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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 24197. (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 3587 | . . . 4 ⊢ ℂ ⊆ ℂ | |
2 | logf1o 24115 | . . . . . . 7 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log | |
3 | f1of 6050 | . . . . . . 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 24113 | . . . . . . 7 ⊢ (𝑥 ∈ ran log → 𝑥 ∈ ℂ) | |
6 | 5 | ssriv 3572 | . . . . . 6 ⊢ ran log ⊆ ℂ |
7 | fss 5969 | . . . . . 6 ⊢ ((log:(ℂ ∖ {0})⟶ran log ∧ ran log ⊆ ℂ) → log:(ℂ ∖ {0})⟶ℂ) | |
8 | 4, 6, 7 | mp2an 704 | . . . . 5 ⊢ log:(ℂ ∖ {0})⟶ℂ |
9 | eqid 2610 | . . . . . 6 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0)) | |
10 | 9 | logdmss 24188 | . . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0}) |
11 | fssres 5983 | . . . . 5 ⊢ ((log:(ℂ ∖ {0})⟶ℂ ∧ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0})) → (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))⟶ℂ) | |
12 | 8, 10, 11 | mp2an 704 | . . . 4 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))⟶ℂ |
13 | difss 3699 | . . . 4 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ | |
14 | dvlog2.s | . . . . 5 ⊢ 𝑆 = (1(ball‘(abs ∘ − ))1) | |
15 | cnxmet 22386 | . . . . . 6 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
16 | ax-1cn 9873 | . . . . . 6 ⊢ 1 ∈ ℂ | |
17 | 1rp 11712 | . . . . . . 7 ⊢ 1 ∈ ℝ+ | |
18 | rpxr 11716 | . . . . . . 7 ⊢ (1 ∈ ℝ+ → 1 ∈ ℝ*) | |
19 | 17, 18 | ax-mp 5 | . . . . . 6 ⊢ 1 ∈ ℝ* |
20 | blssm 22033 | . . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ*) → (1(ball‘(abs ∘ − ))1) ⊆ ℂ) | |
21 | 15, 16, 19, 20 | mp3an 1416 | . . . . 5 ⊢ (1(ball‘(abs ∘ − ))1) ⊆ ℂ |
22 | 14, 21 | eqsstri 3598 | . . . 4 ⊢ 𝑆 ⊆ ℂ |
23 | eqid 2610 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
24 | 23 | cnfldtop 22397 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) ∈ Top |
25 | 23 | cnfldtopon 22396 | . . . . . . . . 9 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
26 | 25 | toponunii 20547 | . . . . . . . 8 ⊢ ℂ = ∪ (TopOpen‘ℂfld) |
27 | 26 | restid 15917 | . . . . . . 7 ⊢ ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)) |
28 | 24, 27 | ax-mp 5 | . . . . . 6 ⊢ ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld) |
29 | 28 | eqcomi 2619 | . . . . 5 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ) |
30 | 23, 29 | dvres 23481 | . . . 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))) |
31 | 1, 12, 13, 22, 30 | mp4an 705 | . . 3 ⊢ (ℂ D ((log ↾ (ℂ ∖ (-∞(,]0))) ↾ 𝑆)) = ((ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) ↾ ((int‘(TopOpen‘ℂfld))‘𝑆)) |
32 | 14 | dvlog2lem 24198 | . . . . 5 ⊢ 𝑆 ⊆ (ℂ ∖ (-∞(,]0)) |
33 | resabs1 5347 | . . . . 5 ⊢ (𝑆 ⊆ (ℂ ∖ (-∞(,]0)) → ((log ↾ (ℂ ∖ (-∞(,]0))) ↾ 𝑆) = (log ↾ 𝑆)) | |
34 | 32, 33 | ax-mp 5 | . . . 4 ⊢ ((log ↾ (ℂ ∖ (-∞(,]0))) ↾ 𝑆) = (log ↾ 𝑆) |
35 | 34 | oveq2i 6560 | . . 3 ⊢ (ℂ D ((log ↾ (ℂ ∖ (-∞(,]0))) ↾ 𝑆)) = (ℂ D (log ↾ 𝑆)) |
36 | 9 | dvlog 24197 | . . . 4 ⊢ (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) |
37 | 23 | cnfldtopn 22395 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − )) |
38 | 37 | blopn 22115 | . . . . . . 7 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ*) → (1(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂfld)) |
39 | 15, 16, 19, 38 | mp3an 1416 | . . . . . 6 ⊢ (1(ball‘(abs ∘ − ))1) ∈ (TopOpen‘ℂfld) |
40 | 14, 39 | eqeltri 2684 | . . . . 5 ⊢ 𝑆 ∈ (TopOpen‘ℂfld) |
41 | isopn3i 20696 | . . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ (TopOpen‘ℂfld)) → ((int‘(TopOpen‘ℂfld))‘𝑆) = 𝑆) | |
42 | 24, 40, 41 | mp2an 704 | . . . 4 ⊢ ((int‘(TopOpen‘ℂfld))‘𝑆) = 𝑆 |
43 | 36, 42 | reseq12i 5315 | . . 3 ⊢ ((ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) ↾ ((int‘(TopOpen‘ℂfld))‘𝑆)) = ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) ↾ 𝑆) |
44 | 31, 35, 43 | 3eqtr3i 2640 | . 2 ⊢ (ℂ D (log ↾ 𝑆)) = ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) ↾ 𝑆) |
45 | resmpt 5369 | . . 3 ⊢ (𝑆 ⊆ (ℂ ∖ (-∞(,]0)) → ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ (1 / 𝑥))) | |
46 | 32, 45 | ax-mp 5 | . 2 ⊢ ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ (1 / 𝑥)) |
47 | 44, 46 | eqtri 2632 | 1 ⊢ (ℂ D (log ↾ 𝑆)) = (𝑥 ∈ 𝑆 ↦ (1 / 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 ↦ cmpt 4643 ran crn 5039 ↾ cres 5040 ∘ ccom 5042 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 -∞cmnf 9951 ℝ*cxr 9952 − cmin 10145 / cdiv 10563 ℝ+crp 11708 (,]cioc 12047 abscabs 13822 ↾t crest 15904 TopOpenctopn 15905 ∞Metcxmt 19552 ballcbl 19554 ℂfldccnfld 19567 Topctop 20517 intcnt 20631 D cdv 23433 logclog 24105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-tan 14641 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 |
This theorem is referenced by: logtayl 24206 efrlim 24496 |
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