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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 9907 | . . . . 5 ⊢ ℝ ∈ {ℝ, ℂ} | |
2 | 1 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ}) |
3 | rpre 11715 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ) | |
4 | 3 | adantl 481 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ) |
5 | 4 | recnd 9947 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ) |
6 | 1cnd 9935 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → 1 ∈ ℂ) | |
7 | recn 9905 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
8 | 7 | adantl 481 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ) |
9 | 1red 9934 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ) | |
10 | 2 | dvmptid 23526 | . . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1)) |
11 | rpssre 11719 | . . . . . 6 ⊢ ℝ+ ⊆ ℝ | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝ+ ⊆ ℝ) |
13 | eqid 2610 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
14 | 13 | tgioo2 22414 | . . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
15 | ioorp 12122 | . . . . . . 7 ⊢ (0(,)+∞) = ℝ+ | |
16 | iooretop 22379 | . . . . . . 7 ⊢ (0(,)+∞) ∈ (topGen‘ran (,)) | |
17 | 15, 16 | eqeltrri 2685 | . . . . . 6 ⊢ ℝ+ ∈ (topGen‘ran (,)) |
18 | 17 | a1i 11 | . . . . 5 ⊢ (⊤ → ℝ+ ∈ (topGen‘ran (,))) |
19 | 2, 8, 9, 10, 12, 14, 13, 18 | dvmptres 23532 | . . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ 𝑥)) = (𝑥 ∈ ℝ+ ↦ 1)) |
20 | relogcl 24126 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ) | |
21 | 20 | adantl 481 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ) |
22 | peano2rem 10227 | . . . . . 6 ⊢ ((log‘𝑥) ∈ ℝ → ((log‘𝑥) − 1) ∈ ℝ) | |
23 | 21, 22 | syl 17 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥) − 1) ∈ ℝ) |
24 | 23 | recnd 9947 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥) − 1) ∈ ℂ) |
25 | rpreccl 11733 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+) | |
26 | 25 | adantl 481 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ+) |
27 | 26 | rpcnd 11750 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℂ) |
28 | 21 | recnd 9947 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ) |
29 | dvrelog 24183 | . . . . . . 7 ⊢ (ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) | |
30 | relogf1o 24117 | . . . . . . . . . . 11 ⊢ (log ↾ ℝ+):ℝ+–1-1-onto→ℝ | |
31 | f1of 6050 | . . . . . . . . . . 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 6159 | . . . . . . . . 9 ⊢ (⊤ → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥))) |
34 | fvres 6117 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → ((log ↾ ℝ+)‘𝑥) = (log‘𝑥)) | |
35 | 34 | mpteq2ia 4668 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ ↦ ((log ↾ ℝ+)‘𝑥)) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) |
36 | 33, 35 | syl6eq 2660 | . . . . . . . 8 ⊢ (⊤ → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) |
37 | 36 | oveq2d 6565 | . . . . . . 7 ⊢ (⊤ → (ℝ D (log ↾ ℝ+)) = (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥)))) |
38 | 29, 37 | syl5reqr 2659 | . . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))) |
39 | 0cnd 9912 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → 0 ∈ ℂ) | |
40 | 1cnd 9935 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ) | |
41 | 0cnd 9912 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 0 ∈ ℂ) | |
42 | 1cnd 9935 | . . . . . . . 8 ⊢ (⊤ → 1 ∈ ℂ) | |
43 | 2, 42 | dvmptc 23527 | . . . . . . 7 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0)) |
44 | 2, 40, 41, 43, 12, 14, 13, 18 | dvmptres 23532 | . . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ 1)) = (𝑥 ∈ ℝ+ ↦ 0)) |
45 | 2, 28, 27, 38, 6, 39, 44 | dvmptsub 23536 | . . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ+ ↦ ((1 / 𝑥) − 0))) |
46 | 27 | subid1d 10260 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((1 / 𝑥) − 0) = (1 / 𝑥)) |
47 | 46 | mpteq2dva 4672 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((1 / 𝑥) − 0)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))) |
48 | 45, 47 | eqtrd 2644 | . . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))) |
49 | 2, 5, 6, 19, 24, 27, 48 | dvmptmul 23530 | . . 3 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ+ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)))) |
50 | 24 | mulid2d 9937 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 · ((log‘𝑥) − 1)) = ((log‘𝑥) − 1)) |
51 | rpne0 11724 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ≠ 0) | |
52 | 51 | adantl 481 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0) |
53 | 5, 52 | recid2d 10676 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((1 / 𝑥) · 𝑥) = 1) |
54 | 50, 53 | oveq12d 6567 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (((log‘𝑥) − 1) + 1)) |
55 | ax-1cn 9873 | . . . . . 6 ⊢ 1 ∈ ℂ | |
56 | npcan 10169 | . . . . . 6 ⊢ (((log‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((log‘𝑥) − 1) + 1) = (log‘𝑥)) | |
57 | 28, 55, 56 | sylancl 693 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥) − 1) + 1) = (log‘𝑥)) |
58 | 54, 57 | eqtrd 2644 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (log‘𝑥)) |
59 | 58 | mpteq2dva 4672 | . . 3 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥))) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) |
60 | 49, 59 | eqtrd 2644 | . 2 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) |
61 | 60 | trud 1484 | 1 ⊢ (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 {cpr 4127 ↦ cmpt 4643 ran crn 5039 ↾ cres 5040 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 +∞cpnf 9950 − cmin 10145 / cdiv 10563 ℝ+crp 11708 (,)cioo 12046 TopOpenctopn 15905 topGenctg 15921 ℂfldccnfld 19567 D cdv 23433 logclog 24105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 |
This theorem is referenced by: logfacbnd3 24748 |
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