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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 | ⊢ (ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrelog 24532 | . . 3 ⊢ (log ↾ ℝ+) = ◡(exp ↾ ℝ) | |
2 | 1 | oveq2i 6825 | . 2 ⊢ (ℝ D (log ↾ ℝ+)) = (ℝ D ◡(exp ↾ ℝ)) |
3 | reeff1o 24420 | . . . . . . . . 9 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ+ | |
4 | f1of 6299 | . . . . . . . . 9 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ → (exp ↾ ℝ):ℝ⟶ℝ+) | |
5 | 3, 4 | ax-mp 5 | . . . . . . . 8 ⊢ (exp ↾ ℝ):ℝ⟶ℝ+ |
6 | rpssre 12056 | . . . . . . . 8 ⊢ ℝ+ ⊆ ℝ | |
7 | fss 6217 | . . . . . . . 8 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ+ ∧ ℝ+ ⊆ ℝ) → (exp ↾ ℝ):ℝ⟶ℝ) | |
8 | 5, 6, 7 | mp2an 710 | . . . . . . 7 ⊢ (exp ↾ ℝ):ℝ⟶ℝ |
9 | ax-resscn 10205 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
10 | efcn 24416 | . . . . . . . . 9 ⊢ exp ∈ (ℂ–cn→ℂ) | |
11 | rescncf 22921 | . . . . . . . . 9 ⊢ (ℝ ⊆ ℂ → (exp ∈ (ℂ–cn→ℂ) → (exp ↾ ℝ) ∈ (ℝ–cn→ℂ))) | |
12 | 9, 10, 11 | mp2 9 | . . . . . . . 8 ⊢ (exp ↾ ℝ) ∈ (ℝ–cn→ℂ) |
13 | cncffvrn 22922 | . . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ (exp ↾ ℝ) ∈ (ℝ–cn→ℂ)) → ((exp ↾ ℝ) ∈ (ℝ–cn→ℝ) ↔ (exp ↾ ℝ):ℝ⟶ℝ)) | |
14 | 9, 12, 13 | mp2an 710 | . . . . . . 7 ⊢ ((exp ↾ ℝ) ∈ (ℝ–cn→ℝ) ↔ (exp ↾ ℝ):ℝ⟶ℝ) |
15 | 8, 14 | mpbir 221 | . . . . . 6 ⊢ (exp ↾ ℝ) ∈ (ℝ–cn→ℝ) |
16 | 15 | a1i 11 | . . . . 5 ⊢ (⊤ → (exp ↾ ℝ) ∈ (ℝ–cn→ℝ)) |
17 | reelprrecn 10240 | . . . . . . . . . 10 ⊢ ℝ ∈ {ℝ, ℂ} | |
18 | eff 15031 | . . . . . . . . . 10 ⊢ exp:ℂ⟶ℂ | |
19 | ssid 3765 | . . . . . . . . . 10 ⊢ ℂ ⊆ ℂ | |
20 | dvef 23962 | . . . . . . . . . . . . 13 ⊢ (ℂ D exp) = exp | |
21 | 20 | dmeqi 5480 | . . . . . . . . . . . 12 ⊢ dom (ℂ D exp) = dom exp |
22 | 18 | fdmi 6213 | . . . . . . . . . . . 12 ⊢ dom exp = ℂ |
23 | 21, 22 | eqtri 2782 | . . . . . . . . . . 11 ⊢ dom (ℂ D exp) = ℂ |
24 | 9, 23 | sseqtr4i 3779 | . . . . . . . . . 10 ⊢ ℝ ⊆ dom (ℂ D exp) |
25 | dvres3 23896 | . . . . . . . . . 10 ⊢ (((ℝ ∈ {ℝ, ℂ} ∧ exp:ℂ⟶ℂ) ∧ (ℂ ⊆ ℂ ∧ ℝ ⊆ dom (ℂ D exp))) → (ℝ D (exp ↾ ℝ)) = ((ℂ D exp) ↾ ℝ)) | |
26 | 17, 18, 19, 24, 25 | mp4an 711 | . . . . . . . . 9 ⊢ (ℝ D (exp ↾ ℝ)) = ((ℂ D exp) ↾ ℝ) |
27 | 20 | reseq1i 5547 | . . . . . . . . 9 ⊢ ((ℂ D exp) ↾ ℝ) = (exp ↾ ℝ) |
28 | 26, 27 | eqtri 2782 | . . . . . . . 8 ⊢ (ℝ D (exp ↾ ℝ)) = (exp ↾ ℝ) |
29 | 28 | dmeqi 5480 | . . . . . . 7 ⊢ dom (ℝ D (exp ↾ ℝ)) = dom (exp ↾ ℝ) |
30 | 5 | fdmi 6213 | . . . . . . 7 ⊢ dom (exp ↾ ℝ) = ℝ |
31 | 29, 30 | eqtri 2782 | . . . . . 6 ⊢ dom (ℝ D (exp ↾ ℝ)) = ℝ |
32 | 31 | a1i 11 | . . . . 5 ⊢ (⊤ → dom (ℝ D (exp ↾ ℝ)) = ℝ) |
33 | 0nrp 12078 | . . . . . . 7 ⊢ ¬ 0 ∈ ℝ+ | |
34 | 28 | rneqi 5507 | . . . . . . . . 9 ⊢ ran (ℝ D (exp ↾ ℝ)) = ran (exp ↾ ℝ) |
35 | f1ofo 6306 | . . . . . . . . . 10 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ → (exp ↾ ℝ):ℝ–onto→ℝ+) | |
36 | forn 6280 | . . . . . . . . . 10 ⊢ ((exp ↾ ℝ):ℝ–onto→ℝ+ → ran (exp ↾ ℝ) = ℝ+) | |
37 | 3, 35, 36 | mp2b 10 | . . . . . . . . 9 ⊢ ran (exp ↾ ℝ) = ℝ+ |
38 | 34, 37 | eqtri 2782 | . . . . . . . 8 ⊢ ran (ℝ D (exp ↾ ℝ)) = ℝ+ |
39 | 38 | eleq2i 2831 | . . . . . . 7 ⊢ (0 ∈ ran (ℝ D (exp ↾ ℝ)) ↔ 0 ∈ ℝ+) |
40 | 33, 39 | mtbir 312 | . . . . . 6 ⊢ ¬ 0 ∈ ran (ℝ D (exp ↾ ℝ)) |
41 | 40 | a1i 11 | . . . . 5 ⊢ (⊤ → ¬ 0 ∈ ran (ℝ D (exp ↾ ℝ))) |
42 | 3 | a1i 11 | . . . . 5 ⊢ (⊤ → (exp ↾ ℝ):ℝ–1-1-onto→ℝ+) |
43 | 16, 32, 41, 42 | dvcnvre 24001 | . . . 4 ⊢ (⊤ → (ℝ D ◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ+ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(◡(exp ↾ ℝ)‘𝑥))))) |
44 | 43 | trud 1642 | . . 3 ⊢ (ℝ D ◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ+ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(◡(exp ↾ ℝ)‘𝑥)))) |
45 | 28 | fveq1i 6354 | . . . . . 6 ⊢ ((ℝ D (exp ↾ ℝ))‘(◡(exp ↾ ℝ)‘𝑥)) = ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘𝑥)) |
46 | f1ocnvfv2 6697 | . . . . . . 7 ⊢ (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ 𝑥 ∈ ℝ+) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘𝑥)) = 𝑥) | |
47 | 3, 46 | mpan 708 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘𝑥)) = 𝑥) |
48 | 45, 47 | syl5eq 2806 | . . . . 5 ⊢ (𝑥 ∈ ℝ+ → ((ℝ D (exp ↾ ℝ))‘(◡(exp ↾ ℝ)‘𝑥)) = 𝑥) |
49 | 48 | oveq2d 6830 | . . . 4 ⊢ (𝑥 ∈ ℝ+ → (1 / ((ℝ D (exp ↾ ℝ))‘(◡(exp ↾ ℝ)‘𝑥))) = (1 / 𝑥)) |
50 | 49 | mpteq2ia 4892 | . . 3 ⊢ (𝑥 ∈ ℝ+ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(◡(exp ↾ ℝ)‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) |
51 | 44, 50 | eqtri 2782 | . 2 ⊢ (ℝ D ◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) |
52 | 2, 51 | eqtri 2782 | 1 ⊢ (ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 = wceq 1632 ⊤wtru 1633 ∈ wcel 2139 ⊆ wss 3715 {cpr 4323 ↦ cmpt 4881 ◡ccnv 5265 dom cdm 5266 ran crn 5267 ↾ cres 5268 ⟶wf 6045 –onto→wfo 6047 –1-1-onto→wf1o 6048 ‘cfv 6049 (class class class)co 6814 ℂcc 10146 ℝcr 10147 0cc0 10148 1c1 10149 / cdiv 10896 ℝ+crp 12045 expce 15011 –cn→ccncf 22900 D cdv 23846 logclog 24521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 ax-addf 10227 ax-mulf 10228 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-supp 7465 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-2o 7731 df-oadd 7734 df-er 7913 df-map 8027 df-pm 8028 df-ixp 8077 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fsupp 8443 df-fi 8484 df-sup 8515 df-inf 8516 df-oi 8582 df-card 8975 df-cda 9202 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-q 12002 df-rp 12046 df-xneg 12159 df-xadd 12160 df-xmul 12161 df-ioo 12392 df-ioc 12393 df-ico 12394 df-icc 12395 df-fz 12540 df-fzo 12680 df-fl 12807 df-mod 12883 df-seq 13016 df-exp 13075 df-fac 13275 df-bc 13304 df-hash 13332 df-shft 14026 df-cj 14058 df-re 14059 df-im 14060 df-sqrt 14194 df-abs 14195 df-limsup 14421 df-clim 14438 df-rlim 14439 df-sum 14636 df-ef 15017 df-sin 15019 df-cos 15020 df-pi 15022 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-starv 16178 df-sca 16179 df-vsca 16180 df-ip 16181 df-tset 16182 df-ple 16183 df-ds 16186 df-unif 16187 df-hom 16188 df-cco 16189 df-rest 16305 df-topn 16306 df-0g 16324 df-gsum 16325 df-topgen 16326 df-pt 16327 df-prds 16330 df-xrs 16384 df-qtop 16389 df-imas 16390 df-xps 16392 df-mre 16468 df-mrc 16469 df-acs 16471 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-submnd 17557 df-mulg 17762 df-cntz 17970 df-cmn 18415 df-psmet 19960 df-xmet 19961 df-met 19962 df-bl 19963 df-mopn 19964 df-fbas 19965 df-fg 19966 df-cnfld 19969 df-top 20921 df-topon 20938 df-topsp 20959 df-bases 20972 df-cld 21045 df-ntr 21046 df-cls 21047 df-nei 21124 df-lp 21162 df-perf 21163 df-cn 21253 df-cnp 21254 df-haus 21341 df-cmp 21412 df-tx 21587 df-hmeo 21780 df-fil 21871 df-fm 21963 df-flim 21964 df-flf 21965 df-xms 22346 df-ms 22347 df-tms 22348 df-cncf 22902 df-limc 23849 df-dv 23850 df-log 24523 |
This theorem is referenced by: relogcn 24604 advlog 24620 advlogexp 24621 logccv 24629 dvcxp1 24701 loglesqrt 24719 logdivsum 25442 log2sumbnd 25453 logdivsqrle 31058 |
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