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Mirrors > Home > MPE Home > Th. List > logneg | Structured version Visualization version GIF version |
Description: The natural logarithm of a negative real number. (Contributed by Mario Carneiro, 13-May-2014.) (Revised by Mario Carneiro, 3-Apr-2015.) |
Ref | Expression |
---|---|
logneg | ⊢ (𝐴 ∈ ℝ+ → (log‘-𝐴) = ((log‘𝐴) + (i · π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relogcl 24763 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ) | |
2 | 1 | recnd 10407 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℂ) |
3 | ax-icn 10333 | . . . . . 6 ⊢ i ∈ ℂ | |
4 | picn 24653 | . . . . . 6 ⊢ π ∈ ℂ | |
5 | 3, 4 | mulcli 10386 | . . . . 5 ⊢ (i · π) ∈ ℂ |
6 | efadd 15230 | . . . . 5 ⊢ (((log‘𝐴) ∈ ℂ ∧ (i · π) ∈ ℂ) → (exp‘((log‘𝐴) + (i · π))) = ((exp‘(log‘𝐴)) · (exp‘(i · π)))) | |
7 | 2, 5, 6 | sylancl 580 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (exp‘((log‘𝐴) + (i · π))) = ((exp‘(log‘𝐴)) · (exp‘(i · π)))) |
8 | efipi 24667 | . . . . . 6 ⊢ (exp‘(i · π)) = -1 | |
9 | 8 | oveq2i 6935 | . . . . 5 ⊢ ((exp‘(log‘𝐴)) · (exp‘(i · π))) = ((exp‘(log‘𝐴)) · -1) |
10 | reeflog 24768 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (exp‘(log‘𝐴)) = 𝐴) | |
11 | 10 | oveq1d 6939 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → ((exp‘(log‘𝐴)) · -1) = (𝐴 · -1)) |
12 | 9, 11 | syl5eq 2826 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → ((exp‘(log‘𝐴)) · (exp‘(i · π))) = (𝐴 · -1)) |
13 | rpcn 12153 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
14 | neg1cn 11500 | . . . . . 6 ⊢ -1 ∈ ℂ | |
15 | mulcom 10360 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ -1 ∈ ℂ) → (𝐴 · -1) = (-1 · 𝐴)) | |
16 | 13, 14, 15 | sylancl 580 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (𝐴 · -1) = (-1 · 𝐴)) |
17 | 13 | mulm1d 10829 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (-1 · 𝐴) = -𝐴) |
18 | 16, 17 | eqtrd 2814 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (𝐴 · -1) = -𝐴) |
19 | 7, 12, 18 | 3eqtrd 2818 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (exp‘((log‘𝐴) + (i · π))) = -𝐴) |
20 | 19 | fveq2d 6452 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘(exp‘((log‘𝐴) + (i · π)))) = (log‘-𝐴)) |
21 | addcl 10356 | . . . . 5 ⊢ (((log‘𝐴) ∈ ℂ ∧ (i · π) ∈ ℂ) → ((log‘𝐴) + (i · π)) ∈ ℂ) | |
22 | 2, 5, 21 | sylancl 580 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → ((log‘𝐴) + (i · π)) ∈ ℂ) |
23 | pipos 24654 | . . . . . . 7 ⊢ 0 < π | |
24 | pire 24652 | . . . . . . . 8 ⊢ π ∈ ℝ | |
25 | lt0neg2 10884 | . . . . . . . 8 ⊢ (π ∈ ℝ → (0 < π ↔ -π < 0)) | |
26 | 24, 25 | ax-mp 5 | . . . . . . 7 ⊢ (0 < π ↔ -π < 0) |
27 | 23, 26 | mpbi 222 | . . . . . 6 ⊢ -π < 0 |
28 | 24 | renegcli 10686 | . . . . . . 7 ⊢ -π ∈ ℝ |
29 | 0re 10380 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
30 | 28, 29, 24 | lttri 10504 | . . . . . 6 ⊢ ((-π < 0 ∧ 0 < π) → -π < π) |
31 | 27, 23, 30 | mp2an 682 | . . . . 5 ⊢ -π < π |
32 | crim 14266 | . . . . . 6 ⊢ (((log‘𝐴) ∈ ℝ ∧ π ∈ ℝ) → (ℑ‘((log‘𝐴) + (i · π))) = π) | |
33 | 1, 24, 32 | sylancl 580 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (ℑ‘((log‘𝐴) + (i · π))) = π) |
34 | 31, 33 | syl5breqr 4926 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → -π < (ℑ‘((log‘𝐴) + (i · π)))) |
35 | 24 | leidi 10911 | . . . . 5 ⊢ π ≤ π |
36 | 33, 35 | syl6eqbr 4927 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (ℑ‘((log‘𝐴) + (i · π))) ≤ π) |
37 | ellogrn 24747 | . . . 4 ⊢ (((log‘𝐴) + (i · π)) ∈ ran log ↔ (((log‘𝐴) + (i · π)) ∈ ℂ ∧ -π < (ℑ‘((log‘𝐴) + (i · π))) ∧ (ℑ‘((log‘𝐴) + (i · π))) ≤ π)) | |
38 | 22, 34, 36, 37 | syl3anbrc 1400 | . . 3 ⊢ (𝐴 ∈ ℝ+ → ((log‘𝐴) + (i · π)) ∈ ran log) |
39 | logef 24769 | . . 3 ⊢ (((log‘𝐴) + (i · π)) ∈ ran log → (log‘(exp‘((log‘𝐴) + (i · π)))) = ((log‘𝐴) + (i · π))) | |
40 | 38, 39 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘(exp‘((log‘𝐴) + (i · π)))) = ((log‘𝐴) + (i · π))) |
41 | 20, 40 | eqtr3d 2816 | 1 ⊢ (𝐴 ∈ ℝ+ → (log‘-𝐴) = ((log‘𝐴) + (i · π))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 class class class wbr 4888 ran crn 5358 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 ℝcr 10273 0cc0 10274 1c1 10275 ici 10276 + caddc 10277 · cmul 10279 < clt 10413 ≤ cle 10414 -cneg 10609 ℝ+crp 12141 ℑcim 14249 expce 15198 πcpi 15203 logclog 24742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-ioo 12495 df-ioc 12496 df-ico 12497 df-icc 12498 df-fz 12648 df-fzo 12789 df-fl 12916 df-mod 12992 df-seq 13124 df-exp 13183 df-fac 13383 df-bc 13412 df-hash 13440 df-shft 14218 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-limsup 14614 df-clim 14631 df-rlim 14632 df-sum 14829 df-ef 15204 df-sin 15206 df-cos 15207 df-pi 15209 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-starv 16357 df-sca 16358 df-vsca 16359 df-ip 16360 df-tset 16361 df-ple 16362 df-ds 16364 df-unif 16365 df-hom 16366 df-cco 16367 df-rest 16473 df-topn 16474 df-0g 16492 df-gsum 16493 df-topgen 16494 df-pt 16495 df-prds 16498 df-xrs 16552 df-qtop 16557 df-imas 16558 df-xps 16560 df-mre 16636 df-mrc 16637 df-acs 16639 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-submnd 17726 df-mulg 17932 df-cntz 18137 df-cmn 18585 df-psmet 20138 df-xmet 20139 df-met 20140 df-bl 20141 df-mopn 20142 df-fbas 20143 df-fg 20144 df-cnfld 20147 df-top 21110 df-topon 21127 df-topsp 21149 df-bases 21162 df-cld 21235 df-ntr 21236 df-cls 21237 df-nei 21314 df-lp 21352 df-perf 21353 df-cn 21443 df-cnp 21444 df-haus 21531 df-tx 21778 df-hmeo 21971 df-fil 22062 df-fm 22154 df-flim 22155 df-flf 22156 df-xms 22537 df-ms 22538 df-tms 22539 df-cncf 23093 df-limc 24071 df-dv 24072 df-log 24744 |
This theorem is referenced by: logm1 24776 lognegb 24777 cxpsqrt 24890 |
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