  RR ) )  =  ( exp  |`  RR )
2928dmeqi 5053 . . . . . . 7  |-  dom  ( RR  _D  ( exp  |`  RR ) )  =  dom  ( exp  |`  RR )
305fdmi 5749 . . . . . . 7  |-  dom  ( exp  |`  RR )  =  RR
3129, 30eqtri 2452 . . . . . 6  |-  dom  ( RR  _D  ( exp  |`  RR ) )  =  RR
3231a1i 11 . . . . 5  |-  ( T. 
->  dom  ( RR  _D  ( exp  |`  RR )
)  =  RR )
33 0nrp 11336 . . . . . . 7  |-  -.  0  e.  RR+
3428rneqi 5078 . . . . . . . . 9  |-  ran  ( RR  _D  ( exp  |`  RR ) )  =  ran  ( exp  |`  RR )
35 f1ofo 5836 . . . . . . . . . 10  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  ->  ( exp  |`  RR ) : RR -onto-> RR+ )
36 forn 5811 . . . . . . . . . 10  |-  ( ( exp  |`  RR ) : RR -onto-> RR+  ->  ran  ( exp  |`  RR )  =  RR+ )
373, 35, 36mp2b 10 . . . . . . . . 9  |-  ran  ( exp  |`  RR )  = 
RR+
3834, 37eqtri 2452 . . . . . . . 8  |-  ran  ( RR  _D  ( exp  |`  RR ) )  =  RR+
3938eleq2i 2501 . . . . . . 7  |-  ( 0  e.  ran  ( RR 
_D  ( exp  |`  RR ) )  <->  0  e.  RR+ )
4033, 39mtbir 301 . . . . . 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 22963 . . . 4  |-  ( T. 
->  ( RR  _D  `' ( exp  |`  RR )
)  =  ( x  e.  RR+  |->  ( 1  /  ( ( RR 
_D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) ) ) ) )
4443trud 1447 . . 3  |-  ( RR 
_D  `' ( exp  |`  RR ) )  =  ( x  e.  RR+  |->  ( 1  /  (
( RR  _D  ( exp  |`  RR ) ) `
 ( `' ( exp  |`  RR ) `  x ) ) ) )
4528fveq1i 5880 . . . . . 6  |-  ( ( RR  _D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) )  =  ( ( exp  |`  RR ) `  ( `' ( exp  |`  RR ) `
 x ) )
46 f1ocnvfv2 6189 . . . . . . 7  |-  ( ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  /\  x  e.  RR+ )  ->  ( ( exp  |`  RR ) `  ( `' ( exp  |`  RR ) `  x
) )  =  x )
473, 46mpan 675 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( exp  |`  RR ) `  ( `' ( exp  |`  RR ) `  x
) )  =  x )
4845, 47syl5eq 2476 . . . . 5  |-  ( x  e.  RR+  ->  ( ( RR  _D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) )  =  x )
4948oveq2d 6319 . . . 4  |-  ( x  e.  RR+  ->  ( 1  /  ( ( RR 
_D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) ) )  =  ( 1  /  x ) )
5049mpteq2ia 4504 . . 3  |-  ( x  e.  RR+  |->  ( 1  /  ( ( RR 
_D  ( exp  |`  RR ) ) `  ( `' ( exp  |`  RR ) `
 x ) ) ) )  =  ( x  e.  RR+  |->  ( 1  /  x ) )
5144, 50eqtri 2452 . 2  |-  ( RR 
_D  `' ( exp  |`  RR ) )  =  ( x  e.  RR+  |->  ( 1  /  x
) )
522, 51eqtri 2452 1  |-  ( RR 
_D  ( log  |`  RR+ )
)  =  ( x  e.  RR+  |->  ( 1  /  x ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    = wceq 1438   T. wtru 1439    e. wcel 1869    C_ wss 3437   {cpr 3999    |-> cmpt 4480   `'ccnv 4850   dom cdm 4851   ran crn 4852    |` cres 4853   -->wf 5595   -onto->wfo 5597   -1-1-onto->wf1o 5598   ` cfv 5599  (class class class)co 6303   CCcc 9539   RRcr 9540   0cc0 9541   1c1 9542    / cdiv 10271   RR+crp 11304   expce 14107   -cn->ccncf 21900    _D cdv 22810   logclog 23496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619  ax-addf 9620  ax-mulf 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-ixp 7529  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-fi 7929  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-q 11267  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-ioo 11641  df-ioc 11642  df-ico 11643  df-icc 11644  df-fz 11787  df-fzo 11918  df-fl 12029  df-mod 12098  df-seq 12215  df-exp 12274  df-fac 12461  df-bc 12489  df-hash 12517  df-shft 13124  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117  df-pi 14119  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19913  df-bases 19914  df-topon 19915  df-topsp 19916  df-cld 20026  df-ntr 20027  df-cls 20028  df-nei 20106  df-lp 20144  df-perf 20145  df-cn 20235  df-cnp 20236  df-haus 20323  df-cmp 20394  df-tx 20569  df-hmeo 20762  df-fil 20853  df-fm 20945  df-flim 20946  df-flf 20947  df-xms 21327  df-ms 21328  df-tms 21329  df-cncf 21902  df-limc 22813  df-dv 22814  df-log 23498
This theorem is referenced by:  relogcn  23575  advlog  23591  advlogexp  23592  logccv  23600  dvcxp1  23672  loglesqrt  23690  logdivsum  24363  log2sumbnd  24374
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