dvrelog - Metamath Proof Explorer

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Theorem dvrelog 25992

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 25921	. . 3 ⊢ (log ↾ ℝ⁺) = ^◡(exp ↾ ℝ)
2	1	oveq2i 7368	. 2 ⊢ (ℝ D (log ↾ ℝ⁺)) = (ℝ D ^◡(exp ↾ ℝ))
3		reeff1o 25806	. . . . . . . . 9 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺
4		f1of 6784	. . . . . . . . 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 12922	. . . . . . . 8 ⊢ ℝ⁺ ⊆ ℝ
7		fss 6685	. . . . . . . 8 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ⁺ ∧ ℝ⁺ ⊆ ℝ) → (exp ↾ ℝ):ℝ⟶ℝ)
8	5, 6, 7	mp2an 690	. . . . . . 7 ⊢ (exp ↾ ℝ):ℝ⟶ℝ
9		ax-resscn 11108	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
10		efcn 25802	. . . . . . . . 9 ⊢ exp ∈ (ℂ–cn→ℂ)
11		rescncf 24260	. . . . . . . . 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 24261	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ (exp ↾ ℝ) ∈ (ℝ–cn→ℂ)) → ((exp ↾ ℝ) ∈ (ℝ–cn→ℝ) ↔ (exp ↾ ℝ):ℝ⟶ℝ))
14	9, 12, 13	mp2an 690	. . . . . . 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 11143	. . . . . . . . . 10 ⊢ ℝ ∈ {ℝ, ℂ}
18		eff 15964	. . . . . . . . . 10 ⊢ exp:ℂ⟶ℂ
19		ssid 3966	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
20		dvef 25344	. . . . . . . . . . . . 13 ⊢ (ℂ D exp) = exp
21	20	dmeqi 5860	. . . . . . . . . . . 12 ⊢ dom (ℂ D exp) = dom exp
22	18	fdmi 6680	. . . . . . . . . . . 12 ⊢ dom exp = ℂ
23	21, 22	eqtri 2764	. . . . . . . . . . 11 ⊢ dom (ℂ D exp) = ℂ
24	9, 23	sseqtrri 3981	. . . . . . . . . 10 ⊢ ℝ ⊆ dom (ℂ D exp)
25		dvres3 25277	. . . . . . . . . 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 691	. . . . . . . . 9 ⊢ (ℝ D (exp ↾ ℝ)) = ((ℂ D exp) ↾ ℝ)
27	20	reseq1i 5933	. . . . . . . . 9 ⊢ ((ℂ D exp) ↾ ℝ) = (exp ↾ ℝ)
28	26, 27	eqtri 2764	. . . . . . . 8 ⊢ (ℝ D (exp ↾ ℝ)) = (exp ↾ ℝ)
29	28	dmeqi 5860	. . . . . . 7 ⊢ dom (ℝ D (exp ↾ ℝ)) = dom (exp ↾ ℝ)
30	5	fdmi 6680	. . . . . . 7 ⊢ dom (exp ↾ ℝ) = ℝ
31	29, 30	eqtri 2764	. . . . . 6 ⊢ dom (ℝ D (exp ↾ ℝ)) = ℝ
32	31	a1i 11	. . . . 5 ⊢ (⊤ → dom (ℝ D (exp ↾ ℝ)) = ℝ)
33		0nrp 12950	. . . . . . 7 ⊢ ¬ 0 ∈ ℝ⁺
34	28	rneqi 5892	. . . . . . . . 9 ⊢ ran (ℝ D (exp ↾ ℝ)) = ran (exp ↾ ℝ)
35		f1ofo 6791	. . . . . . . . . 10 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ → (exp ↾ ℝ):ℝ–onto→ℝ⁺)
36		forn 6759	. . . . . . . . . 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 2764	. . . . . . . 8 ⊢ ran (ℝ D (exp ↾ ℝ)) = ℝ⁺
39	38	eleq2i 2829	. . . . . . 7 ⊢ (0 ∈ ran (ℝ D (exp ↾ ℝ)) ↔ 0 ∈ ℝ⁺)
40	33, 39	mtbir 322	. . . . . 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 25383	. . . 4 ⊢ (⊤ → (ℝ D ^◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ⁺ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)))))
44	43	mptru 1548	. . 3 ⊢ (ℝ D ^◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ⁺ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥))))
45	28	fveq1i 6843	. . . . . 6 ⊢ ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)) = ((exp ↾ ℝ)‘(^◡(exp ↾ ℝ)‘𝑥))
46		f1ocnvfv2 7223	. . . . . . 7 ⊢ (((exp ↾ ℝ):ℝ–1-1-onto→ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → ((exp ↾ ℝ)‘(^◡(exp ↾ ℝ)‘𝑥)) = 𝑥)
47	3, 46	mpan 688	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((exp ↾ ℝ)‘(^◡(exp ↾ ℝ)‘𝑥)) = 𝑥)
48	45, 47	eqtrid 2788	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)) = 𝑥)
49	48	oveq2d 7373	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥))) = (1 / 𝑥))
50	49	mpteq2ia 5208	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(^◡(exp ↾ ℝ)‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
51	44, 50	eqtri 2764	. 2 ⊢ (ℝ D ^◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
52	2, 51	eqtri 2764	1 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ⊆ wss 3910 {cpr 4588 ↦ cmpt 5188 ^◡ccnv 5632 dom cdm 5633 ran crn 5634 ↾ cres 5635 ⟶wf 6492 –onto→wfo 6494 –1-1-onto→wf1o 6495 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 / cdiv 11812 ℝ⁺crp 12915 expce 15944 –cn→ccncf 24239 D cdv 25227 logclog 25910

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-log 25912

This theorem is referenced by: relogcn 25993 advlog 26009 advlogexp 26010 logccv 26018 dvcxp1 26093 loglesqrt 26111 logdivsum 26881 log2sumbnd 26892 logdivsqrle 33263 dvrelog2 40521 dvrelog3 40522

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