dvrelog - Metamath Proof Explorer

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Theorem dvrelog 26668

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 26596	. . 3 ⊢ (log ↾ ℝ⁺) = ^◡(exp ↾ ℝ)
2	1	oveq2i 7392	. 2 ⊢ (ℝ D (log ↾ ℝ⁺)) = (ℝ D ^◡(exp ↾ ℝ))
3		reeff1o 26476	. . . . . . . . 9 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺
4		f1of 6791	. . . . . . . . 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 12987	. . . . . . . 8 ⊢ ℝ⁺ ⊆ ℝ
7		fss 6693	. . . . . . . 8 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ⁺ ∧ ℝ⁺ ⊆ ℝ) → (exp ↾ ℝ):ℝ⟶ℝ)
8	5, 6, 7	mp2an 700	. . . . . . 7 ⊢ (exp ↾ ℝ):ℝ⟶ℝ
9		ax-resscn 11116	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
10		efcn 26472	. . . . . . . . 9 ⊢ exp ∈ (ℂ–cn→ℂ)
11		rescncf 24928	. . . . . . . . 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		cncfcdm 24929	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ (exp ↾ ℝ) ∈ (ℝ–cn→ℂ)) → ((exp ↾ ℝ) ∈ (ℝ–cn→ℝ) ↔ (exp ↾ ℝ):ℝ⟶ℝ))
14	9, 12, 13	mp2an 700	. . . . . . 7 ⊢ ((exp ↾ ℝ) ∈ (ℝ–cn→ℝ) ↔ (exp ↾ ℝ):ℝ⟶ℝ)
15	8, 14	mpbir 233	. . . . . 6 ⊢ (exp ↾ ℝ) ∈ (ℝ–cn→ℝ)
16	15	a1i 11	. . . . 5 ⊢ (⊤ → (exp ↾ ℝ) ∈ (ℝ–cn→ℝ))
17		reelprrecn 11151	. . . . . . . . . 10 ⊢ ℝ ∈ {ℝ, ℂ}
18		eff 16083	. . . . . . . . . 10 ⊢ exp:ℂ⟶ℂ
19		ssid 3949	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
20		dvef 26011	. . . . . . . . . . . . 13 ⊢ (ℂ D exp) = exp
21	20	dmeqi 5869	. . . . . . . . . . . 12 ⊢ dom (ℂ D exp) = dom exp
22	18	fdmi 6688	. . . . . . . . . . . 12 ⊢ dom exp = ℂ
23	21, 22	eqtri 2775	. . . . . . . . . . 11 ⊢ dom (ℂ D exp) = ℂ
24	9, 23	sseqtrri 3976	. . . . . . . . . 10 ⊢ ℝ ⊆ dom (ℂ D exp)
25		dvres3 25944	. . . . . . . . . 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 701	. . . . . . . . 9 ⊢ (ℝ D (exp ↾ ℝ)) = ((ℂ D exp) ↾ ℝ)
27	20	reseq1i 5950	. . . . . . . . 9 ⊢ ((ℂ D exp) ↾ ℝ) = (exp ↾ ℝ)
28	26, 27	eqtri 2775	. . . . . . . 8 ⊢ (ℝ D (exp ↾ ℝ)) = (exp ↾ ℝ)
29	28	dmeqi 5869	. . . . . . 7 ⊢ dom (ℝ D (exp ↾ ℝ)) = dom (exp ↾ ℝ)
30	5	fdmi 6688	. . . . . . 7 ⊢ dom (exp ↾ ℝ) = ℝ
31	29, 30	eqtri 2775	. . . . . 6 ⊢ dom (ℝ D (exp ↾ ℝ)) = ℝ
32	31	a1i 11	. . . . 5 ⊢ (⊤ → dom (ℝ D (exp ↾ ℝ)) = ℝ)
33		0nrp 13016	. . . . . . 7 ⊢ ¬ 0 ∈ ℝ⁺
34	28	rneqi 5902	. . . . . . . . 9 ⊢ ran (ℝ D (exp ↾ ℝ)) = ran (exp ↾ ℝ)
35		f1ofo 6799	. . . . . . . . . 10 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ → (exp ↾ ℝ):ℝ–onto→ℝ⁺)
36		forn 6766	. . . . . . . . . 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 2775	. . . . . . . 8 ⊢ ran (ℝ D (exp ↾ ℝ)) = ℝ⁺
39	38	eleq2i 2844	. . . . . . 7 ⊢ (0 ∈ ran (ℝ D (exp ↾ ℝ)) ↔ 0 ∈ ℝ⁺)
40	33, 39	mtbir 325	. . . . . 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 26050	. . . 4 ⊢ (⊤ → (ℝ D ^◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ⁺ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)))))
44	43	mptru 1557	. . 3 ⊢ (ℝ D ^◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ⁺ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥))))
45	28	fveq1i 6853	. . . . . 6 ⊢ ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)) = ((exp ↾ ℝ)‘(^◡(exp ↾ ℝ)‘𝑥))
46		f1ocnvfv2 7246	. . . . . . 7 ⊢ (((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → ((exp ↾ ℝ)‘(^◡(exp ↾ ℝ)‘𝑥)) = 𝑥)
47	3, 46	mpan 698	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((exp ↾ ℝ)‘(^◡(exp ↾ ℝ)‘𝑥)) = 𝑥)
48	45, 47	eqtrid 2799	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)) = 𝑥)
49	48	oveq2d 7397	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥))) = (1 / 𝑥))
50	49	mpteq2ia 5185	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
51	44, 50	eqtri 2775	. 2 ⊢ (ℝ D ^◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
52	2, 51	eqtri 2775	1 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1550 ⊤wtru 1551 ∈ wcel 2132 ⊆ wss 3895 {cpr 4574 ↦ cmpt 5171 ^◡ccnv 5635 dom cdm 5636 ran crn 5637 ↾ cres 5638 ⟶wf 6502 –onto→wfo 6504 –1-1-onto→wf1o 6505 ‘cfv 6506 (class class class)co 7381 ℂcc 11057 ℝcr 11058 0cc0 11059 1c1 11060 / cdiv 11830 ℝ⁺crp 12979 expce 16063 –cn→ccncf 24907 D cdv 25894 logclog 26585

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-inf2 9582 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rmo 3357 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4856 df-int 4896 df-iun 4941 df-iin 4942 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-se 5590 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-isom 6515 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-of 7645 df-om 7832 df-1st 7955 df-2nd 7956 df-supp 8125 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-1o 8421 df-2o 8422 df-er 8662 df-map 8794 df-pm 8795 df-ixp 8865 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-fsupp 9294 df-fi 9343 df-sup 9374 df-inf 9375 df-oi 9444 df-card 9883 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208 df-sub 11402 df-neg 11403 df-div 11831 df-nn 12197 df-2 12266 df-3 12267 df-4 12268 df-5 12269 df-6 12270 df-7 12271 df-8 12272 df-9 12273 df-n0 12468 df-z 12555 df-dec 12675 df-uz 12826 df-q 12936 df-rp 12980 df-xneg 13100 df-xadd 13101 df-xmul 13102 df-ioo 13339 df-ioc 13340 df-ico 13341 df-icc 13342 df-fz 13499 df-fzo 13646 df-fl 13788 df-mod 13866 df-seq 14001 df-exp 14061 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15066 df-cj 15098 df-re 15099 df-im 15100 df-sqrt 15234 df-abs 15235 df-limsup 15470 df-clim 15487 df-rlim 15488 df-sum 15686 df-ef 16069 df-sin 16071 df-cos 16072 df-pi 16074 df-struct 17155 df-sets 17172 df-slot 17190 df-ndx 17202 df-base 17218 df-ress 17239 df-plusg 17271 df-mulr 17272 df-starv 17273 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-hom 17282 df-cco 17283 df-rest 17423 df-topn 17424 df-0g 17442 df-gsum 17443 df-topgen 17444 df-pt 17445 df-prds 17448 df-xrs 17504 df-qtop 17509 df-imas 17510 df-xps 17512 df-mre 17586 df-mrc 17587 df-acs 17589 df-mgm 18646 df-sgrp 18725 df-mnd 18741 df-submnd 18790 df-mulg 19082 df-cntz 19329 df-cmn 19794 df-psmet 21385 df-xmet 21386 df-met 21387 df-bl 21388 df-mopn 21389 df-fbas 21390 df-fg 21391 df-cnfld 21394 df-top 22923 df-topon 22940 df-topsp 22962 df-bases 22975 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24349 df-ms 24350 df-tms 24351 df-cncf 24909 df-limc 25897 df-dv 25898 df-log 26587

This theorem is referenced by: relogcn 26669 advlog 26685 advlogexp 26686 logccv 26694 dvcxp1 26771 loglesqrt 26792 logdivsum 27563 log2sumbnd 27574 logdivsqrle 34891 dvrelog2 42619 dvrelog3 42620

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