dvrelog - Metamath Proof Explorer

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Theorem dvrelog 24128

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 24061	. . 3 ⊢ (log ↾ ℝ⁺) = ^◡(exp ↾ ℝ)
2	1	oveq2i 6538	. 2 ⊢ (ℝ D (log ↾ ℝ⁺)) = (ℝ D ^◡(exp ↾ ℝ))
3		reeff1o 23950	. . . . . . . . 9 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺
4		f1of 6035	. . . . . . . . 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 11678	. . . . . . . 8 ⊢ ℝ⁺ ⊆ ℝ
7		fss 5955	. . . . . . . 8 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ⁺ ∧ ℝ⁺ ⊆ ℝ) → (exp ↾ ℝ):ℝ⟶ℝ)
8	5, 6, 7	mp2an 704	. . . . . . 7 ⊢ (exp ↾ ℝ):ℝ⟶ℝ
9		ax-resscn 9850	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
10		efcn 23946	. . . . . . . . 9 ⊢ exp ∈ (ℂ–cn→ℂ)
11		rescncf 22456	. . . . . . . . 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 22457	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ (exp ↾ ℝ) ∈ (ℝ–cn→ℂ)) → ((exp ↾ ℝ) ∈ (ℝ–cn→ℝ) ↔ (exp ↾ ℝ):ℝ⟶ℝ))
14	9, 12, 13	mp2an 704	. . . . . . 7 ⊢ ((exp ↾ ℝ) ∈ (ℝ–cn→ℝ) ↔ (exp ↾ ℝ):ℝ⟶ℝ)
15	8, 14	mpbir 220	. . . . . 6 ⊢ (exp ↾ ℝ) ∈ (ℝ–cn→ℝ)
16	15	a1i 11	. . . . 5 ⊢ (⊤ → (exp ↾ ℝ) ∈ (ℝ–cn→ℝ))
17		reelprrecn 9885	. . . . . . . . . 10 ⊢ ℝ ∈ {ℝ, ℂ}
18		eff 14600	. . . . . . . . . 10 ⊢ exp:ℂ⟶ℂ
19		ssid 3587	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
20		dvef 23492	. . . . . . . . . . . . 13 ⊢ (ℂ D exp) = exp
21	20	dmeqi 5234	. . . . . . . . . . . 12 ⊢ dom (ℂ D exp) = dom exp
22	18	fdmi 5951	. . . . . . . . . . . 12 ⊢ dom exp = ℂ
23	21, 22	eqtri 2632	. . . . . . . . . . 11 ⊢ dom (ℂ D exp) = ℂ
24	9, 23	sseqtr4i 3601	. . . . . . . . . 10 ⊢ ℝ ⊆ dom (ℂ D exp)
25		dvres3 23428	. . . . . . . . . 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 705	. . . . . . . . 9 ⊢ (ℝ D (exp ↾ ℝ)) = ((ℂ D exp) ↾ ℝ)
27	20	reseq1i 5300	. . . . . . . . 9 ⊢ ((ℂ D exp) ↾ ℝ) = (exp ↾ ℝ)
28	26, 27	eqtri 2632	. . . . . . . 8 ⊢ (ℝ D (exp ↾ ℝ)) = (exp ↾ ℝ)
29	28	dmeqi 5234	. . . . . . 7 ⊢ dom (ℝ D (exp ↾ ℝ)) = dom (exp ↾ ℝ)
30	5	fdmi 5951	. . . . . . 7 ⊢ dom (exp ↾ ℝ) = ℝ
31	29, 30	eqtri 2632	. . . . . 6 ⊢ dom (ℝ D (exp ↾ ℝ)) = ℝ
32	31	a1i 11	. . . . 5 ⊢ (⊤ → dom (ℝ D (exp ↾ ℝ)) = ℝ)
33		0nrp 11700	. . . . . . 7 ⊢ ¬ 0 ∈ ℝ⁺
34	28	rneqi 5260	. . . . . . . . 9 ⊢ ran (ℝ D (exp ↾ ℝ)) = ran (exp ↾ ℝ)
35		f1ofo 6042	. . . . . . . . . 10 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ → (exp ↾ ℝ):ℝ–onto→ℝ⁺)
36		forn 6016	. . . . . . . . . 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 2632	. . . . . . . 8 ⊢ ran (ℝ D (exp ↾ ℝ)) = ℝ⁺
39	38	eleq2i 2680	. . . . . . 7 ⊢ (0 ∈ ran (ℝ D (exp ↾ ℝ)) ↔ 0 ∈ ℝ⁺)
40	33, 39	mtbir 312	. . . . . 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 23531	. . . 4 ⊢ (⊤ → (ℝ D ^◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ⁺ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)))))
44	43	trud 1484	. . 3 ⊢ (ℝ D ^◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ⁺ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥))))
45	28	fveq1i 6089	. . . . . 6 ⊢ ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)) = ((exp ↾ ℝ)‘(^◡(exp ↾ ℝ)‘𝑥))
46		f1ocnvfv2 6411	. . . . . . 7 ⊢ (((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → ((exp ↾ ℝ)‘(^◡(exp ↾ ℝ)‘𝑥)) = 𝑥)
47	3, 46	mpan 702	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((exp ↾ ℝ)‘(^◡(exp ↾ ℝ)‘𝑥)) = 𝑥)
48	45, 47	syl5eq 2656	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)) = 𝑥)
49	48	oveq2d 6543	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥))) = (1 / 𝑥))
50	49	mpteq2ia 4663	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
51	44, 50	eqtri 2632	. 2 ⊢ (ℝ D ^◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
52	2, 51	eqtri 2632	1 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 195 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ⊆ wss 3540 {cpr 4127 ↦ cmpt 4638 ^◡ccnv 5027 dom cdm 5028 ran crn 5029 ↾ cres 5030 ⟶wf 5786 –onto→wfo 5788 –1-1-onto→wf1o 5789 ‘cfv 5790 (class class class)co 6527 ℂcc 9791 ℝcr 9792 0cc0 9793 1c1 9794 / cdiv 10536 ℝ⁺crp 11667 expce 14580 –cn→ccncf 22435 D cdv 23378 logclog 24050

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4694 ax-sep 4704 ax-nul 4712 ax-pow 4764 ax-pr 4828 ax-un 6825 ax-inf2 8399 ax-cnex 9849 ax-resscn 9850 ax-1cn 9851 ax-icn 9852 ax-addcl 9853 ax-addrcl 9854 ax-mulcl 9855 ax-mulrcl 9856 ax-mulcom 9857 ax-addass 9858 ax-mulass 9859 ax-distr 9860 ax-i2m1 9861 ax-1ne0 9862 ax-1rid 9863 ax-rnegex 9864 ax-rrecex 9865 ax-cnre 9866 ax-pre-lttri 9867 ax-pre-lttrn 9868 ax-pre-ltadd 9869 ax-pre-mulgt0 9870 ax-pre-sup 9871 ax-addf 9872 ax-mulf 9873

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4368 df-int 4406 df-iun 4452 df-iin 4453 df-br 4579 df-opab 4639 df-mpt 4640 df-tr 4676 df-eprel 4939 df-id 4943 df-po 4949 df-so 4950 df-fr 4987 df-se 4988 df-we 4989 df-xp 5034 df-rel 5035 df-cnv 5036 df-co 5037 df-dm 5038 df-rn 5039 df-res 5040 df-ima 5041 df-pred 5583 df-ord 5629 df-on 5630 df-lim 5631 df-suc 5632 df-iota 5754 df-fun 5792 df-fn 5793 df-f 5794 df-f1 5795 df-fo 5796 df-f1o 5797 df-fv 5798 df-isom 5799 df-riota 6489 df-ov 6530 df-oprab 6531 df-mpt2 6532 df-of 6773 df-om 6936 df-1st 7037 df-2nd 7038 df-supp 7161 df-wrecs 7272 df-recs 7333 df-rdg 7371 df-1o 7425 df-2o 7426 df-oadd 7429 df-er 7607 df-map 7724 df-pm 7725 df-ixp 7773 df-en 7820 df-dom 7821 df-sdom 7822 df-fin 7823 df-fsupp 8137 df-fi 8178 df-sup 8209 df-inf 8210 df-oi 8276 df-card 8626 df-cda 8851 df-pnf 9933 df-mnf 9934 df-xr 9935 df-ltxr 9936 df-le 9937 df-sub 10120 df-neg 10121 df-div 10537 df-nn 10871 df-2 10929 df-3 10930 df-4 10931 df-5 10932 df-6 10933 df-7 10934 df-8 10935 df-9 10936 df-n0 11143 df-z 11214 df-dec 11329 df-uz 11523 df-q 11624 df-rp 11668 df-xneg 11781 df-xadd 11782 df-xmul 11783 df-ioo 12009 df-ioc 12010 df-ico 12011 df-icc 12012 df-fz 12156 df-fzo 12293 df-fl 12413 df-mod 12489 df-seq 12622 df-exp 12681 df-fac 12881 df-bc 12910 df-hash 12938 df-shft 13604 df-cj 13636 df-re 13637 df-im 13638 df-sqrt 13772 df-abs 13773 df-limsup 13999 df-clim 14016 df-rlim 14017 df-sum 14214 df-ef 14586 df-sin 14588 df-cos 14589 df-pi 14591 df-struct 15646 df-ndx 15647 df-slot 15648 df-base 15649 df-sets 15650 df-ress 15651 df-plusg 15730 df-mulr 15731 df-starv 15732 df-sca 15733 df-vsca 15734 df-ip 15735 df-tset 15736 df-ple 15737 df-ds 15740 df-unif 15741 df-hom 15742 df-cco 15743 df-rest 15855 df-topn 15856 df-0g 15874 df-gsum 15875 df-topgen 15876 df-pt 15877 df-prds 15880 df-xrs 15934 df-qtop 15939 df-imas 15940 df-xps 15942 df-mre 16018 df-mrc 16019 df-acs 16021 df-mgm 17014 df-sgrp 17056 df-mnd 17067 df-submnd 17108 df-mulg 17313 df-cntz 17522 df-cmn 17967 df-psmet 19508 df-xmet 19509 df-met 19510 df-bl 19511 df-mopn 19512 df-fbas 19513 df-fg 19514 df-cnfld 19517 df-top 20469 df-bases 20470 df-topon 20471 df-topsp 20472 df-cld 20581 df-ntr 20582 df-cls 20583 df-nei 20660 df-lp 20698 df-perf 20699 df-cn 20789 df-cnp 20790 df-haus 20877 df-cmp 20948 df-tx 21123 df-hmeo 21316 df-fil 21408 df-fm 21500 df-flim 21501 df-flf 21502 df-xms 21883 df-ms 21884 df-tms 21885 df-cncf 22437 df-limc 23381 df-dv 23382 df-log 24052

This theorem is referenced by: relogcn 24129 advlog 24145 advlogexp 24146 logccv 24154 dvcxp1 24226 loglesqrt 24244 logdivsum 24967 log2sumbnd 24978

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