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Theorem advlog 26830

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 11198	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
3		rpre 13031	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
4	3	adantl 486	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
5	4	recnd 11243	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
6		1cnd 11208	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
7		recn 11196	. . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
8	7	adantl 486	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
9		1red 11215	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
10	2	dvmptid 26127	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
11		rpssre 13030	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
12	11	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
13		tgioo4 24973	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
14		eqid 2762	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
15		ioorp 13458	. . . . . . 7 ⊢ (0(,)+∞) = ℝ⁺
16		iooretop 24933	. . . . . . 7 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
17	15, 16	eqeltrri 2859	. . . . . 6 ⊢ ℝ⁺ ∈ (topGen‘ran (,))
18	17	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ (topGen‘ran (,)))
19	2, 8, 9, 10, 12, 13, 14, 18	dvmptres 26133	. . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ 1))
20		relogcl 26751	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
21	20	adantl 486	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
22		peano2rem 11531	. . . . . 6 ⊢ ((log‘𝑥) ∈ ℝ → ((log‘𝑥) − 1) ∈ ℝ)
23	21, 22	syl 18	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − 1) ∈ ℝ)
24	23	recnd 11243	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − 1) ∈ ℂ)
25		rpreccl 13050	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
26	25	adantl 486	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
27	26	rpcnd 13068	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
28	21	recnd 11243	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
29		relogf1o 26742	. . . . . . . . . . 11 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
30		f1of 6820	. . . . . . . . . . 11 ⊢ ((log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
31	29, 30	mp1i 14	. . . . . . . . . 10 ⊢ (⊤ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
32	31	feqmptd 6949	. . . . . . . . 9 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
33		fvres 6900	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
34	33	mpteq2ia 5205	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
35	32, 34	eqtrdi 2813	. . . . . . . 8 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
36	35	oveq2d 7428	. . . . . . 7 ⊢ (⊤ → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
37		dvrelog 26813	. . . . . . 7 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
38	36, 37	eqtr3di 2812	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
39		0cnd 11205	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 0 ∈ ℂ)
40		1cnd 11208	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
41		0cnd 11205	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 0 ∈ ℂ)
42		1cnd 11208	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℂ)
43	2, 42	dvmptc 26128	. . . . . . 7 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0))
44	2, 40, 41, 43, 12, 13, 14, 18	dvmptres 26133	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ 1)) = (𝑥 ∈ ℝ⁺ ↦ 0))
45	2, 28, 27, 38, 6, 39, 44	dvmptsub 26137	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ⁺ ↦ ((1 / 𝑥) − 0)))
46	27	subid1d 11564	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) − 0) = (1 / 𝑥))
47	46	mpteq2dva 5203	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((1 / 𝑥) − 0)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
48	45, 47	eqtrd 2797	. . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
49	2, 5, 6, 19, 24, 27, 48	dvmptmul 26131	. . 3 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥))))
50	24	mullidd 11233	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 · ((log‘𝑥) − 1)) = ((log‘𝑥) − 1))
51		rpne0 13039	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
52	51	adantl 486	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
53	5, 52	recid2d 11993	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) · 𝑥) = 1)
54	50, 53	oveq12d 7430	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (((log‘𝑥) − 1) + 1))
55		ax-1cn 11164	. . . . . 6 ⊢ 1 ∈ ℂ
56		npcan 11472	. . . . . 6 ⊢ (((log‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((log‘𝑥) − 1) + 1) = (log‘𝑥))
57	28, 55, 56	sylancl 597	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((log‘𝑥) − 1) + 1) = (log‘𝑥))
58	54, 57	eqtrd 2797	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (log‘𝑥))
59	58	mpteq2dva 5203	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
60	49, 59	eqtrd 2797	. 2 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
61	60	mptru 1576	1 ⊢ (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 400 = wceq 1569 ⊤wtru 1570 ∈ wcel 2142 ≠ wne 2957 ⊆ wss 3904 {cpr 4590 ↦ cmpt 5191 ran crn 5661 ↾ cres 5662 ⟶wf 6532 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7412 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 +∞cpnf 11246 − cmin 11447 / cdiv 11877 ℝ⁺crp 13022 (,)cioo 13378 TopOpenctopn 17480 topGenctg 17496 ℂ_fldccnfld 21533 D cdv 26033 logclog 26730

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-er 8692 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-fl 13832 df-mod 13910 df-seq 14045 df-exp 14105 df-fac 14317 df-bc 14346 df-hash 14374 df-shft 15111 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-limsup 15529 df-clim 15546 df-rlim 15547 df-sum 15745 df-ef 16127 df-sin 16129 df-cos 16130 df-pi 16132 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-submnd 18848 df-mulg 19140 df-cntz 19393 df-cmn 19858 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-fbas 21530 df-fg 21531 df-cnfld 21534 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-nei 23266 df-lp 23304 df-perf 23305 df-cn 23395 df-cnp 23396 df-haus 23483 df-cmp 23555 df-tx 23730 df-hmeo 23923 df-fil 24014 df-fm 24106 df-flim 24107 df-flf 24108 df-xms 24488 df-ms 24489 df-tms 24490 df-cncf 25048 df-limc 26036 df-dv 26037 df-log 26732

This theorem is used by: logfacbnd3 27398

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