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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 7446 | . . 3 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥)))) |
4 | reelprrecn 11244 | . . . . 5 ⊢ ℝ ∈ {ℝ, ℂ} | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ}) |
6 | rpcn 13042 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℂ) | |
7 | 6 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ) |
8 | rpne0 13048 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ≠ 0) | |
9 | 8 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0) |
10 | 7, 9 | logcld 26626 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ) |
11 | 1red 11259 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 1 ∈ ℝ) | |
12 | rpre 13040 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ) | |
13 | 12 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ) |
14 | 11, 13, 9 | redivcld 12092 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ) |
15 | logf1o 26620 | . . . . . . . . . 10 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log | |
16 | f1of 6848 | . . . . . . . . . 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 13067 | . . . . . . . . . . . 12 ⊢ ¬ 0 ∈ ℝ+ | |
20 | disjsn 4715 | . . . . . . . . . . . 12 ⊢ ((ℝ+ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℝ+) | |
21 | 19, 20 | mpbir 231 | . . . . . . . . . . 11 ⊢ (ℝ+ ∩ {0}) = ∅ |
22 | disjdif2 4485 | . . . . . . . . . . 11 ⊢ ((ℝ+ ∩ {0}) = ∅ → (ℝ+ ∖ {0}) = ℝ+) | |
23 | 21, 22 | ax-mp 5 | . . . . . . . . . 10 ⊢ (ℝ+ ∖ {0}) = ℝ+ |
24 | rpssre 13039 | . . . . . . . . . . . 12 ⊢ ℝ+ ⊆ ℝ | |
25 | ax-resscn 11209 | . . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ | |
26 | 24, 25 | sstri 4004 | . . . . . . . . . . 11 ⊢ ℝ+ ⊆ ℂ |
27 | ssdif 4153 | . . . . . . . . . . 11 ⊢ (ℝ+ ⊆ ℂ → (ℝ+ ∖ {0}) ⊆ (ℂ ∖ {0})) | |
28 | 26, 27 | ax-mp 5 | . . . . . . . . . 10 ⊢ (ℝ+ ∖ {0}) ⊆ (ℂ ∖ {0}) |
29 | 23, 28 | eqsstrri 4030 | . . . . . . . . 9 ⊢ ℝ+ ⊆ (ℂ ∖ {0}) |
30 | 29 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ℝ+ ⊆ (ℂ ∖ {0})) |
31 | 18, 30 | feqresmpt 6977 | . . . . . . 7 ⊢ (𝜑 → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) |
32 | 31 | eqcomd 2740 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) = (log ↾ ℝ+)) |
33 | 32 | oveq2d 7446 | . . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) = (ℝ D (log ↾ ℝ+))) |
34 | dvrelog 26693 | . . . . . 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 2774 | . . . 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 13424 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑦 ∈ (𝐴(,)𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 < 𝐵))) | |
40 | 37, 38, 39 | syl2anc 584 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 < 𝐵))) |
41 | 40 | biimpa 476 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦 ∈ ℝ ∧ 𝐴 < 𝑦 ∧ 𝑦 < 𝐵)) |
42 | 41 | simp1d 1141 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ) |
43 | 0red 11261 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ) | |
44 | 43 | rexrd 11308 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ∈ ℝ*) |
45 | 37 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ*) |
46 | 42 | rexrd 11308 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ*) |
47 | dvrelog3.3 | . . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐴) | |
48 | 47 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 ≤ 𝐴) |
49 | 41 | simp2d 1142 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝑦) |
50 | 44, 45, 46, 48, 49 | xrlelttrd 13198 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 0 < 𝑦) |
51 | 42, 50 | jca 511 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝑦 ∈ ℝ ∧ 0 < 𝑦)) |
52 | elrp 13033 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ+ ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) | |
53 | 51, 52 | sylibr 234 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ+) |
54 | 53 | ex 412 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) → 𝑦 ∈ ℝ+)) |
55 | 54 | ssrdv 4000 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ+) |
56 | eqid 2734 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
57 | 56 | tgioo2 24838 | . . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
58 | retop 24797 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
59 | 58 | a1i 11 | . . . . 5 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top) |
60 | iooretop 24801 | . . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) | |
61 | 60 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,))) |
62 | isopn3i 23105 | . . . . 5 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴(,)𝐵) ∈ (topGen‘ran (,))) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)) | |
63 | 59, 61, 62 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)) |
64 | 5, 10, 14, 36, 55, 57, 56, 63 | dvmptres2 26014 | . . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))) |
65 | 3, 64 | eqtrd 2774 | . 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 2740 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) = 𝐺) |
69 | 65, 68 | eqtrd 2774 | 1 ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∖ cdif 3959 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 {csn 4630 {cpr 4632 class class class wbr 5147 ↦ cmpt 5230 ran crn 5689 ↾ cres 5690 ⟶wf 6558 –1-1-onto→wf1o 6561 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 ℝ*cxr 11291 < clt 11292 ≤ cle 11293 / cdiv 11917 ℝ+crp 13031 (,)cioo 13383 TopOpenctopn 17467 topGenctg 17483 ℂfldccnfld 21381 Topctop 22914 intcnt 23040 D cdv 25912 logclog 26610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-cmp 23410 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-limc 25915 df-dv 25916 df-log 26612 |
This theorem is referenced by: dvrelog2b 42047 |
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