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Mirrors > Home > MPE Home > Th. List > advlog | Structured version Visualization version GIF version |
Description: The antiderivative of the logarithm. (Contributed by Mario Carneiro, 21-May-2016.) |
Ref | Expression |
---|---|
advlog | ⊢ (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reelprrecn 10314 | . . . . 5 ⊢ ℝ ∈ {ℝ, ℂ} | |
2 | 1 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ}) |
3 | rpre 12078 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ) | |
4 | 3 | adantl 474 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ) |
5 | 4 | recnd 10355 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ) |
6 | 1cnd 10321 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → 1 ∈ ℂ) | |
7 | recn 10312 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
8 | 7 | adantl 474 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ) |
9 | 1red 10327 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ) | |
10 | 2 | dvmptid 24058 | . . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1)) |
11 | rpssre 12077 | . . . . . 6 ⊢ ℝ+ ⊆ ℝ | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝ+ ⊆ ℝ) |
13 | eqid 2797 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
14 | 13 | tgioo2 22931 | . . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
15 | ioorp 12496 | . . . . . . 7 ⊢ (0(,)+∞) = ℝ+ | |
16 | iooretop 22894 | . . . . . . 7 ⊢ (0(,)+∞) ∈ (topGen‘ran (,)) | |
17 | 15, 16 | eqeltrri 2873 | . . . . . 6 ⊢ ℝ+ ∈ (topGen‘ran (,)) |
18 | 17 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝ+ ∈ (topGen‘ran (,))) |
19 | 2, 8, 9, 10, 12, 14, 13, 18 | dvmptres 24064 | . . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ 𝑥)) = (𝑥 ∈ ℝ+ ↦ 1)) |
20 | relogcl 24660 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ) | |
21 | 20 | adantl 474 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ) |
22 | peano2rem 10638 | . . . . . 6 ⊢ ((log‘𝑥) ∈ ℝ → ((log‘𝑥) − 1) ∈ ℝ) | |
23 | 21, 22 | syl 17 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥) − 1) ∈ ℝ) |
24 | 23 | recnd 10355 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥) − 1) ∈ ℂ) |
25 | rpreccl 12098 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+) | |
26 | 25 | adantl 474 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ+) |
27 | 26 | rpcnd 12115 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℂ) |
28 | 21 | recnd 10355 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ) |
29 | dvrelog 24721 | . . . . . . 7 ⊢ (ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) | |
30 | relogf1o 24651 | . . . . . . . . . . 11 ⊢ (log ↾ ℝ+):ℝ+–1-1-onto→ℝ | |
31 | f1of 6354 | . . . . . . . . . . 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 6472 | . . . . . . . . 9 ⊢ (⊤ → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥))) |
34 | fvres 6428 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → ((log ↾ ℝ+)‘𝑥) = (log‘𝑥)) | |
35 | 34 | mpteq2ia 4931 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥)) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) |
36 | 33, 35 | syl6eq 2847 | . . . . . . . 8 ⊢ (⊤ → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) |
37 | 36 | oveq2d 6892 | . . . . . . 7 ⊢ (⊤ → (ℝ D (log ↾ ℝ+)) = (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥)))) |
38 | 29, 37 | syl5reqr 2846 | . . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))) |
39 | 0cnd 10319 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → 0 ∈ ℂ) | |
40 | 1cnd 10321 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ) | |
41 | 0cnd 10319 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 0 ∈ ℂ) | |
42 | 1cnd 10321 | . . . . . . . 8 ⊢ (⊤ → 1 ∈ ℂ) | |
43 | 2, 42 | dvmptc 24059 | . . . . . . 7 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0)) |
44 | 2, 40, 41, 43, 12, 14, 13, 18 | dvmptres 24064 | . . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ 1)) = (𝑥 ∈ ℝ+ ↦ 0)) |
45 | 2, 28, 27, 38, 6, 39, 44 | dvmptsub 24068 | . . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ+ ↦ ((1 / 𝑥) − 0))) |
46 | 27 | subid1d 10671 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((1 / 𝑥) − 0) = (1 / 𝑥)) |
47 | 46 | mpteq2dva 4935 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((1 / 𝑥) − 0)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))) |
48 | 45, 47 | eqtrd 2831 | . . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))) |
49 | 2, 5, 6, 19, 24, 27, 48 | dvmptmul 24062 | . . 3 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ+ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)))) |
50 | 24 | mulid2d 10345 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 · ((log‘𝑥) − 1)) = ((log‘𝑥) − 1)) |
51 | rpne0 12088 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ≠ 0) | |
52 | 51 | adantl 474 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0) |
53 | 5, 52 | recid2d 11087 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((1 / 𝑥) · 𝑥) = 1) |
54 | 50, 53 | oveq12d 6894 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (((log‘𝑥) − 1) + 1)) |
55 | ax-1cn 10280 | . . . . . 6 ⊢ 1 ∈ ℂ | |
56 | npcan 10580 | . . . . . 6 ⊢ (((log‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((log‘𝑥) − 1) + 1) = (log‘𝑥)) | |
57 | 28, 55, 56 | sylancl 581 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥) − 1) + 1) = (log‘𝑥)) |
58 | 54, 57 | eqtrd 2831 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (log‘𝑥)) |
59 | 58 | mpteq2dva 4935 | . . 3 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥))) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) |
60 | 49, 59 | eqtrd 2831 | . 2 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) |
61 | 60 | mptru 1661 | 1 ⊢ (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 385 = wceq 1653 ⊤wtru 1654 ∈ wcel 2157 ≠ wne 2969 ⊆ wss 3767 {cpr 4368 ↦ cmpt 4920 ran crn 5311 ↾ cres 5312 ⟶wf 6095 –1-1-onto→wf1o 6098 ‘cfv 6099 (class class class)co 6876 ℂcc 10220 ℝcr 10221 0cc0 10222 1c1 10223 + caddc 10225 · cmul 10227 +∞cpnf 10358 − cmin 10554 / cdiv 10974 ℝ+crp 12070 (,)cioo 12420 TopOpenctopn 16394 topGenctg 16410 ℂfldccnfld 20065 D cdv 23965 logclog 24639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-inf2 8786 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-pre-sup 10300 ax-addf 10301 ax-mulf 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-iin 4711 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-se 5270 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-isom 6108 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-of 7129 df-om 7298 df-1st 7399 df-2nd 7400 df-supp 7531 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-2o 7798 df-oadd 7801 df-er 7980 df-map 8095 df-pm 8096 df-ixp 8147 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-fsupp 8516 df-fi 8557 df-sup 8588 df-inf 8589 df-oi 8655 df-card 9049 df-cda 9276 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-z 11663 df-dec 11780 df-uz 11927 df-q 12030 df-rp 12071 df-xneg 12189 df-xadd 12190 df-xmul 12191 df-ioo 12424 df-ioc 12425 df-ico 12426 df-icc 12427 df-fz 12577 df-fzo 12717 df-fl 12844 df-mod 12920 df-seq 13052 df-exp 13111 df-fac 13310 df-bc 13339 df-hash 13367 df-shft 14145 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-limsup 14540 df-clim 14557 df-rlim 14558 df-sum 14755 df-ef 15131 df-sin 15133 df-cos 15134 df-pi 15136 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-starv 16279 df-sca 16280 df-vsca 16281 df-ip 16282 df-tset 16283 df-ple 16284 df-ds 16286 df-unif 16287 df-hom 16288 df-cco 16289 df-rest 16395 df-topn 16396 df-0g 16414 df-gsum 16415 df-topgen 16416 df-pt 16417 df-prds 16420 df-xrs 16474 df-qtop 16479 df-imas 16480 df-xps 16482 df-mre 16558 df-mrc 16559 df-acs 16561 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-submnd 17648 df-mulg 17854 df-cntz 18059 df-cmn 18507 df-psmet 20057 df-xmet 20058 df-met 20059 df-bl 20060 df-mopn 20061 df-fbas 20062 df-fg 20063 df-cnfld 20066 df-top 21024 df-topon 21041 df-topsp 21063 df-bases 21076 df-cld 21149 df-ntr 21150 df-cls 21151 df-nei 21228 df-lp 21266 df-perf 21267 df-cn 21357 df-cnp 21358 df-haus 21445 df-cmp 21516 df-tx 21691 df-hmeo 21884 df-fil 21975 df-fm 22067 df-flim 22068 df-flf 22069 df-xms 22450 df-ms 22451 df-tms 22452 df-cncf 23006 df-limc 23968 df-dv 23969 df-log 24641 |
This theorem is referenced by: logfacbnd3 25297 |
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