dvrelog - Metamath Proof Explorer

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Theorem dvrelog 24383

Description: The derivative of the real logarithm function. (Contributed by Mario Carneiro, 24-Feb-2015.)

**Assertion**
Ref	Expression
dvrelog	⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))

**Proof of Theorem dvrelog**
Step	Hyp	Ref	Expression
1		dfrelog 24312	. . 3 ⊢ (log ↾ ℝ⁺) = ^◡(exp ↾ ℝ)
2	1	oveq2i 6661	. 2 ⊢ (ℝ D (log ↾ ℝ⁺)) = (ℝ D ^◡(exp ↾ ℝ))
3		reeff1o 24201	. . . . . . . . 9 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺
4		f1of 6137	. . . . . . . . 9 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ → (exp ↾ ℝ):ℝ⟶ℝ⁺)
5	3, 4	ax-mp 5	. . . . . . . 8 ⊢ (exp ↾ ℝ):ℝ⟶ℝ⁺
6		rpssre 11843	. . . . . . . 8 ⊢ ℝ⁺ ⊆ ℝ
7		fss 6056	. . . . . . . 8 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ⁺ ∧ ℝ⁺ ⊆ ℝ) → (exp ↾ ℝ):ℝ⟶ℝ)
8	5, 6, 7	mp2an 708	. . . . . . 7 ⊢ (exp ↾ ℝ):ℝ⟶ℝ
9		ax-resscn 9993	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
10		efcn 24197	. . . . . . . . 9 ⊢ exp ∈ (ℂ–cn→ℂ)
11		rescncf 22700	. . . . . . . . 9 ⊢ (ℝ ⊆ ℂ → (exp ∈ (ℂ–cn→ℂ) → (exp ↾ ℝ) ∈ (ℝ–cn→ℂ)))
12	9, 10, 11	mp2 9	. . . . . . . 8 ⊢ (exp ↾ ℝ) ∈ (ℝ–cn→ℂ)
13		cncffvrn 22701	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ (exp ↾ ℝ) ∈ (ℝ–cn→ℂ)) → ((exp ↾ ℝ) ∈ (ℝ–cn→ℝ) ↔ (exp ↾ ℝ):ℝ⟶ℝ))
14	9, 12, 13	mp2an 708	. . . . . . 7 ⊢ ((exp ↾ ℝ) ∈ (ℝ–cn→ℝ) ↔ (exp ↾ ℝ):ℝ⟶ℝ)
15	8, 14	mpbir 221	. . . . . 6 ⊢ (exp ↾ ℝ) ∈ (ℝ–cn→ℝ)
16	15	a1i 11	. . . . 5 ⊢ (⊤ → (exp ↾ ℝ) ∈ (ℝ–cn→ℝ))
17		reelprrecn 10028	. . . . . . . . . 10 ⊢ ℝ ∈ {ℝ, ℂ}
18		eff 14812	. . . . . . . . . 10 ⊢ exp:ℂ⟶ℂ
19		ssid 3624	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
20		dvef 23743	. . . . . . . . . . . . 13 ⊢ (ℂ D exp) = exp
21	20	dmeqi 5325	. . . . . . . . . . . 12 ⊢ dom (ℂ D exp) = dom exp
22	18	fdmi 6052	. . . . . . . . . . . 12 ⊢ dom exp = ℂ
23	21, 22	eqtri 2644	. . . . . . . . . . 11 ⊢ dom (ℂ D exp) = ℂ
24	9, 23	sseqtr4i 3638	. . . . . . . . . 10 ⊢ ℝ ⊆ dom (ℂ D exp)
25		dvres3 23677	. . . . . . . . . 10 ⊢ (((ℝ ∈ {ℝ, ℂ} ∧ exp:ℂ⟶ℂ) ∧ (ℂ ⊆ ℂ ∧ ℝ ⊆ dom (ℂ D exp))) → (ℝ D (exp ↾ ℝ)) = ((ℂ D exp) ↾ ℝ))
26	17, 18, 19, 24, 25	mp4an 709	. . . . . . . . 9 ⊢ (ℝ D (exp ↾ ℝ)) = ((ℂ D exp) ↾ ℝ)
27	20	reseq1i 5392	. . . . . . . . 9 ⊢ ((ℂ D exp) ↾ ℝ) = (exp ↾ ℝ)
28	26, 27	eqtri 2644	. . . . . . . 8 ⊢ (ℝ D (exp ↾ ℝ)) = (exp ↾ ℝ)
29	28	dmeqi 5325	. . . . . . 7 ⊢ dom (ℝ D (exp ↾ ℝ)) = dom (exp ↾ ℝ)
30	5	fdmi 6052	. . . . . . 7 ⊢ dom (exp ↾ ℝ) = ℝ
31	29, 30	eqtri 2644	. . . . . 6 ⊢ dom (ℝ D (exp ↾ ℝ)) = ℝ
32	31	a1i 11	. . . . 5 ⊢ (⊤ → dom (ℝ D (exp ↾ ℝ)) = ℝ)
33		0nrp 11865	. . . . . . 7 ⊢ ¬ 0 ∈ ℝ⁺
34	28	rneqi 5352	. . . . . . . . 9 ⊢ ran (ℝ D (exp ↾ ℝ)) = ran (exp ↾ ℝ)
35		f1ofo 6144	. . . . . . . . . 10 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ → (exp ↾ ℝ):ℝ–onto→ℝ⁺)
36		forn 6118	. . . . . . . . . 10 ⊢ ((exp ↾ ℝ):ℝ–onto→ℝ⁺ → ran (exp ↾ ℝ) = ℝ⁺)
37	3, 35, 36	mp2b 10	. . . . . . . . 9 ⊢ ran (exp ↾ ℝ) = ℝ⁺
38	34, 37	eqtri 2644	. . . . . . . 8 ⊢ ran (ℝ D (exp ↾ ℝ)) = ℝ⁺
39	38	eleq2i 2693	. . . . . . 7 ⊢ (0 ∈ ran (ℝ D (exp ↾ ℝ)) ↔ 0 ∈ ℝ⁺)
40	33, 39	mtbir 313	. . . . . 6 ⊢ ¬ 0 ∈ ran (ℝ D (exp ↾ ℝ))
41	40	a1i 11	. . . . 5 ⊢ (⊤ → ¬ 0 ∈ ran (ℝ D (exp ↾ ℝ)))
42	3	a1i 11	. . . . 5 ⊢ (⊤ → (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺)
43	16, 32, 41, 42	dvcnvre 23782	. . . 4 ⊢ (⊤ → (ℝ D ^◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ⁺ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)))))
44	43	trud 1493	. . 3 ⊢ (ℝ D ^◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ⁺ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥))))
45	28	fveq1i 6192	. . . . . 6 ⊢ ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)) = ((exp ↾ ℝ)‘(^◡(exp ↾ ℝ)‘𝑥))
46		f1ocnvfv2 6533	. . . . . . 7 ⊢ (((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → ((exp ↾ ℝ)‘(^◡(exp ↾ ℝ)‘𝑥)) = 𝑥)
47	3, 46	mpan 706	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((exp ↾ ℝ)‘(^◡(exp ↾ ℝ)‘𝑥)) = 𝑥)
48	45, 47	syl5eq 2668	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)) = 𝑥)
49	48	oveq2d 6666	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥))) = (1 / 𝑥))
50	49	mpteq2ia 4740	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
51	44, 50	eqtri 2644	. 2 ⊢ (ℝ D ^◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
52	2, 51	eqtri 2644	1 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 196 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ⊆ wss 3574 {cpr 4179 ↦ cmpt 4729 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 ↾ cres 5116 ⟶wf 5884 –onto→wfo 5886 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 / cdiv 10684 ℝ⁺crp 11832 expce 14792 –cn→ccncf 22679 D cdv 23627 logclog 24301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303

This theorem is referenced by: relogcn 24384 advlog 24400 advlogexp 24401 logccv 24409 dvcxp1 24481 loglesqrt 24499 logdivsum 25222 log2sumbnd 25233 logdivsqrle 30728

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