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Theorem advlog 24922

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 10482	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
3		rpre 12251	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
4	3	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
5	4	recnd 10522	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
6		1cnd 10489	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
7		recn 10480	. . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
8	7	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
9		1red 10495	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
10	2	dvmptid 24241	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
11		rpssre 12250	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
12	11	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
13		eqid 2797	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
14	13	tgioo2 23098	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
15		ioorp 12668	. . . . . . 7 ⊢ (0(,)+∞) = ℝ⁺
16		iooretop 23061	. . . . . . 7 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
17	15, 16	eqeltrri 2882	. . . . . 6 ⊢ ℝ⁺ ∈ (topGen‘ran (,))
18	17	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ (topGen‘ran (,)))
19	2, 8, 9, 10, 12, 14, 13, 18	dvmptres 24247	. . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ 1))
20		relogcl 24844	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
21	20	adantl 482	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
22		peano2rem 10807	. . . . . 6 ⊢ ((log‘𝑥) ∈ ℝ → ((log‘𝑥) − 1) ∈ ℝ)
23	21, 22	syl 17	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − 1) ∈ ℝ)
24	23	recnd 10522	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − 1) ∈ ℂ)
25		rpreccl 12269	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
26	25	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
27	26	rpcnd 12287	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
28	21	recnd 10522	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
29		dvrelog 24905	. . . . . . 7 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
30		relogf1o 24835	. . . . . . . . . . 11 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
31		f1of 6490	. . . . . . . . . . 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 6608	. . . . . . . . 9 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
34		fvres 6564	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
35	34	mpteq2ia 5058	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
36	33, 35	syl6eq 2849	. . . . . . . 8 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
37	36	oveq2d 7039	. . . . . . 7 ⊢ (⊤ → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
38	29, 37	syl5reqr 2848	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
39		0cnd 10487	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 0 ∈ ℂ)
40		1cnd 10489	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
41		0cnd 10487	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 0 ∈ ℂ)
42		1cnd 10489	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℂ)
43	2, 42	dvmptc 24242	. . . . . . 7 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0))
44	2, 40, 41, 43, 12, 14, 13, 18	dvmptres 24247	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ 1)) = (𝑥 ∈ ℝ⁺ ↦ 0))
45	2, 28, 27, 38, 6, 39, 44	dvmptsub 24251	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ⁺ ↦ ((1 / 𝑥) − 0)))
46	27	subid1d 10840	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) − 0) = (1 / 𝑥))
47	46	mpteq2dva 5062	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((1 / 𝑥) − 0)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
48	45, 47	eqtrd 2833	. . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
49	2, 5, 6, 19, 24, 27, 48	dvmptmul 24245	. . 3 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥))))
50	24	mulid2d 10512	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 · ((log‘𝑥) − 1)) = ((log‘𝑥) − 1))
51		rpne0 12259	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
52	51	adantl 482	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
53	5, 52	recid2d 11266	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) · 𝑥) = 1)
54	50, 53	oveq12d 7041	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (((log‘𝑥) − 1) + 1))
55		ax-1cn 10448	. . . . . 6 ⊢ 1 ∈ ℂ
56		npcan 10749	. . . . . 6 ⊢ (((log‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((log‘𝑥) − 1) + 1) = (log‘𝑥))
57	28, 55, 56	sylancl 586	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((log‘𝑥) − 1) + 1) = (log‘𝑥))
58	54, 57	eqtrd 2833	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (log‘𝑥))
59	58	mpteq2dva 5062	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
60	49, 59	eqtrd 2833	. 2 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
61	60	mptru 1532	1 ⊢ (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 396 = wceq 1525 ⊤wtru 1526 ∈ wcel 2083 ≠ wne 2986 ⊆ wss 3865 {cpr 4480 ↦ cmpt 5047 ran crn 5451 ↾ cres 5452 ⟶wf 6228 –1-1-onto→wf1o 6231 ‘cfv 6232 (class class class)co 7023 ℂcc 10388 ℝcr 10389 0cc0 10390 1c1 10391 + caddc 10393 · cmul 10395 +∞cpnf 10525 − cmin 10723 / cdiv 11151 ℝ⁺crp 12243 (,)cioo 12592 TopOpenctopn 16528 topGenctg 16544 ℂ_fldccnfld 20231 D cdv 24148 logclog 24823

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-inf2 8957 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 ax-addf 10469 ax-mulf 10470

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-of 7274 df-om 7444 df-1st 7552 df-2nd 7553 df-supp 7689 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-2o 7961 df-oadd 7964 df-er 8146 df-map 8265 df-pm 8266 df-ixp 8318 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-fsupp 8687 df-fi 8728 df-sup 8759 df-inf 8760 df-oi 8827 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-z 11836 df-dec 11953 df-uz 12098 df-q 12202 df-rp 12244 df-xneg 12361 df-xadd 12362 df-xmul 12363 df-ioo 12596 df-ioc 12597 df-ico 12598 df-icc 12599 df-fz 12747 df-fzo 12888 df-fl 13016 df-mod 13092 df-seq 13224 df-exp 13284 df-fac 13488 df-bc 13517 df-hash 13545 df-shft 14264 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-limsup 14666 df-clim 14683 df-rlim 14684 df-sum 14881 df-ef 15258 df-sin 15260 df-cos 15261 df-pi 15263 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-starv 16413 df-sca 16414 df-vsca 16415 df-ip 16416 df-tset 16417 df-ple 16418 df-ds 16420 df-unif 16421 df-hom 16422 df-cco 16423 df-rest 16529 df-topn 16530 df-0g 16548 df-gsum 16549 df-topgen 16550 df-pt 16551 df-prds 16554 df-xrs 16608 df-qtop 16613 df-imas 16614 df-xps 16616 df-mre 16690 df-mrc 16691 df-acs 16693 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-submnd 17779 df-mulg 17986 df-cntz 18192 df-cmn 18639 df-psmet 20223 df-xmet 20224 df-met 20225 df-bl 20226 df-mopn 20227 df-fbas 20228 df-fg 20229 df-cnfld 20232 df-top 21190 df-topon 21207 df-topsp 21229 df-bases 21242 df-cld 21315 df-ntr 21316 df-cls 21317 df-nei 21394 df-lp 21432 df-perf 21433 df-cn 21523 df-cnp 21524 df-haus 21611 df-cmp 21683 df-tx 21858 df-hmeo 22051 df-fil 22142 df-fm 22234 df-flim 22235 df-flf 22236 df-xms 22617 df-ms 22618 df-tms 22619 df-cncf 23173 df-limc 24151 df-dv 24152 df-log 24825

This theorem is referenced by: logfacbnd3 25485

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