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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 7186 | . . 3 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥)))) |
4 | reelprrecn 10707 | . . . . 5 ⊢ ℝ ∈ {ℝ, ℂ} | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ}) |
6 | rpcn 12482 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℂ) | |
7 | 6 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ) |
8 | rpne0 12488 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ≠ 0) | |
9 | 8 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0) |
10 | 7, 9 | logcld 25314 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ) |
11 | 1red 10720 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 1 ∈ ℝ) | |
12 | rpre 12480 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ) | |
13 | 12 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ) |
14 | 11, 13, 9 | redivcld 11546 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ) |
15 | logf1o 25308 | . . . . . . . . . 10 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log | |
16 | f1of 6618 | . . . . . . . . . 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 12507 | . . . . . . . . . . . 12 ⊢ ¬ 0 ∈ ℝ+ | |
20 | disjsn 4602 | . . . . . . . . . . . 12 ⊢ ((ℝ+ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℝ+) | |
21 | 19, 20 | mpbir 234 | . . . . . . . . . . 11 ⊢ (ℝ+ ∩ {0}) = ∅ |
22 | disjdif2 4369 | . . . . . . . . . . 11 ⊢ ((ℝ+ ∩ {0}) = ∅ → (ℝ+ ∖ {0}) = ℝ+) | |
23 | 21, 22 | ax-mp 5 | . . . . . . . . . 10 ⊢ (ℝ+ ∖ {0}) = ℝ+ |
24 | rpssre 12479 | . . . . . . . . . . . 12 ⊢ ℝ+ ⊆ ℝ | |
25 | ax-resscn 10672 | . . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ | |
26 | 24, 25 | sstri 3886 | . . . . . . . . . . 11 ⊢ ℝ+ ⊆ ℂ |
27 | ssdif 4030 | . . . . . . . . . . 11 ⊢ (ℝ+ ⊆ ℂ → (ℝ+ ∖ {0}) ⊆ (ℂ ∖ {0})) | |
28 | 26, 27 | ax-mp 5 | . . . . . . . . . 10 ⊢ (ℝ+ ∖ {0}) ⊆ (ℂ ∖ {0}) |
29 | 23, 28 | eqsstrri 3912 | . . . . . . . . 9 ⊢ ℝ+ ⊆ (ℂ ∖ {0}) |
30 | 29 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ℝ+ ⊆ (ℂ ∖ {0})) |
31 | 18, 30 | feqresmpt 6738 | . . . . . . 7 ⊢ (𝜑 → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) |
32 | 31 | eqcomd 2744 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) = (log ↾ ℝ+)) |
33 | 32 | oveq2d 7186 | . . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) = (ℝ D (log ↾ ℝ+))) |
34 | dvrelog 25380 | . . . . . 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 2773 | . . . 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 12862 | . . . . . . . . . . 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 1143 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ) |
43 | 0red 10722 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ) | |
44 | 43 | rexrd 10769 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ*) |
45 | 37 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ*) |
46 | 42 | rexrd 10769 | . . . . . . . . 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 1144 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑦) |
50 | 44, 45, 46, 48, 49 | xrlelttrd 12636 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 < 𝑦) |
51 | 42, 50 | jca 515 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦 ∈ ℝ ∧ 0 < 𝑦)) |
52 | elrp 12474 | . . . . . . 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 3883 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ+) |
56 | eqid 2738 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
57 | 56 | tgioo2 23555 | . . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
58 | retop 23514 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
59 | 58 | a1i 11 | . . . . 5 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top) |
60 | iooretop 23518 | . . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) | |
61 | 60 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,))) |
62 | isopn3i 21833 | . . . . 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 24714 | . . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))) |
65 | 3, 64 | eqtrd 2773 | . 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 2744 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) = 𝐺) |
69 | 65, 68 | eqtrd 2773 | 1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∖ cdif 3840 ∩ cin 3842 ⊆ wss 3843 ∅c0 4211 {csn 4516 {cpr 4518 class class class wbr 5030 ↦ cmpt 5110 ran crn 5526 ↾ cres 5527 ⟶wf 6335 –1-1-onto→wf1o 6338 ‘cfv 6339 (class class class)co 7170 ℂcc 10613 ℝcr 10614 0cc0 10615 1c1 10616 ℝ*cxr 10752 < clt 10753 ≤ cle 10754 / cdiv 11375 ℝ+crp 12472 (,)cioo 12821 TopOpenctopn 16798 topGenctg 16814 ℂfldccnfld 20217 Topctop 21644 intcnt 21768 D cdv 24615 logclog 25298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-er 8320 df-map 8439 df-pm 8440 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-fi 8948 df-sup 8979 df-inf 8980 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-q 12431 df-rp 12473 df-xneg 12590 df-xadd 12591 df-xmul 12592 df-ioo 12825 df-ioc 12826 df-ico 12827 df-icc 12828 df-fz 12982 df-fzo 13125 df-fl 13253 df-mod 13329 df-seq 13461 df-exp 13522 df-fac 13726 df-bc 13755 df-hash 13783 df-shft 14516 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-limsup 14918 df-clim 14935 df-rlim 14936 df-sum 15136 df-ef 15513 df-sin 15515 df-cos 15516 df-pi 15518 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-hom 16692 df-cco 16693 df-rest 16799 df-topn 16800 df-0g 16818 df-gsum 16819 df-topgen 16820 df-pt 16821 df-prds 16824 df-xrs 16878 df-qtop 16883 df-imas 16884 df-xps 16886 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-mulg 18343 df-cntz 18565 df-cmn 19026 df-psmet 20209 df-xmet 20210 df-met 20211 df-bl 20212 df-mopn 20213 df-fbas 20214 df-fg 20215 df-cnfld 20218 df-top 21645 df-topon 21662 df-topsp 21684 df-bases 21697 df-cld 21770 df-ntr 21771 df-cls 21772 df-nei 21849 df-lp 21887 df-perf 21888 df-cn 21978 df-cnp 21979 df-haus 22066 df-cmp 22138 df-tx 22313 df-hmeo 22506 df-fil 22597 df-fm 22689 df-flim 22690 df-flf 22691 df-xms 23073 df-ms 23074 df-tms 23075 df-cncf 23630 df-limc 24618 df-dv 24619 df-log 25300 |
This theorem is referenced by: dvrelog2b 39693 |
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