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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 26508 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ) | |
2 | 1 | recnd 11272 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℂ) |
3 | ax-icn 11197 | . . . . . 6 ⊢ i ∈ ℂ | |
4 | picn 26393 | . . . . . 6 ⊢ π ∈ ℂ | |
5 | 3, 4 | mulcli 11251 | . . . . 5 ⊢ (i · π) ∈ ℂ |
6 | efadd 16070 | . . . . 5 ⊢ (((log‘𝐴) ∈ ℂ ∧ (i · π) ∈ ℂ) → (exp‘((log‘𝐴) + (i · π))) = ((exp‘(log‘𝐴)) · (exp‘(i · π)))) | |
7 | 2, 5, 6 | sylancl 585 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (exp‘((log‘𝐴) + (i · π))) = ((exp‘(log‘𝐴)) · (exp‘(i · π)))) |
8 | efipi 26407 | . . . . . 6 ⊢ (exp‘(i · π)) = -1 | |
9 | 8 | oveq2i 7431 | . . . . 5 ⊢ ((exp‘(log‘𝐴)) · (exp‘(i · π))) = ((exp‘(log‘𝐴)) · -1) |
10 | reeflog 26513 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (exp‘(log‘𝐴)) = 𝐴) | |
11 | 10 | oveq1d 7435 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → ((exp‘(log‘𝐴)) · -1) = (𝐴 · -1)) |
12 | 9, 11 | eqtrid 2780 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → ((exp‘(log‘𝐴)) · (exp‘(i · π))) = (𝐴 · -1)) |
13 | rpcn 13016 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
14 | neg1cn 12356 | . . . . . 6 ⊢ -1 ∈ ℂ | |
15 | mulcom 11224 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ -1 ∈ ℂ) → (𝐴 · -1) = (-1 · 𝐴)) | |
16 | 13, 14, 15 | sylancl 585 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (𝐴 · -1) = (-1 · 𝐴)) |
17 | 13 | mulm1d 11696 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (-1 · 𝐴) = -𝐴) |
18 | 16, 17 | eqtrd 2768 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (𝐴 · -1) = -𝐴) |
19 | 7, 12, 18 | 3eqtrd 2772 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (exp‘((log‘𝐴) + (i · π))) = -𝐴) |
20 | 19 | fveq2d 6901 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘(exp‘((log‘𝐴) + (i · π)))) = (log‘-𝐴)) |
21 | addcl 11220 | . . . . 5 ⊢ (((log‘𝐴) ∈ ℂ ∧ (i · π) ∈ ℂ) → ((log‘𝐴) + (i · π)) ∈ ℂ) | |
22 | 2, 5, 21 | sylancl 585 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → ((log‘𝐴) + (i · π)) ∈ ℂ) |
23 | pipos 26394 | . . . . . . 7 ⊢ 0 < π | |
24 | pire 26392 | . . . . . . . 8 ⊢ π ∈ ℝ | |
25 | lt0neg2 11751 | . . . . . . . 8 ⊢ (π ∈ ℝ → (0 < π ↔ -π < 0)) | |
26 | 24, 25 | ax-mp 5 | . . . . . . 7 ⊢ (0 < π ↔ -π < 0) |
27 | 23, 26 | mpbi 229 | . . . . . 6 ⊢ -π < 0 |
28 | 24 | renegcli 11551 | . . . . . . 7 ⊢ -π ∈ ℝ |
29 | 0re 11246 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
30 | 28, 29, 24 | lttri 11370 | . . . . . 6 ⊢ ((-π < 0 ∧ 0 < π) → -π < π) |
31 | 27, 23, 30 | mp2an 691 | . . . . 5 ⊢ -π < π |
32 | crim 15094 | . . . . . 6 ⊢ (((log‘𝐴) ∈ ℝ ∧ π ∈ ℝ) → (ℑ‘((log‘𝐴) + (i · π))) = π) | |
33 | 1, 24, 32 | sylancl 585 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (ℑ‘((log‘𝐴) + (i · π))) = π) |
34 | 31, 33 | breqtrrid 5186 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → -π < (ℑ‘((log‘𝐴) + (i · π)))) |
35 | 24 | leidi 11778 | . . . . 5 ⊢ π ≤ π |
36 | 33, 35 | eqbrtrdi 5187 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (ℑ‘((log‘𝐴) + (i · π))) ≤ π) |
37 | ellogrn 26492 | . . . 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 1341 | . . 3 ⊢ (𝐴 ∈ ℝ+ → ((log‘𝐴) + (i · π)) ∈ ran log) |
39 | logef 26514 | . . 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 2770 | 1 ⊢ (𝐴 ∈ ℝ+ → (log‘-𝐴) = ((log‘𝐴) + (i · π))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 ran crn 5679 ‘cfv 6548 (class class class)co 7420 ℂcc 11136 ℝcr 11137 0cc0 11138 1c1 11139 ici 11140 + caddc 11141 · cmul 11143 < clt 11278 ≤ cle 11279 -cneg 11475 ℝ+crp 13006 ℑcim 15077 expce 16037 πcpi 16042 logclog 26487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 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 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8846 df-pm 8847 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-fac 14265 df-bc 14294 df-hash 14322 df-shft 15046 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-sum 15665 df-ef 16043 df-sin 16045 df-cos 16046 df-pi 16048 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-mulg 19023 df-cntz 19267 df-cmn 19736 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-fbas 21275 df-fg 21276 df-cnfld 21279 df-top 22795 df-topon 22812 df-topsp 22834 df-bases 22848 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-lp 23039 df-perf 23040 df-cn 23130 df-cnp 23131 df-haus 23218 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-xms 24225 df-ms 24226 df-tms 24227 df-cncf 24797 df-limc 25794 df-dv 25795 df-log 26489 |
This theorem is referenced by: logm1 26522 lognegb 26523 cxpsqrt 26636 |
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