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Theorem advlog 24620

Description: The antiderivative of the logarithm. (Contributed by Mario Carneiro, 21-May-2016.)

**Assertion**
Ref	Expression
advlog	⊢ (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))

**Proof of Theorem advlog**
Step	Hyp	Ref	Expression
1		reelprrecn 10229	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
3		rpre 12041	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
4	3	adantl 467	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
5	4	recnd 10269	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
6		1cnd 10257	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
7		recn 10227	. . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
8	7	adantl 467	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
9		1red 10256	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
10	2	dvmptid 23939	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
11		rpssre 12045	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
12	11	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
13		eqid 2770	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
14	13	tgioo2 22825	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
15		ioorp 12455	. . . . . . 7 ⊢ (0(,)+∞) = ℝ⁺
16		iooretop 22788	. . . . . . 7 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
17	15, 16	eqeltrri 2846	. . . . . 6 ⊢ ℝ⁺ ∈ (topGen‘ran (,))
18	17	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ (topGen‘ran (,)))
19	2, 8, 9, 10, 12, 14, 13, 18	dvmptres 23945	. . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ 1))
20		relogcl 24542	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
21	20	adantl 467	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
22		peano2rem 10549	. . . . . 6 ⊢ ((log‘𝑥) ∈ ℝ → ((log‘𝑥) − 1) ∈ ℝ)
23	21, 22	syl 17	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − 1) ∈ ℝ)
24	23	recnd 10269	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − 1) ∈ ℂ)
25		rpreccl 12059	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
26	25	adantl 467	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
27	26	rpcnd 12076	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
28	21	recnd 10269	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
29		dvrelog 24603	. . . . . . 7 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
30		relogf1o 24533	. . . . . . . . . . 11 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
31		f1of 6278	. . . . . . . . . . 11 ⊢ ((log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
32	30, 31	mp1i 13	. . . . . . . . . 10 ⊢ (⊤ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
33	32	feqmptd 6391	. . . . . . . . 9 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
34		fvres 6348	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
35	34	mpteq2ia 4872	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
36	33, 35	syl6eq 2820	. . . . . . . 8 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
37	36	oveq2d 6808	. . . . . . 7 ⊢ (⊤ → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
38	29, 37	syl5reqr 2819	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
39		0cnd 10234	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 0 ∈ ℂ)
40		1cnd 10257	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
41		0cnd 10234	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 0 ∈ ℂ)
42		1cnd 10257	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℂ)
43	2, 42	dvmptc 23940	. . . . . . 7 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0))
44	2, 40, 41, 43, 12, 14, 13, 18	dvmptres 23945	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ 1)) = (𝑥 ∈ ℝ⁺ ↦ 0))
45	2, 28, 27, 38, 6, 39, 44	dvmptsub 23949	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ⁺ ↦ ((1 / 𝑥) − 0)))
46	27	subid1d 10582	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) − 0) = (1 / 𝑥))
47	46	mpteq2dva 4876	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((1 / 𝑥) − 0)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
48	45, 47	eqtrd 2804	. . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
49	2, 5, 6, 19, 24, 27, 48	dvmptmul 23943	. . 3 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥))))
50	24	mulid2d 10259	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 · ((log‘𝑥) − 1)) = ((log‘𝑥) − 1))
51		rpne0 12050	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
52	51	adantl 467	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
53	5, 52	recid2d 10998	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) · 𝑥) = 1)
54	50, 53	oveq12d 6810	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (((log‘𝑥) − 1) + 1))
55		ax-1cn 10195	. . . . . 6 ⊢ 1 ∈ ℂ
56		npcan 10491	. . . . . 6 ⊢ (((log‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((log‘𝑥) − 1) + 1) = (log‘𝑥))
57	28, 55, 56	sylancl 566	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((log‘𝑥) − 1) + 1) = (log‘𝑥))
58	54, 57	eqtrd 2804	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (log‘𝑥))
59	58	mpteq2dva 4876	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
60	49, 59	eqtrd 2804	. 2 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
61	60	trud 1640	1 ⊢ (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 382 = wceq 1630 ⊤wtru 1631 ∈ wcel 2144 ≠ wne 2942 ⊆ wss 3721 {cpr 4316 ↦ cmpt 4861 ran crn 5250 ↾ cres 5251 ⟶wf 6027 –1-1-onto→wf1o 6030 ‘cfv 6031 (class class class)co 6792 ℂcc 10135 ℝcr 10136 0cc0 10137 1c1 10138 + caddc 10140 · cmul 10142 +∞cpnf 10272 − cmin 10467 / cdiv 10885 ℝ⁺crp 12034 (,)cioo 12379 TopOpenctopn 16289 topGenctg 16305 ℂ_fldccnfld 19960 D cdv 23846 logclog 24521

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-inf2 8701 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-pre-sup 10215 ax-addf 10216 ax-mulf 10217

This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-fal 1636 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-int 4610 df-iun 4654 df-iin 4655 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-of 7043 df-om 7212 df-1st 7314 df-2nd 7315 df-supp 7446 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-2o 7713 df-oadd 7716 df-er 7895 df-map 8010 df-pm 8011 df-ixp 8062 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-fsupp 8431 df-fi 8472 df-sup 8503 df-inf 8504 df-oi 8570 df-card 8964 df-cda 9191 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-div 10886 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-7 11285 df-8 11286 df-9 11287 df-n0 11494 df-z 11579 df-dec 11695 df-uz 11888 df-q 11991 df-rp 12035 df-xneg 12150 df-xadd 12151 df-xmul 12152 df-ioo 12383 df-ioc 12384 df-ico 12385 df-icc 12386 df-fz 12533 df-fzo 12673 df-fl 12800 df-mod 12876 df-seq 13008 df-exp 13067 df-fac 13264 df-bc 13293 df-hash 13321 df-shft 14014 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-limsup 14409 df-clim 14426 df-rlim 14427 df-sum 14624 df-ef 15003 df-sin 15005 df-cos 15006 df-pi 15008 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-starv 16163 df-sca 16164 df-vsca 16165 df-ip 16166 df-tset 16167 df-ple 16168 df-ds 16171 df-unif 16172 df-hom 16173 df-cco 16174 df-rest 16290 df-topn 16291 df-0g 16309 df-gsum 16310 df-topgen 16311 df-pt 16312 df-prds 16315 df-xrs 16369 df-qtop 16374 df-imas 16375 df-xps 16377 df-mre 16453 df-mrc 16454 df-acs 16456 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-submnd 17543 df-mulg 17748 df-cntz 17956 df-cmn 18401 df-psmet 19952 df-xmet 19953 df-met 19954 df-bl 19955 df-mopn 19956 df-fbas 19957 df-fg 19958 df-cnfld 19961 df-top 20918 df-topon 20935 df-topsp 20957 df-bases 20970 df-cld 21043 df-ntr 21044 df-cls 21045 df-nei 21122 df-lp 21160 df-perf 21161 df-cn 21251 df-cnp 21252 df-haus 21339 df-cmp 21410 df-tx 21585 df-hmeo 21778 df-fil 21869 df-fm 21961 df-flim 21962 df-flf 21963 df-xms 22344 df-ms 22345 df-tms 22346 df-cncf 22900 df-limc 23849 df-dv 23850 df-log 24523

This theorem is referenced by: logfacbnd3 25168

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