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Mirrors > Home > MPE Home > Th. List > dvrelog | Structured version Visualization version GIF version |
Description: The derivative of the real logarithm function. (Contributed by Mario Carneiro, 24-Feb-2015.) |
Ref | Expression |
---|---|
dvrelog | β’ (β D (log βΎ β+)) = (π₯ β β+ β¦ (1 / π₯)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrelog 25944 | . . 3 β’ (log βΎ β+) = β‘(exp βΎ β) | |
2 | 1 | oveq2i 7372 | . 2 β’ (β D (log βΎ β+)) = (β D β‘(exp βΎ β)) |
3 | reeff1o 25829 | . . . . . . . . 9 β’ (exp βΎ β):ββ1-1-ontoββ+ | |
4 | f1of 6788 | . . . . . . . . 9 β’ ((exp βΎ β):ββ1-1-ontoββ+ β (exp βΎ β):ββΆβ+) | |
5 | 3, 4 | ax-mp 5 | . . . . . . . 8 β’ (exp βΎ β):ββΆβ+ |
6 | rpssre 12930 | . . . . . . . 8 β’ β+ β β | |
7 | fss 6689 | . . . . . . . 8 β’ (((exp βΎ β):ββΆβ+ β§ β+ β β) β (exp βΎ β):ββΆβ) | |
8 | 5, 6, 7 | mp2an 691 | . . . . . . 7 β’ (exp βΎ β):ββΆβ |
9 | ax-resscn 11116 | . . . . . . . 8 β’ β β β | |
10 | efcn 25825 | . . . . . . . . 9 β’ exp β (ββcnββ) | |
11 | rescncf 24283 | . . . . . . . . 9 β’ (β β β β (exp β (ββcnββ) β (exp βΎ β) β (ββcnββ))) | |
12 | 9, 10, 11 | mp2 9 | . . . . . . . 8 β’ (exp βΎ β) β (ββcnββ) |
13 | cncfcdm 24284 | . . . . . . . 8 β’ ((β β β β§ (exp βΎ β) β (ββcnββ)) β ((exp βΎ β) β (ββcnββ) β (exp βΎ β):ββΆβ)) | |
14 | 9, 12, 13 | mp2an 691 | . . . . . . 7 β’ ((exp βΎ β) β (ββcnββ) β (exp βΎ β):ββΆβ) |
15 | 8, 14 | mpbir 230 | . . . . . 6 β’ (exp βΎ β) β (ββcnββ) |
16 | 15 | a1i 11 | . . . . 5 β’ (β€ β (exp βΎ β) β (ββcnββ)) |
17 | reelprrecn 11151 | . . . . . . . . . 10 β’ β β {β, β} | |
18 | eff 15972 | . . . . . . . . . 10 β’ exp:ββΆβ | |
19 | ssid 3970 | . . . . . . . . . 10 β’ β β β | |
20 | dvef 25367 | . . . . . . . . . . . . 13 β’ (β D exp) = exp | |
21 | 20 | dmeqi 5864 | . . . . . . . . . . . 12 β’ dom (β D exp) = dom exp |
22 | 18 | fdmi 6684 | . . . . . . . . . . . 12 β’ dom exp = β |
23 | 21, 22 | eqtri 2761 | . . . . . . . . . . 11 β’ dom (β D exp) = β |
24 | 9, 23 | sseqtrri 3985 | . . . . . . . . . 10 β’ β β dom (β D exp) |
25 | dvres3 25300 | . . . . . . . . . 10 β’ (((β β {β, β} β§ exp:ββΆβ) β§ (β β β β§ β β dom (β D exp))) β (β D (exp βΎ β)) = ((β D exp) βΎ β)) | |
26 | 17, 18, 19, 24, 25 | mp4an 692 | . . . . . . . . 9 β’ (β D (exp βΎ β)) = ((β D exp) βΎ β) |
27 | 20 | reseq1i 5937 | . . . . . . . . 9 β’ ((β D exp) βΎ β) = (exp βΎ β) |
28 | 26, 27 | eqtri 2761 | . . . . . . . 8 β’ (β D (exp βΎ β)) = (exp βΎ β) |
29 | 28 | dmeqi 5864 | . . . . . . 7 β’ dom (β D (exp βΎ β)) = dom (exp βΎ β) |
30 | 5 | fdmi 6684 | . . . . . . 7 β’ dom (exp βΎ β) = β |
31 | 29, 30 | eqtri 2761 | . . . . . 6 β’ dom (β D (exp βΎ β)) = β |
32 | 31 | a1i 11 | . . . . 5 β’ (β€ β dom (β D (exp βΎ β)) = β) |
33 | 0nrp 12958 | . . . . . . 7 β’ Β¬ 0 β β+ | |
34 | 28 | rneqi 5896 | . . . . . . . . 9 β’ ran (β D (exp βΎ β)) = ran (exp βΎ β) |
35 | f1ofo 6795 | . . . . . . . . . 10 β’ ((exp βΎ β):ββ1-1-ontoββ+ β (exp βΎ β):ββontoββ+) | |
36 | forn 6763 | . . . . . . . . . 10 β’ ((exp βΎ β):ββontoββ+ β ran (exp βΎ β) = β+) | |
37 | 3, 35, 36 | mp2b 10 | . . . . . . . . 9 β’ ran (exp βΎ β) = β+ |
38 | 34, 37 | eqtri 2761 | . . . . . . . 8 β’ ran (β D (exp βΎ β)) = β+ |
39 | 38 | eleq2i 2826 | . . . . . . 7 β’ (0 β ran (β D (exp βΎ β)) β 0 β β+) |
40 | 33, 39 | mtbir 323 | . . . . . 6 β’ Β¬ 0 β ran (β D (exp βΎ β)) |
41 | 40 | a1i 11 | . . . . 5 β’ (β€ β Β¬ 0 β ran (β D (exp βΎ β))) |
42 | 3 | a1i 11 | . . . . 5 β’ (β€ β (exp βΎ β):ββ1-1-ontoββ+) |
43 | 16, 32, 41, 42 | dvcnvre 25406 | . . . 4 β’ (β€ β (β D β‘(exp βΎ β)) = (π₯ β β+ β¦ (1 / ((β D (exp βΎ β))β(β‘(exp βΎ β)βπ₯))))) |
44 | 43 | mptru 1549 | . . 3 β’ (β D β‘(exp βΎ β)) = (π₯ β β+ β¦ (1 / ((β D (exp βΎ β))β(β‘(exp βΎ β)βπ₯)))) |
45 | 28 | fveq1i 6847 | . . . . . 6 β’ ((β D (exp βΎ β))β(β‘(exp βΎ β)βπ₯)) = ((exp βΎ β)β(β‘(exp βΎ β)βπ₯)) |
46 | f1ocnvfv2 7227 | . . . . . . 7 β’ (((exp βΎ β):ββ1-1-ontoββ+ β§ π₯ β β+) β ((exp βΎ β)β(β‘(exp βΎ β)βπ₯)) = π₯) | |
47 | 3, 46 | mpan 689 | . . . . . 6 β’ (π₯ β β+ β ((exp βΎ β)β(β‘(exp βΎ β)βπ₯)) = π₯) |
48 | 45, 47 | eqtrid 2785 | . . . . 5 β’ (π₯ β β+ β ((β D (exp βΎ β))β(β‘(exp βΎ β)βπ₯)) = π₯) |
49 | 48 | oveq2d 7377 | . . . 4 β’ (π₯ β β+ β (1 / ((β D (exp βΎ β))β(β‘(exp βΎ β)βπ₯))) = (1 / π₯)) |
50 | 49 | mpteq2ia 5212 | . . 3 β’ (π₯ β β+ β¦ (1 / ((β D (exp βΎ β))β(β‘(exp βΎ β)βπ₯)))) = (π₯ β β+ β¦ (1 / π₯)) |
51 | 44, 50 | eqtri 2761 | . 2 β’ (β D β‘(exp βΎ β)) = (π₯ β β+ β¦ (1 / π₯)) |
52 | 2, 51 | eqtri 2761 | 1 β’ (β D (log βΎ β+)) = (π₯ β β+ β¦ (1 / π₯)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wb 205 = wceq 1542 β€wtru 1543 β wcel 2107 β wss 3914 {cpr 4592 β¦ cmpt 5192 β‘ccnv 5636 dom cdm 5637 ran crn 5638 βΎ cres 5639 βΆwf 6496 βontoβwfo 6498 β1-1-ontoβwf1o 6499 βcfv 6500 (class class class)co 7361 βcc 11057 βcr 11058 0cc0 11059 1c1 11060 / cdiv 11820 β+crp 12923 expce 15952 βcnβccncf 24262 D cdv 25250 logclog 25933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-pm 8774 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-fi 9355 df-sup 9386 df-inf 9387 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13434 df-fzo 13577 df-fl 13706 df-mod 13784 df-seq 13916 df-exp 13977 df-fac 14183 df-bc 14212 df-hash 14240 df-shft 14961 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-limsup 15362 df-clim 15379 df-rlim 15380 df-sum 15580 df-ef 15958 df-sin 15960 df-cos 15961 df-pi 15963 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-hom 17165 df-cco 17166 df-rest 17312 df-topn 17313 df-0g 17331 df-gsum 17332 df-topgen 17333 df-pt 17334 df-prds 17337 df-xrs 17392 df-qtop 17397 df-imas 17398 df-xps 17400 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-mulg 18881 df-cntz 19105 df-cmn 19572 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-fbas 20816 df-fg 20817 df-cnfld 20820 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-cld 22393 df-ntr 22394 df-cls 22395 df-nei 22472 df-lp 22510 df-perf 22511 df-cn 22601 df-cnp 22602 df-haus 22689 df-cmp 22761 df-tx 22936 df-hmeo 23129 df-fil 23220 df-fm 23312 df-flim 23313 df-flf 23314 df-xms 23696 df-ms 23697 df-tms 23698 df-cncf 24264 df-limc 25253 df-dv 25254 df-log 25935 |
This theorem is referenced by: relogcn 26016 advlog 26032 advlogexp 26033 logccv 26041 dvcxp1 26116 loglesqrt 26134 logdivsum 26904 log2sumbnd 26915 logdivsqrle 33327 dvrelog2 40571 dvrelog3 40572 |
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