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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 26630. (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 3999 | . . . 4 ⊢ ℂ ⊆ ℂ | |
2 | logf1o 26543 | . . . . . . 7 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log | |
3 | f1of 6838 | . . . . . . 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 26541 | . . . . . . 7 ⊢ (𝑥 ∈ ran log → 𝑥 ∈ ℂ) | |
6 | 5 | ssriv 3980 | . . . . . 6 ⊢ ran log ⊆ ℂ |
7 | fss 6739 | . . . . . 6 ⊢ ((log:(ℂ ∖ {0})⟶ran log ∧ ran log ⊆ ℂ) → log:(ℂ ∖ {0})⟶ℂ) | |
8 | 4, 6, 7 | mp2an 690 | . . . . 5 ⊢ log:(ℂ ∖ {0})⟶ℂ |
9 | eqid 2725 | . . . . . 6 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0)) | |
10 | 9 | logdmss 26621 | . . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0}) |
11 | fssres 6763 | . . . . 5 ⊢ ((log:(ℂ ∖ {0})⟶ℂ ∧ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})) → (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))⟶ℂ) | |
12 | 8, 10, 11 | mp2an 690 | . . . 4 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))⟶ℂ |
13 | difss 4128 | . . . 4 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ | |
14 | dvlog2.s | . . . . 5 ⊢ 𝑆 = (1(ball‘(abs ∘ − ))1) | |
15 | cnxmet 24733 | . . . . . 6 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
16 | ax-1cn 11198 | . . . . . 6 ⊢ 1 ∈ ℂ | |
17 | 1xr 11305 | . . . . . 6 ⊢ 1 ∈ ℝ* | |
18 | blssm 24368 | . . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ*) → (1(ball‘(abs ∘ − ))1) ⊆ ℂ) | |
19 | 15, 16, 17, 18 | mp3an 1457 | . . . . 5 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ ℂ |
20 | 14, 19 | eqsstri 4011 | . . . 4 ⊢ 𝑆 ⊆ ℂ |
21 | eqid 2725 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
22 | 21 | cnfldtopon 24743 | . . . . . 6 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
23 | 22 | toponrestid 22867 | . . . . 5 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ) |
24 | 21, 23 | dvres 25884 | . . . 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 691 | . . 3 ⊢ (ℂ D ((log ↾ (ℂ ∖ (-∞(,]0))) ↾ 𝑆)) = ((ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) ↾ ((int‘(TopOpen‘ℂfld))‘𝑆)) |
26 | 14 | dvlog2lem 26631 | . . . . 5 ⊢ 𝑆 ⊆ (ℂ ∖ (-∞(,]0)) |
27 | resabs1 6012 | . . . . 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 7430 | . . 3 ⊢ (ℂ D ((log ↾ (ℂ ∖ (-∞(,]0))) ↾ 𝑆)) = (ℂ D (log ↾ 𝑆)) |
30 | 9 | dvlog 26630 | . . . 4 ⊢ (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) |
31 | 21 | cnfldtop 24744 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ Top |
32 | 21 | cnfldtopn 24742 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − )) |
33 | 32 | blopn 24453 | . . . . . . 7 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ*) → (1(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂfld)) |
34 | 15, 16, 17, 33 | mp3an 1457 | . . . . . 6 ⊢ (1(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂfld) |
35 | 14, 34 | eqeltri 2821 | . . . . 5 ⊢ 𝑆 ∈ (TopOpen‘ℂfld) |
36 | isopn3i 23030 | . . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ (TopOpen‘ℂfld)) → ((int‘(TopOpen‘ℂfld))‘𝑆) = 𝑆) | |
37 | 31, 35, 36 | mp2an 690 | . . . 4 ⊢ ((int‘(TopOpen‘ℂfld))‘𝑆) = 𝑆 |
38 | 30, 37 | reseq12i 5983 | . . 3 ⊢ ((ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) ↾ ((int‘(TopOpen‘ℂfld))‘𝑆)) = ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) ↾ 𝑆) |
39 | 25, 29, 38 | 3eqtr3i 2761 | . 2 ⊢ (ℂ D (log ↾ 𝑆)) = ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) ↾ 𝑆) |
40 | resmpt 6042 | . . 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 2753 | 1 ⊢ (ℂ D (log ↾ 𝑆)) = (𝑥 ∈ 𝑆 ↦ (1 / 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ∖ cdif 3941 ⊆ wss 3944 {csn 4630 ↦ cmpt 5232 ran crn 5679 ↾ cres 5680 ∘ ccom 5682 ⟶wf 6545 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 0cc0 11140 1c1 11141 -∞cmnf 11278 ℝ*cxr 11279 − cmin 11476 / cdiv 11903 (,]cioc 13360 abscabs 15217 TopOpenctopn 17406 ∞Metcxmet 21281 ballcbl 21283 ℂfldccnfld 21296 Topctop 22839 intcnt 22965 D cdv 25836 logclog 26533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-ioc 13364 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-mod 13871 df-seq 14003 df-exp 14063 df-fac 14269 df-bc 14298 df-hash 14326 df-shft 15050 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-limsup 15451 df-clim 15468 df-rlim 15469 df-sum 15669 df-ef 16047 df-sin 16049 df-cos 16050 df-tan 16051 df-pi 16052 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-rest 17407 df-topn 17408 df-0g 17426 df-gsum 17427 df-topgen 17428 df-pt 17429 df-prds 17432 df-xrs 17487 df-qtop 17492 df-imas 17493 df-xps 17495 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-mulg 19032 df-cntz 19280 df-cmn 19749 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-cmp 23335 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24270 df-ms 24271 df-tms 24272 df-cncf 24842 df-limc 25839 df-dv 25840 df-log 26535 |
This theorem is referenced by: logtayl 26639 efrlim 26946 efrlimOLD 26947 |
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