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Theorem advlog 26634

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 11119	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
3		rpre 12940	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
4	3	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
5	4	recnd 11162	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
6		1cnd 11128	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
7		recn 11117	. . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
8	7	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
9		1red 11134	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
10	2	dvmptid 25933	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
11		rpssre 12939	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
12	11	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
13		tgioo4 24779	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
14		eqid 2737	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
15		ioorp 13367	. . . . . . 7 ⊢ (0(,)+∞) = ℝ⁺
16		iooretop 24739	. . . . . . 7 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
17	15, 16	eqeltrri 2834	. . . . . 6 ⊢ ℝ⁺ ∈ (topGen‘ran (,))
18	17	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ (topGen‘ran (,)))
19	2, 8, 9, 10, 12, 13, 14, 18	dvmptres 25939	. . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ 1))
20		relogcl 26555	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
21	20	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
22		peano2rem 11450	. . . . . 6 ⊢ ((log‘𝑥) ∈ ℝ → ((log‘𝑥) − 1) ∈ ℝ)
23	21, 22	syl 17	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − 1) ∈ ℝ)
24	23	recnd 11162	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − 1) ∈ ℂ)
25		rpreccl 12959	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
26	25	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
27	26	rpcnd 12977	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
28	21	recnd 11162	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
29		relogf1o 26546	. . . . . . . . . . 11 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
30		f1of 6772	. . . . . . . . . . 11 ⊢ ((log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
31	29, 30	mp1i 13	. . . . . . . . . 10 ⊢ (⊤ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
32	31	feqmptd 6900	. . . . . . . . 9 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
33		fvres 6851	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
34	33	mpteq2ia 5181	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
35	32, 34	eqtrdi 2788	. . . . . . . 8 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
36	35	oveq2d 7374	. . . . . . 7 ⊢ (⊤ → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
37		dvrelog 26617	. . . . . . 7 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
38	36, 37	eqtr3di 2787	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
39		0cnd 11126	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 0 ∈ ℂ)
40		1cnd 11128	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
41		0cnd 11126	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 0 ∈ ℂ)
42		1cnd 11128	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℂ)
43	2, 42	dvmptc 25934	. . . . . . 7 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0))
44	2, 40, 41, 43, 12, 13, 14, 18	dvmptres 25939	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ 1)) = (𝑥 ∈ ℝ⁺ ↦ 0))
45	2, 28, 27, 38, 6, 39, 44	dvmptsub 25943	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ⁺ ↦ ((1 / 𝑥) − 0)))
46	27	subid1d 11483	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) − 0) = (1 / 𝑥))
47	46	mpteq2dva 5179	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((1 / 𝑥) − 0)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
48	45, 47	eqtrd 2772	. . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
49	2, 5, 6, 19, 24, 27, 48	dvmptmul 25937	. . 3 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥))))
50	24	mullidd 11152	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 · ((log‘𝑥) − 1)) = ((log‘𝑥) − 1))
51		rpne0 12948	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
52	51	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
53	5, 52	recid2d 11916	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) · 𝑥) = 1)
54	50, 53	oveq12d 7376	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (((log‘𝑥) − 1) + 1))
55		ax-1cn 11085	. . . . . 6 ⊢ 1 ∈ ℂ
56		npcan 11391	. . . . . 6 ⊢ (((log‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((log‘𝑥) − 1) + 1) = (log‘𝑥))
57	28, 55, 56	sylancl 587	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((log‘𝑥) − 1) + 1) = (log‘𝑥))
58	54, 57	eqtrd 2772	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (log‘𝑥))
59	58	mpteq2dva 5179	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
60	49, 59	eqtrd 2772	. 2 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
61	60	mptru 1549	1 ⊢ (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2933 ⊆ wss 3890 {cpr 4570 ↦ cmpt 5167 ran crn 5623 ↾ cres 5624 ⟶wf 6486 –1-1-onto→wf1o 6489 ‘cfv 6490 (class class class)co 7358 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 + caddc 11030 · cmul 11032 +∞cpnf 11165 − cmin 11366 / cdiv 11796 ℝ⁺crp 12931 (,)cioo 13287 TopOpenctopn 17373 topGenctg 17389 ℂ_fldccnfld 21342 D cdv 25839 logclog 26534

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-ioc 13292 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-fac 14225 df-bc 14254 df-hash 14282 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15638 df-ef 16021 df-sin 16023 df-cos 16024 df-pi 16026 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18741 df-mulg 19033 df-cntz 19281 df-cmn 19746 df-psmet 21334 df-xmet 21335 df-met 21336 df-bl 21337 df-mopn 21338 df-fbas 21339 df-fg 21340 df-cnfld 21343 df-top 22868 df-topon 22885 df-topsp 22907 df-bases 22920 df-cld 22993 df-ntr 22994 df-cls 22995 df-nei 23072 df-lp 23110 df-perf 23111 df-cn 23201 df-cnp 23202 df-haus 23289 df-cmp 23361 df-tx 23536 df-hmeo 23729 df-fil 23820 df-fm 23912 df-flim 23913 df-flf 23914 df-xms 24294 df-ms 24295 df-tms 24296 df-cncf 24854 df-limc 25842 df-dv 25843 df-log 26536

This theorem is referenced by: logfacbnd3 27205

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