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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvrelog2 | Structured version Visualization version GIF version |
Description: The derivative of the logarithm, ftc2 24748 version. (Contributed by metakunt, 11-Aug-2024.) |
Ref | Expression |
---|---|
dvrelog2.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
dvrelog2.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
dvrelog2.3 | ⊢ (𝜑 → 0 < 𝐴) |
dvrelog2.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
dvrelog2.5 | ⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (log‘𝑥)) |
dvrelog2.6 | ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) |
Ref | Expression |
---|---|
dvrelog2 | ⊢ (𝜑 → (ℝ D 𝐹) = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvrelog2.5 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (log‘𝑥)) | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (log‘𝑥))) |
3 | 2 | oveq2d 7171 | . . 3 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (log‘𝑥)))) |
4 | reelprrecn 10672 | . . . . 5 ⊢ ℝ ∈ {ℝ, ℂ} | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ}) |
6 | rpssre 12442 | . . . . . . . 8 ⊢ ℝ+ ⊆ ℝ | |
7 | ax-resscn 10637 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
8 | 6, 7 | sstri 3903 | . . . . . . 7 ⊢ ℝ+ ⊆ ℂ |
9 | 8 | sseli 3890 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℂ) |
10 | 9 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ) |
11 | rpne0 12451 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ≠ 0) | |
12 | 11 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0) |
13 | 10, 12 | logcld 25266 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ) |
14 | 1red 10685 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 1 ∈ ℝ) | |
15 | 6 | sseli 3890 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ) |
16 | 14, 15, 11 | redivcld 11511 | . . . . 5 ⊢ (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ) |
17 | 16 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ) |
18 | logf1o 25260 | . . . . . . . . . 10 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log | |
19 | f1of 6606 | . . . . . . . . . 10 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log) | |
20 | 18, 19 | ax-mp 5 | . . . . . . . . 9 ⊢ log:(ℂ ∖ {0})⟶ran log |
21 | 20 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → log:(ℂ ∖ {0})⟶ran log) |
22 | 0nrp 12470 | . . . . . . . . . . . 12 ⊢ ¬ 0 ∈ ℝ+ | |
23 | disjsn 4607 | . . . . . . . . . . . 12 ⊢ ((ℝ+ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℝ+) | |
24 | 22, 23 | mpbir 234 | . . . . . . . . . . 11 ⊢ (ℝ+ ∩ {0}) = ∅ |
25 | disjdif2 4379 | . . . . . . . . . . 11 ⊢ ((ℝ+ ∩ {0}) = ∅ → (ℝ+ ∖ {0}) = ℝ+) | |
26 | 24, 25 | ax-mp 5 | . . . . . . . . . 10 ⊢ (ℝ+ ∖ {0}) = ℝ+ |
27 | ssdif 4047 | . . . . . . . . . . 11 ⊢ (ℝ+ ⊆ ℂ → (ℝ+ ∖ {0}) ⊆ (ℂ ∖ {0})) | |
28 | 8, 27 | ax-mp 5 | . . . . . . . . . 10 ⊢ (ℝ+ ∖ {0}) ⊆ (ℂ ∖ {0}) |
29 | 26, 28 | eqsstrri 3929 | . . . . . . . . 9 ⊢ ℝ+ ⊆ (ℂ ∖ {0}) |
30 | 29 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ℝ+ ⊆ (ℂ ∖ {0})) |
31 | 21, 30 | feqresmpt 6726 | . . . . . . 7 ⊢ (𝜑 → (log ↾ ℝ+) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) |
32 | 31 | eqcomd 2764 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) = (log ↾ ℝ+)) |
33 | 32 | oveq2d 7171 | . . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ+ ↦ (log‘𝑥))) = (ℝ D (log ↾ ℝ+))) |
34 | dvrelog 25332 | . . . . . 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 | dvrelog2.1 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
38 | dvrelog2.2 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
39 | elicc2 12849 | . . . . . . . . . . 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 10687 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 0 ∈ ℝ) | |
44 | 37 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ) |
45 | dvrelog2.3 | . . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝐴) | |
46 | 45 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 0 < 𝐴) |
47 | 41 | simp2d 1140 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑦) |
48 | 43, 44, 42, 46, 47 | ltletrd 10843 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 0 < 𝑦) |
49 | 42, 48 | jca 515 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑦 ∈ ℝ ∧ 0 < 𝑦)) |
50 | elrp 12437 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ+ ↔ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) | |
51 | 49, 50 | sylibr 237 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴[,]𝐵)) → 𝑦 ∈ ℝ+) |
52 | 51 | ex 416 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐴[,]𝐵) → 𝑦 ∈ ℝ+)) |
53 | 52 | ssrdv 3900 | . . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ+) |
54 | eqid 2758 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
55 | 54 | tgioo2 23509 | . . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
56 | iccntr 23527 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) | |
57 | 37, 38, 56 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) |
58 | 5, 13, 17, 36, 53, 55, 54, 57 | dvmptres2 24666 | . . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝐴[,]𝐵) ↦ (log‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))) |
59 | 3, 58 | eqtrd 2793 | . 2 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))) |
60 | dvrelog2.6 | . . . 4 ⊢ 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) | |
61 | 60 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))) |
62 | 61 | eqcomd 2764 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥)) = 𝐺) |
63 | 59, 62 | 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 5035 ↦ cmpt 5115 ran crn 5528 ↾ cres 5529 ⟶wf 6335 –1-1-onto→wf1o 6338 ‘cfv 6339 (class class class)co 7155 ℂcc 10578 ℝcr 10579 0cc0 10580 1c1 10581 < clt 10718 ≤ cle 10719 / cdiv 11340 ℝ+crp 12435 (,)cioo 12784 [,]cicc 12787 TopOpenctopn 16758 topGenctg 16774 ℂfldccnfld 20171 intcnt 21722 D cdv 24567 logclog 25250 |
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 5159 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-inf2 9142 ax-cnex 10636 ax-resscn 10637 ax-1cn 10638 ax-icn 10639 ax-addcl 10640 ax-addrcl 10641 ax-mulcl 10642 ax-mulrcl 10643 ax-mulcom 10644 ax-addass 10645 ax-mulass 10646 ax-distr 10647 ax-i2m1 10648 ax-1ne0 10649 ax-1rid 10650 ax-rnegex 10651 ax-rrecex 10652 ax-cnre 10653 ax-pre-lttri 10654 ax-pre-lttrn 10655 ax-pre-ltadd 10656 ax-pre-mulgt0 10657 ax-pre-sup 10658 ax-addf 10659 ax-mulf 10660 |
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 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-se 5487 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 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 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7410 df-om 7585 df-1st 7698 df-2nd 7699 df-supp 7841 df-wrecs 7962 df-recs 8023 df-rdg 8061 df-1o 8117 df-2o 8118 df-er 8304 df-map 8423 df-pm 8424 df-ixp 8485 df-en 8533 df-dom 8534 df-sdom 8535 df-fin 8536 df-fsupp 8872 df-fi 8913 df-sup 8944 df-inf 8945 df-oi 9012 df-card 9406 df-pnf 10720 df-mnf 10721 df-xr 10722 df-ltxr 10723 df-le 10724 df-sub 10915 df-neg 10916 df-div 11341 df-nn 11680 df-2 11742 df-3 11743 df-4 11744 df-5 11745 df-6 11746 df-7 11747 df-8 11748 df-9 11749 df-n0 11940 df-z 12026 df-dec 12143 df-uz 12288 df-q 12394 df-rp 12436 df-xneg 12553 df-xadd 12554 df-xmul 12555 df-ioo 12788 df-ioc 12789 df-ico 12790 df-icc 12791 df-fz 12945 df-fzo 13088 df-fl 13216 df-mod 13292 df-seq 13424 df-exp 13485 df-fac 13689 df-bc 13718 df-hash 13746 df-shft 14479 df-cj 14511 df-re 14512 df-im 14513 df-sqrt 14647 df-abs 14648 df-limsup 14881 df-clim 14898 df-rlim 14899 df-sum 15096 df-ef 15474 df-sin 15476 df-cos 15477 df-pi 15479 df-struct 16548 df-ndx 16549 df-slot 16550 df-base 16552 df-sets 16553 df-ress 16554 df-plusg 16641 df-mulr 16642 df-starv 16643 df-sca 16644 df-vsca 16645 df-ip 16646 df-tset 16647 df-ple 16648 df-ds 16650 df-unif 16651 df-hom 16652 df-cco 16653 df-rest 16759 df-topn 16760 df-0g 16778 df-gsum 16779 df-topgen 16780 df-pt 16781 df-prds 16784 df-xrs 16838 df-qtop 16843 df-imas 16844 df-xps 16846 df-mre 16920 df-mrc 16921 df-acs 16923 df-mgm 17923 df-sgrp 17972 df-mnd 17983 df-submnd 18028 df-mulg 18297 df-cntz 18519 df-cmn 18980 df-psmet 20163 df-xmet 20164 df-met 20165 df-bl 20166 df-mopn 20167 df-fbas 20168 df-fg 20169 df-cnfld 20172 df-top 21599 df-topon 21616 df-topsp 21638 df-bases 21651 df-cld 21724 df-ntr 21725 df-cls 21726 df-nei 21803 df-lp 21841 df-perf 21842 df-cn 21932 df-cnp 21933 df-haus 22020 df-cmp 22092 df-tx 22267 df-hmeo 22460 df-fil 22551 df-fm 22643 df-flim 22644 df-flf 22645 df-xms 23027 df-ms 23028 df-tms 23029 df-cncf 23584 df-limc 24570 df-dv 24571 df-log 25252 |
This theorem is referenced by: (None) |
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