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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 10629 | . . . . 5 ⊢ ℝ ∈ {ℝ, ℂ} | |
2 | 1 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ}) |
3 | rpre 12398 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ) | |
4 | 3 | adantl 484 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ) |
5 | 4 | recnd 10669 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ) |
6 | 1cnd 10636 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → 1 ∈ ℂ) | |
7 | recn 10627 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
8 | 7 | adantl 484 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ) |
9 | 1red 10642 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ) | |
10 | 2 | dvmptid 24554 | . . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1)) |
11 | rpssre 12397 | . . . . . 6 ⊢ ℝ+ ⊆ ℝ | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝ+ ⊆ ℝ) |
13 | eqid 2821 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
14 | 13 | tgioo2 23411 | . . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
15 | ioorp 12815 | . . . . . . 7 ⊢ (0(,)+∞) = ℝ+ | |
16 | iooretop 23374 | . . . . . . 7 ⊢ (0(,)+∞) ∈ (topGen‘ran (,)) | |
17 | 15, 16 | eqeltrri 2910 | . . . . . 6 ⊢ ℝ+ ∈ (topGen‘ran (,)) |
18 | 17 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝ+ ∈ (topGen‘ran (,))) |
19 | 2, 8, 9, 10, 12, 14, 13, 18 | dvmptres 24560 | . . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ 𝑥)) = (𝑥 ∈ ℝ+ ↦ 1)) |
20 | relogcl 25159 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ) | |
21 | 20 | adantl 484 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ) |
22 | peano2rem 10953 | . . . . . 6 ⊢ ((log‘𝑥) ∈ ℝ → ((log‘𝑥) − 1) ∈ ℝ) | |
23 | 21, 22 | syl 17 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥) − 1) ∈ ℝ) |
24 | 23 | recnd 10669 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥) − 1) ∈ ℂ) |
25 | rpreccl 12416 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+) | |
26 | 25 | adantl 484 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ+) |
27 | 26 | rpcnd 12434 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℂ) |
28 | 21 | recnd 10669 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ) |
29 | dvrelog 25220 | . . . . . . 7 ⊢ (ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) | |
30 | relogf1o 25150 | . . . . . . . . . . 11 ⊢ (log ↾ ℝ+):ℝ+–1-1-onto→ℝ | |
31 | f1of 6615 | . . . . . . . . . . 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 6733 | . . . . . . . . 9 ⊢ (⊤ → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥))) |
34 | fvres 6689 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → ((log ↾ ℝ+)‘𝑥) = (log‘𝑥)) | |
35 | 34 | mpteq2ia 5157 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥)) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) |
36 | 33, 35 | syl6eq 2872 | . . . . . . . 8 ⊢ (⊤ → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) |
37 | 36 | oveq2d 7172 | . . . . . . 7 ⊢ (⊤ → (ℝ D (log ↾ ℝ+)) = (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥)))) |
38 | 29, 37 | syl5reqr 2871 | . . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))) |
39 | 0cnd 10634 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → 0 ∈ ℂ) | |
40 | 1cnd 10636 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ) | |
41 | 0cnd 10634 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 0 ∈ ℂ) | |
42 | 1cnd 10636 | . . . . . . . 8 ⊢ (⊤ → 1 ∈ ℂ) | |
43 | 2, 42 | dvmptc 24555 | . . . . . . 7 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0)) |
44 | 2, 40, 41, 43, 12, 14, 13, 18 | dvmptres 24560 | . . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ 1)) = (𝑥 ∈ ℝ+ ↦ 0)) |
45 | 2, 28, 27, 38, 6, 39, 44 | dvmptsub 24564 | . . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ+ ↦ ((1 / 𝑥) − 0))) |
46 | 27 | subid1d 10986 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((1 / 𝑥) − 0) = (1 / 𝑥)) |
47 | 46 | mpteq2dva 5161 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((1 / 𝑥) − 0)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))) |
48 | 45, 47 | eqtrd 2856 | . . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))) |
49 | 2, 5, 6, 19, 24, 27, 48 | dvmptmul 24558 | . . 3 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ+ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)))) |
50 | 24 | mulid2d 10659 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 · ((log‘𝑥) − 1)) = ((log‘𝑥) − 1)) |
51 | rpne0 12406 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ≠ 0) | |
52 | 51 | adantl 484 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0) |
53 | 5, 52 | recid2d 11412 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((1 / 𝑥) · 𝑥) = 1) |
54 | 50, 53 | oveq12d 7174 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (((log‘𝑥) − 1) + 1)) |
55 | ax-1cn 10595 | . . . . . 6 ⊢ 1 ∈ ℂ | |
56 | npcan 10895 | . . . . . 6 ⊢ (((log‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((log‘𝑥) − 1) + 1) = (log‘𝑥)) | |
57 | 28, 55, 56 | sylancl 588 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥) − 1) + 1) = (log‘𝑥)) |
58 | 54, 57 | eqtrd 2856 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (log‘𝑥)) |
59 | 58 | mpteq2dva 5161 | . . 3 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥))) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) |
60 | 49, 59 | eqtrd 2856 | . 2 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) |
61 | 60 | mptru 1544 | 1 ⊢ (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ≠ wne 3016 ⊆ wss 3936 {cpr 4569 ↦ cmpt 5146 ran crn 5556 ↾ cres 5557 ⟶wf 6351 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 +∞cpnf 10672 − cmin 10870 / cdiv 11297 ℝ+crp 12390 (,)cioo 12739 TopOpenctopn 16695 topGenctg 16711 ℂfldccnfld 20545 D cdv 24461 logclog 25138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140 |
This theorem is referenced by: logfacbnd3 25799 |
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