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Mirrors > Home > MPE Home > Th. List > dvlog2 | Structured version Visualization version GIF version |
Description: The derivative of the complex logarithm function on the open unit ball centered at 1, a sometimes easier region to work with than the ℂ ∖ (-∞, 0] of dvlog 25242. (Contributed by Mario Carneiro, 1-Mar-2015.) |
Ref | Expression |
---|---|
dvlog2.s | ⊢ 𝑆 = (1(ball‘(abs ∘ − ))1) |
Ref | Expression |
---|---|
dvlog2 | ⊢ (ℂ D (log ↾ 𝑆)) = (𝑥 ∈ 𝑆 ↦ (1 / 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3937 | . . . 4 ⊢ ℂ ⊆ ℂ | |
2 | logf1o 25156 | . . . . . . 7 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log | |
3 | f1of 6590 | . . . . . . 7 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})⟶ran log) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ log:(ℂ ∖ {0})⟶ran log |
5 | logrncn 25154 | . . . . . . 7 ⊢ (𝑥 ∈ ran log → 𝑥 ∈ ℂ) | |
6 | 5 | ssriv 3919 | . . . . . 6 ⊢ ran log ⊆ ℂ |
7 | fss 6501 | . . . . . 6 ⊢ ((log:(ℂ ∖ {0})⟶ran log ∧ ran log ⊆ ℂ) → log:(ℂ ∖ {0})⟶ℂ) | |
8 | 4, 6, 7 | mp2an 691 | . . . . 5 ⊢ log:(ℂ ∖ {0})⟶ℂ |
9 | eqid 2798 | . . . . . 6 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0)) | |
10 | 9 | logdmss 25233 | . . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0}) |
11 | fssres 6518 | . . . . 5 ⊢ ((log:(ℂ ∖ {0})⟶ℂ ∧ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})) → (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))⟶ℂ) | |
12 | 8, 10, 11 | mp2an 691 | . . . 4 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))⟶ℂ |
13 | difss 4059 | . . . 4 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ | |
14 | dvlog2.s | . . . . 5 ⊢ 𝑆 = (1(ball‘(abs ∘ − ))1) | |
15 | cnxmet 23378 | . . . . . 6 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
16 | ax-1cn 10584 | . . . . . 6 ⊢ 1 ∈ ℂ | |
17 | 1xr 10689 | . . . . . 6 ⊢ 1 ∈ ℝ* | |
18 | blssm 23025 | . . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ*) → (1(ball‘(abs ∘ − ))1) ⊆ ℂ) | |
19 | 15, 16, 17, 18 | mp3an 1458 | . . . . 5 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ ℂ |
20 | 14, 19 | eqsstri 3949 | . . . 4 ⊢ 𝑆 ⊆ ℂ |
21 | eqid 2798 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
22 | 21 | cnfldtopon 23388 | . . . . . 6 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
23 | 22 | toponrestid 21526 | . . . . 5 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ) |
24 | 21, 23 | dvres 24514 | . . . 4 ⊢ (((ℂ ⊆ ℂ ∧ (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))⟶ℂ) ∧ ((ℂ ∖ (-∞(,]0)) ⊆ ℂ ∧ 𝑆 ⊆ ℂ)) → (ℂ D ((log ↾ (ℂ ∖ (-∞(,]0))) ↾ 𝑆)) = ((ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) ↾ ((int‘(TopOpen‘ℂfld))‘𝑆))) |
25 | 1, 12, 13, 20, 24 | mp4an 692 | . . 3 ⊢ (ℂ D ((log ↾ (ℂ ∖ (-∞(,]0))) ↾ 𝑆)) = ((ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) ↾ ((int‘(TopOpen‘ℂfld))‘𝑆)) |
26 | 14 | dvlog2lem 25243 | . . . . 5 ⊢ 𝑆 ⊆ (ℂ ∖ (-∞(,]0)) |
27 | resabs1 5848 | . . . . 5 ⊢ (𝑆 ⊆ (ℂ ∖ (-∞(,]0)) → ((log ↾ (ℂ ∖ (-∞(,]0))) ↾ 𝑆) = (log ↾ 𝑆)) | |
28 | 26, 27 | ax-mp 5 | . . . 4 ⊢ ((log ↾ (ℂ ∖ (-∞(,]0))) ↾ 𝑆) = (log ↾ 𝑆) |
29 | 28 | oveq2i 7146 | . . 3 ⊢ (ℂ D ((log ↾ (ℂ ∖ (-∞(,]0))) ↾ 𝑆)) = (ℂ D (log ↾ 𝑆)) |
30 | 9 | dvlog 25242 | . . . 4 ⊢ (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) |
31 | 21 | cnfldtop 23389 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ Top |
32 | 21 | cnfldtopn 23387 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − )) |
33 | 32 | blopn 23107 | . . . . . . 7 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ*) → (1(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂfld)) |
34 | 15, 16, 17, 33 | mp3an 1458 | . . . . . 6 ⊢ (1(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂfld) |
35 | 14, 34 | eqeltri 2886 | . . . . 5 ⊢ 𝑆 ∈ (TopOpen‘ℂfld) |
36 | isopn3i 21687 | . . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ (TopOpen‘ℂfld)) → ((int‘(TopOpen‘ℂfld))‘𝑆) = 𝑆) | |
37 | 31, 35, 36 | mp2an 691 | . . . 4 ⊢ ((int‘(TopOpen‘ℂfld))‘𝑆) = 𝑆 |
38 | 30, 37 | reseq12i 5816 | . . 3 ⊢ ((ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) ↾ ((int‘(TopOpen‘ℂfld))‘𝑆)) = ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) ↾ 𝑆) |
39 | 25, 29, 38 | 3eqtr3i 2829 | . 2 ⊢ (ℂ D (log ↾ 𝑆)) = ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) ↾ 𝑆) |
40 | resmpt 5872 | . . 3 ⊢ (𝑆 ⊆ (ℂ ∖ (-∞(,]0)) → ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ (1 / 𝑥))) | |
41 | 26, 40 | ax-mp 5 | . 2 ⊢ ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ (1 / 𝑥)) |
42 | 39, 41 | eqtri 2821 | 1 ⊢ (ℂ D (log ↾ 𝑆)) = (𝑥 ∈ 𝑆 ↦ (1 / 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ⊆ wss 3881 {csn 4525 ↦ cmpt 5110 ran crn 5520 ↾ cres 5521 ∘ ccom 5523 ⟶wf 6320 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 -∞cmnf 10662 ℝ*cxr 10663 − cmin 10859 / cdiv 11286 (,]cioc 12727 abscabs 14585 TopOpenctopn 16687 ∞Metcxmet 20076 ballcbl 20078 ℂfldccnfld 20091 Topctop 21498 intcnt 21622 D cdv 24466 logclog 25146 |
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 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
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 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-tan 15417 df-pi 15418 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 df-log 25148 |
This theorem is referenced by: logtayl 25251 efrlim 25555 |
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