Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 25234. (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 3989 | . . . 4 ⊢ ℂ ⊆ ℂ | |
2 | logf1o 25148 | . . . . . . 7 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log | |
3 | f1of 6615 | . . . . . . 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 25146 | . . . . . . 7 ⊢ (𝑥 ∈ ran log → 𝑥 ∈ ℂ) | |
6 | 5 | ssriv 3971 | . . . . . 6 ⊢ ran log ⊆ ℂ |
7 | fss 6527 | . . . . . 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 2821 | . . . . . 6 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0)) | |
10 | 9 | logdmss 25225 | . . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ (ℂ ∖ {0}) |
11 | fssres 6544 | . . . . 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 4108 | . . . 4 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ | |
14 | dvlog2.s | . . . . 5 ⊢ 𝑆 = (1(ball‘(abs ∘ − ))1) | |
15 | cnxmet 23381 | . . . . . 6 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | |
16 | ax-1cn 10595 | . . . . . 6 ⊢ 1 ∈ ℂ | |
17 | 1xr 10700 | . . . . . 6 ⊢ 1 ∈ ℝ* | |
18 | blssm 23028 | . . . . . 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 4001 | . . . 4 ⊢ 𝑆 ⊆ ℂ |
21 | eqid 2821 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
22 | 21 | cnfldtopon 23391 | . . . . . 6 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
23 | 22 | toponrestid 21529 | . . . . 5 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ) |
24 | 21, 23 | dvres 24509 | . . . 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 25235 | . . . . 5 ⊢ 𝑆 ⊆ (ℂ ∖ (-∞(,]0)) |
27 | resabs1 5883 | . . . . 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 7167 | . . 3 ⊢ (ℂ D ((log ↾ (ℂ ∖ (-∞(,]0))) ↾ 𝑆)) = (ℂ D (log ↾ 𝑆)) |
30 | 9 | dvlog 25234 | . . . 4 ⊢ (ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) = (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) |
31 | 21 | cnfldtop 23392 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ Top |
32 | 21 | cnfldtopn 23390 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − )) |
33 | 32 | blopn 23110 | . . . . . . 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 2909 | . . . . 5 ⊢ 𝑆 ∈ (TopOpen‘ℂfld) |
36 | isopn3i 21690 | . . . . 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 5851 | . . 3 ⊢ ((ℂ D (log ↾ (ℂ ∖ (-∞(,]0)))) ↾ ((int‘(TopOpen‘ℂfld))‘𝑆)) = ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) ↾ 𝑆) |
39 | 25, 29, 38 | 3eqtr3i 2852 | . 2 ⊢ (ℂ D (log ↾ 𝑆)) = ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (1 / 𝑥)) ↾ 𝑆) |
40 | resmpt 5905 | . . 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 2844 | 1 ⊢ (ℂ D (log ↾ 𝑆)) = (𝑥 ∈ 𝑆 ↦ (1 / 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 ⊆ wss 3936 {csn 4567 ↦ cmpt 5146 ran crn 5556 ↾ cres 5557 ∘ ccom 5559 ⟶wf 6351 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 0cc0 10537 1c1 10538 -∞cmnf 10673 ℝ*cxr 10674 − cmin 10870 / cdiv 11297 (,]cioc 12740 abscabs 14593 TopOpenctopn 16695 ∞Metcxmet 20530 ballcbl 20532 ℂfldccnfld 20545 Topctop 21501 intcnt 21625 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-tan 15425 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: logtayl 25243 efrlim 25547 |
Copyright terms: Public domain | W3C validator |