dvrelog - Metamath Proof Explorer

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Theorem dvrelog 25788

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 25717	. . 3 ⊢ (log ↾ ℝ⁺) = ^◡(exp ↾ ℝ)
2	1	oveq2i 7280	. 2 ⊢ (ℝ D (log ↾ ℝ⁺)) = (ℝ D ^◡(exp ↾ ℝ))
3		reeff1o 25602	. . . . . . . . 9 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺
4		f1of 6713	. . . . . . . . 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 12734	. . . . . . . 8 ⊢ ℝ⁺ ⊆ ℝ
7		fss 6614	. . . . . . . 8 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ⁺ ∧ ℝ⁺ ⊆ ℝ) → (exp ↾ ℝ):ℝ⟶ℝ)
8	5, 6, 7	mp2an 689	. . . . . . 7 ⊢ (exp ↾ ℝ):ℝ⟶ℝ
9		ax-resscn 10927	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
10		efcn 25598	. . . . . . . . 9 ⊢ exp ∈ (ℂ–cn→ℂ)
11		rescncf 24056	. . . . . . . . 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 24057	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ (exp ↾ ℝ) ∈ (ℝ–cn→ℂ)) → ((exp ↾ ℝ) ∈ (ℝ–cn→ℝ) ↔ (exp ↾ ℝ):ℝ⟶ℝ))
14	9, 12, 13	mp2an 689	. . . . . . 7 ⊢ ((exp ↾ ℝ) ∈ (ℝ–cn→ℝ) ↔ (exp ↾ ℝ):ℝ⟶ℝ)
15	8, 14	mpbir 230	. . . . . 6 ⊢ (exp ↾ ℝ) ∈ (ℝ–cn→ℝ)
16	15	a1i 11	. . . . 5 ⊢ (⊤ → (exp ↾ ℝ) ∈ (ℝ–cn→ℝ))
17		reelprrecn 10962	. . . . . . . . . 10 ⊢ ℝ ∈ {ℝ, ℂ}
18		eff 15787	. . . . . . . . . 10 ⊢ exp:ℂ⟶ℂ
19		ssid 3948	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
20		dvef 25140	. . . . . . . . . . . . 13 ⊢ (ℂ D exp) = exp
21	20	dmeqi 5811	. . . . . . . . . . . 12 ⊢ dom (ℂ D exp) = dom exp
22	18	fdmi 6609	. . . . . . . . . . . 12 ⊢ dom exp = ℂ
23	21, 22	eqtri 2768	. . . . . . . . . . 11 ⊢ dom (ℂ D exp) = ℂ
24	9, 23	sseqtrri 3963	. . . . . . . . . 10 ⊢ ℝ ⊆ dom (ℂ D exp)
25		dvres3 25073	. . . . . . . . . 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 690	. . . . . . . . 9 ⊢ (ℝ D (exp ↾ ℝ)) = ((ℂ D exp) ↾ ℝ)
27	20	reseq1i 5885	. . . . . . . . 9 ⊢ ((ℂ D exp) ↾ ℝ) = (exp ↾ ℝ)
28	26, 27	eqtri 2768	. . . . . . . 8 ⊢ (ℝ D (exp ↾ ℝ)) = (exp ↾ ℝ)
29	28	dmeqi 5811	. . . . . . 7 ⊢ dom (ℝ D (exp ↾ ℝ)) = dom (exp ↾ ℝ)
30	5	fdmi 6609	. . . . . . 7 ⊢ dom (exp ↾ ℝ) = ℝ
31	29, 30	eqtri 2768	. . . . . 6 ⊢ dom (ℝ D (exp ↾ ℝ)) = ℝ
32	31	a1i 11	. . . . 5 ⊢ (⊤ → dom (ℝ D (exp ↾ ℝ)) = ℝ)
33		0nrp 12762	. . . . . . 7 ⊢ ¬ 0 ∈ ℝ⁺
34	28	rneqi 5844	. . . . . . . . 9 ⊢ ran (ℝ D (exp ↾ ℝ)) = ran (exp ↾ ℝ)
35		f1ofo 6720	. . . . . . . . . 10 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ → (exp ↾ ℝ):ℝ–onto→ℝ⁺)
36		forn 6688	. . . . . . . . . 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 2768	. . . . . . . 8 ⊢ ran (ℝ D (exp ↾ ℝ)) = ℝ⁺
39	38	eleq2i 2832	. . . . . . 7 ⊢ (0 ∈ ran (ℝ D (exp ↾ ℝ)) ↔ 0 ∈ ℝ⁺)
40	33, 39	mtbir 323	. . . . . 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 25179	. . . 4 ⊢ (⊤ → (ℝ D ^◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ⁺ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)))))
44	43	mptru 1549	. . 3 ⊢ (ℝ D ^◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ⁺ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥))))
45	28	fveq1i 6770	. . . . . 6 ⊢ ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)) = ((exp ↾ ℝ)‘(^◡(exp ↾ ℝ)‘𝑥))
46		f1ocnvfv2 7144	. . . . . . 7 ⊢ (((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → ((exp ↾ ℝ)‘(^◡(exp ↾ ℝ)‘𝑥)) = 𝑥)
47	3, 46	mpan 687	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((exp ↾ ℝ)‘(^◡(exp ↾ ℝ)‘𝑥)) = 𝑥)
48	45, 47	eqtrid 2792	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)) = 𝑥)
49	48	oveq2d 7285	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥))) = (1 / 𝑥))
50	49	mpteq2ia 5182	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
51	44, 50	eqtri 2768	. 2 ⊢ (ℝ D ^◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
52	2, 51	eqtri 2768	1 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1542 ⊤wtru 1543 ∈ wcel 2110 ⊆ wss 3892 {cpr 4569 ↦ cmpt 5162 ^◡ccnv 5588 dom cdm 5589 ran crn 5590 ↾ cres 5591 ⟶wf 6427 –onto→wfo 6429 –1-1-onto→wf1o 6430 ‘cfv 6431 (class class class)co 7269 ℂcc 10868 ℝcr 10869 0cc0 10870 1c1 10871 / cdiv 11630 ℝ⁺crp 12727 expce 15767 –cn→ccncf 24035 D cdv 25023 logclog 25706

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-inf2 9375 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 ax-addf 10949 ax-mulf 10950

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7705 df-1st 7822 df-2nd 7823 df-supp 7967 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-2o 8287 df-er 8479 df-map 8598 df-pm 8599 df-ixp 8667 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-fsupp 9105 df-fi 9146 df-sup 9177 df-inf 9178 df-oi 9245 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12435 df-uz 12580 df-q 12686 df-rp 12728 df-xneg 12845 df-xadd 12846 df-xmul 12847 df-ioo 13080 df-ioc 13081 df-ico 13082 df-icc 13083 df-fz 13237 df-fzo 13380 df-fl 13508 df-mod 13586 df-seq 13718 df-exp 13779 df-fac 13984 df-bc 14013 df-hash 14041 df-shft 14774 df-cj 14806 df-re 14807 df-im 14808 df-sqrt 14942 df-abs 14943 df-limsup 15176 df-clim 15193 df-rlim 15194 df-sum 15394 df-ef 15773 df-sin 15775 df-cos 15776 df-pi 15778 df-struct 16844 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-ress 16938 df-plusg 16971 df-mulr 16972 df-starv 16973 df-sca 16974 df-vsca 16975 df-ip 16976 df-tset 16977 df-ple 16978 df-ds 16980 df-unif 16981 df-hom 16982 df-cco 16983 df-rest 17129 df-topn 17130 df-0g 17148 df-gsum 17149 df-topgen 17150 df-pt 17151 df-prds 17154 df-xrs 17209 df-qtop 17214 df-imas 17215 df-xps 17217 df-mre 17291 df-mrc 17292 df-acs 17294 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-submnd 18427 df-mulg 18697 df-cntz 18919 df-cmn 19384 df-psmet 20585 df-xmet 20586 df-met 20587 df-bl 20588 df-mopn 20589 df-fbas 20590 df-fg 20591 df-cnfld 20594 df-top 22039 df-topon 22056 df-topsp 22078 df-bases 22092 df-cld 22166 df-ntr 22167 df-cls 22168 df-nei 22245 df-lp 22283 df-perf 22284 df-cn 22374 df-cnp 22375 df-haus 22462 df-cmp 22534 df-tx 22709 df-hmeo 22902 df-fil 22993 df-fm 23085 df-flim 23086 df-flf 23087 df-xms 23469 df-ms 23470 df-tms 23471 df-cncf 24037 df-limc 25026 df-dv 25027 df-log 25708

This theorem is referenced by: relogcn 25789 advlog 25805 advlogexp 25806 logccv 25814 dvcxp1 25889 loglesqrt 25907 logdivsum 26677 log2sumbnd 26688 logdivsqrle 32624 dvrelog2 40067 dvrelog3 40068

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