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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvrelog3 | Structured version Visualization version GIF version |
Description: The derivative of the logarithm on an open interval. (Contributed by metakunt, 11-Aug-2024.) |
Ref | Expression |
---|---|
dvrelog3.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
dvrelog3.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
dvrelog3.3 | ⊢ (𝜑 → 0 ≤ 𝐴) |
dvrelog3.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
dvrelog3.5 | ⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥)) |
dvrelog3.6 | ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) |
Ref | Expression |
---|---|
dvrelog3 | ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvrelog3.5 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥)) | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))) |
3 | 2 | oveq2d 7172 | . . 3 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥)))) |
4 | reelprrecn 10680 | . . . . 5 ⊢ ℝ ∈ {ℝ, ℂ} | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ}) |
6 | rpcn 12453 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℂ) | |
7 | 6 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ) |
8 | rpne0 12459 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ≠ 0) | |
9 | 8 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0) |
10 | 7, 9 | logcld 25274 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ) |
11 | 1red 10693 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 1 ∈ ℝ) | |
12 | rpre 12451 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ) | |
13 | 12 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ) |
14 | 11, 13, 9 | redivcld 11519 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ) |
15 | logf1o 25268 | . . . . . . . . . 10 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log | |
16 | f1of 6607 | . . . . . . . . . 10 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log) | |
17 | 15, 16 | ax-mp 5 | . . . . . . . . 9 ⊢ log:(ℂ ∖ {0})⟶ran log |
18 | 17 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → log:(ℂ ∖ {0})⟶ran log) |
19 | 0nrp 12478 | . . . . . . . . . . . 12 ⊢ ¬ 0 ∈ ℝ+ | |
20 | disjsn 4607 | . . . . . . . . . . . 12 ⊢ ((ℝ+ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℝ+) | |
21 | 19, 20 | mpbir 234 | . . . . . . . . . . 11 ⊢ (ℝ+ ∩ {0}) = ∅ |
22 | disjdif2 4379 | . . . . . . . . . . 11 ⊢ ((ℝ+ ∩ {0}) = ∅ → (ℝ+ ∖ {0}) = ℝ+) | |
23 | 21, 22 | ax-mp 5 | . . . . . . . . . 10 ⊢ (ℝ+ ∖ {0}) = ℝ+ |
24 | rpssre 12450 | . . . . . . . . . . . 12 ⊢ ℝ+ ⊆ ℝ | |
25 | ax-resscn 10645 | . . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ | |
26 | 24, 25 | sstri 3903 | . . . . . . . . . . 11 ⊢ ℝ+ ⊆ ℂ |
27 | ssdif 4047 | . . . . . . . . . . 11 ⊢ (ℝ+ ⊆ ℂ → (ℝ+ ∖ {0}) ⊆ (ℂ ∖ {0})) | |
28 | 26, 27 | ax-mp 5 | . . . . . . . . . 10 ⊢ (ℝ+ ∖ {0}) ⊆ (ℂ ∖ {0}) |
29 | 23, 28 | eqsstrri 3929 | . . . . . . . . 9 ⊢ ℝ+ ⊆ (ℂ ∖ {0}) |
30 | 29 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ℝ+ ⊆ (ℂ ∖ {0})) |
31 | 18, 30 | feqresmpt 6727 | . . . . . . 7 ⊢ (𝜑 → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) |
32 | 31 | eqcomd 2764 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) = (log ↾ ℝ+)) |
33 | 32 | oveq2d 7172 | . . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) = (ℝ D (log ↾ ℝ+))) |
34 | dvrelog 25340 | . . . . . 6 ⊢ (ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) | |
35 | 34 | a1i 11 | . . . . 5 ⊢ (𝜑 → (ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))) |
36 | 33, 35 | eqtrd 2793 | . . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))) |
37 | dvrelog3.1 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
38 | dvrelog3.2 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
39 | elioo2 12833 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑦 ∈ (𝐴(,)𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 < 𝐵))) | |
40 | 37, 38, 39 | syl2anc 587 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 < 𝐵))) |
41 | 40 | biimpa 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 < 𝐵)) |
42 | 41 | simp1d 1139 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ) |
43 | 0red 10695 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ) | |
44 | 43 | rexrd 10742 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ*) |
45 | 37 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ*) |
46 | 42 | rexrd 10742 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ*) |
47 | dvrelog3.3 | . . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐴) | |
48 | 47 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ≤ 𝐴) |
49 | 41 | simp2d 1140 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑦) |
50 | 44, 45, 46, 48, 49 | xrlelttrd 12607 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 < 𝑦) |
51 | 42, 50 | jca 515 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦 ∈ ℝ ∧ 0 < 𝑦)) |
52 | elrp 12445 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ+ ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) | |
53 | 51, 52 | sylibr 237 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ+) |
54 | 53 | ex 416 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) → 𝑦 ∈ ℝ+)) |
55 | 54 | ssrdv 3900 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ+) |
56 | eqid 2758 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
57 | 56 | tgioo2 23517 | . . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
58 | retop 23476 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
59 | 58 | a1i 11 | . . . . 5 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top) |
60 | iooretop 23480 | . . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) | |
61 | 60 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,))) |
62 | isopn3i 21795 | . . . . 5 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴(,)𝐵) ∈ (topGen‘ran (,))) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)) | |
63 | 59, 61, 62 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)) |
64 | 5, 10, 14, 36, 55, 57, 56, 63 | dvmptres2 24674 | . . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))) |
65 | 3, 64 | eqtrd 2793 | . 2 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))) |
66 | dvrelog3.6 | . . . 4 ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) | |
67 | 66 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))) |
68 | 67 | eqcomd 2764 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) = 𝐺) |
69 | 65, 68 | eqtrd 2793 | 1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 ∖ cdif 3857 ∩ cin 3859 ⊆ wss 3860 ∅c0 4227 {csn 4525 {cpr 4527 class class class wbr 5036 ↦ cmpt 5116 ran crn 5529 ↾ cres 5530 ⟶wf 6336 –1-1-onto→wf1o 6339 ‘cfv 6340 (class class class)co 7156 ℂcc 10586 ℝcr 10587 0cc0 10588 1c1 10589 ℝ*cxr 10725 < clt 10726 ≤ cle 10727 / cdiv 11348 ℝ+crp 12443 (,)cioo 12792 TopOpenctopn 16766 topGenctg 16782 ℂfldccnfld 20179 Topctop 21606 intcnt 21730 D cdv 24575 logclog 25258 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-inf2 9150 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 ax-pre-sup 10666 ax-addf 10667 ax-mulf 10668 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-se 5488 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7411 df-om 7586 df-1st 7699 df-2nd 7700 df-supp 7842 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-1o 8118 df-2o 8119 df-er 8305 df-map 8424 df-pm 8425 df-ixp 8493 df-en 8541 df-dom 8542 df-sdom 8543 df-fin 8544 df-fsupp 8880 df-fi 8921 df-sup 8952 df-inf 8953 df-oi 9020 df-card 9414 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-div 11349 df-nn 11688 df-2 11750 df-3 11751 df-4 11752 df-5 11753 df-6 11754 df-7 11755 df-8 11756 df-9 11757 df-n0 11948 df-z 12034 df-dec 12151 df-uz 12296 df-q 12402 df-rp 12444 df-xneg 12561 df-xadd 12562 df-xmul 12563 df-ioo 12796 df-ioc 12797 df-ico 12798 df-icc 12799 df-fz 12953 df-fzo 13096 df-fl 13224 df-mod 13300 df-seq 13432 df-exp 13493 df-fac 13697 df-bc 13726 df-hash 13754 df-shft 14487 df-cj 14519 df-re 14520 df-im 14521 df-sqrt 14655 df-abs 14656 df-limsup 14889 df-clim 14906 df-rlim 14907 df-sum 15104 df-ef 15482 df-sin 15484 df-cos 15485 df-pi 15487 df-struct 16556 df-ndx 16557 df-slot 16558 df-base 16560 df-sets 16561 df-ress 16562 df-plusg 16649 df-mulr 16650 df-starv 16651 df-sca 16652 df-vsca 16653 df-ip 16654 df-tset 16655 df-ple 16656 df-ds 16658 df-unif 16659 df-hom 16660 df-cco 16661 df-rest 16767 df-topn 16768 df-0g 16786 df-gsum 16787 df-topgen 16788 df-pt 16789 df-prds 16792 df-xrs 16846 df-qtop 16851 df-imas 16852 df-xps 16854 df-mre 16928 df-mrc 16929 df-acs 16931 df-mgm 17931 df-sgrp 17980 df-mnd 17991 df-submnd 18036 df-mulg 18305 df-cntz 18527 df-cmn 18988 df-psmet 20171 df-xmet 20172 df-met 20173 df-bl 20174 df-mopn 20175 df-fbas 20176 df-fg 20177 df-cnfld 20180 df-top 21607 df-topon 21624 df-topsp 21646 df-bases 21659 df-cld 21732 df-ntr 21733 df-cls 21734 df-nei 21811 df-lp 21849 df-perf 21850 df-cn 21940 df-cnp 21941 df-haus 22028 df-cmp 22100 df-tx 22275 df-hmeo 22468 df-fil 22559 df-fm 22651 df-flim 22652 df-flf 22653 df-xms 23035 df-ms 23036 df-tms 23037 df-cncf 23592 df-limc 24578 df-dv 24579 df-log 25260 |
This theorem is referenced by: dvrelog2b 39666 |
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