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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 25149 | . . 3 ⊢ (log ↾ ℝ+) = ◡(exp ↾ ℝ) | |
2 | 1 | oveq2i 7167 | . 2 ⊢ (ℝ D (log ↾ ℝ+)) = (ℝ D ◡(exp ↾ ℝ)) |
3 | reeff1o 25035 | . . . . . . . . 9 ⊢ (exp ↾ ℝ):ℝ–1-1-onto→ℝ+ | |
4 | f1of 6615 | . . . . . . . . 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 12397 | . . . . . . . 8 ⊢ ℝ+ ⊆ ℝ | |
7 | fss 6527 | . . . . . . . 8 ⊢ (((exp ↾ ℝ):ℝ⟶ℝ+ ∧ ℝ+ ⊆ ℝ) → (exp ↾ ℝ):ℝ⟶ℝ) | |
8 | 5, 6, 7 | mp2an 690 | . . . . . . 7 ⊢ (exp ↾ ℝ):ℝ⟶ℝ |
9 | ax-resscn 10594 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
10 | efcn 25031 | . . . . . . . . 9 ⊢ exp ∈ (ℂ–cn→ℂ) | |
11 | rescncf 23505 | . . . . . . . . 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 23506 | . . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ (exp ↾ ℝ) ∈ (ℝ–cn→ℂ)) → ((exp ↾ ℝ) ∈ (ℝ–cn→ℝ) ↔ (exp ↾ ℝ):ℝ⟶ℝ)) | |
14 | 9, 12, 13 | mp2an 690 | . . . . . . 7 ⊢ ((exp ↾ ℝ) ∈ (ℝ–cn→ℝ) ↔ (exp ↾ ℝ):ℝ⟶ℝ) |
15 | 8, 14 | mpbir 233 | . . . . . 6 ⊢ (exp ↾ ℝ) ∈ (ℝ–cn→ℝ) |
16 | 15 | a1i 11 | . . . . 5 ⊢ (⊤ → (exp ↾ ℝ) ∈ (ℝ–cn→ℝ)) |
17 | reelprrecn 10629 | . . . . . . . . . 10 ⊢ ℝ ∈ {ℝ, ℂ} | |
18 | eff 15435 | . . . . . . . . . 10 ⊢ exp:ℂ⟶ℂ | |
19 | ssid 3989 | . . . . . . . . . 10 ⊢ ℂ ⊆ ℂ | |
20 | dvef 24577 | . . . . . . . . . . . . 13 ⊢ (ℂ D exp) = exp | |
21 | 20 | dmeqi 5773 | . . . . . . . . . . . 12 ⊢ dom (ℂ D exp) = dom exp |
22 | 18 | fdmi 6524 | . . . . . . . . . . . 12 ⊢ dom exp = ℂ |
23 | 21, 22 | eqtri 2844 | . . . . . . . . . . 11 ⊢ dom (ℂ D exp) = ℂ |
24 | 9, 23 | sseqtrri 4004 | . . . . . . . . . 10 ⊢ ℝ ⊆ dom (ℂ D exp) |
25 | dvres3 24511 | . . . . . . . . . 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 691 | . . . . . . . . 9 ⊢ (ℝ D (exp ↾ ℝ)) = ((ℂ D exp) ↾ ℝ) |
27 | 20 | reseq1i 5849 | . . . . . . . . 9 ⊢ ((ℂ D exp) ↾ ℝ) = (exp ↾ ℝ) |
28 | 26, 27 | eqtri 2844 | . . . . . . . 8 ⊢ (ℝ D (exp ↾ ℝ)) = (exp ↾ ℝ) |
29 | 28 | dmeqi 5773 | . . . . . . 7 ⊢ dom (ℝ D (exp ↾ ℝ)) = dom (exp ↾ ℝ) |
30 | 5 | fdmi 6524 | . . . . . . 7 ⊢ dom (exp ↾ ℝ) = ℝ |
31 | 29, 30 | eqtri 2844 | . . . . . 6 ⊢ dom (ℝ D (exp ↾ ℝ)) = ℝ |
32 | 31 | a1i 11 | . . . . 5 ⊢ (⊤ → dom (ℝ D (exp ↾ ℝ)) = ℝ) |
33 | 0nrp 12425 | . . . . . . 7 ⊢ ¬ 0 ∈ ℝ+ | |
34 | 28 | rneqi 5807 | . . . . . . . . 9 ⊢ ran (ℝ D (exp ↾ ℝ)) = ran (exp ↾ ℝ) |
35 | f1ofo 6622 | . . . . . . . . . 10 ⊢ ((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ → (exp ↾ ℝ):ℝ–onto→ℝ+) | |
36 | forn 6593 | . . . . . . . . . 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 2844 | . . . . . . . 8 ⊢ ran (ℝ D (exp ↾ ℝ)) = ℝ+ |
39 | 38 | eleq2i 2904 | . . . . . . 7 ⊢ (0 ∈ ran (ℝ D (exp ↾ ℝ)) ↔ 0 ∈ ℝ+) |
40 | 33, 39 | mtbir 325 | . . . . . 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 24616 | . . . 4 ⊢ (⊤ → (ℝ D ◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ+ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(◡(exp ↾ ℝ)‘𝑥))))) |
44 | 43 | mptru 1544 | . . 3 ⊢ (ℝ D ◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ+ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(◡(exp ↾ ℝ)‘𝑥)))) |
45 | 28 | fveq1i 6671 | . . . . . 6 ⊢ ((ℝ D (exp ↾ ℝ))‘(◡(exp ↾ ℝ)‘𝑥)) = ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘𝑥)) |
46 | f1ocnvfv2 7034 | . . . . . . 7 ⊢ (((exp ↾ ℝ):ℝ–1-1-onto→ℝ+ ∧ 𝑥 ∈ ℝ+) → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘𝑥)) = 𝑥) | |
47 | 3, 46 | mpan 688 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → ((exp ↾ ℝ)‘(◡(exp ↾ ℝ)‘𝑥)) = 𝑥) |
48 | 45, 47 | syl5eq 2868 | . . . . 5 ⊢ (𝑥 ∈ ℝ+ → ((ℝ D (exp ↾ ℝ))‘(◡(exp ↾ ℝ)‘𝑥)) = 𝑥) |
49 | 48 | oveq2d 7172 | . . . 4 ⊢ (𝑥 ∈ ℝ+ → (1 / ((ℝ D (exp ↾ ℝ))‘(◡(exp ↾ ℝ)‘𝑥))) = (1 / 𝑥)) |
50 | 49 | mpteq2ia 5157 | . . 3 ⊢ (𝑥 ∈ ℝ+ ↦ (1 / ((ℝ D (exp ↾ ℝ))‘(◡(exp ↾ ℝ)‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) |
51 | 44, 50 | eqtri 2844 | . 2 ⊢ (ℝ D ◡(exp ↾ ℝ)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) |
52 | 2, 51 | eqtri 2844 | 1 ⊢ (ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ⊆ wss 3936 {cpr 4569 ↦ cmpt 5146 ◡ccnv 5554 dom cdm 5555 ran crn 5556 ↾ cres 5557 ⟶wf 6351 –onto→wfo 6353 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 / cdiv 11297 ℝ+crp 12390 expce 15415 –cn→ccncf 23484 D cdv 24461 logclog 25138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140 |
This theorem is referenced by: relogcn 25221 advlog 25237 advlogexp 25238 logccv 25246 dvcxp1 25321 loglesqrt 25339 logdivsum 26109 log2sumbnd 26120 logdivsqrle 31921 |
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