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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 26618 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ) | |
| 2 | 1 | recnd 11290 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℂ) | 
| 3 | ax-icn 11215 | . . . . . 6 ⊢ i ∈ ℂ | |
| 4 | picn 26502 | . . . . . 6 ⊢ π ∈ ℂ | |
| 5 | 3, 4 | mulcli 11269 | . . . . 5 ⊢ (i · π) ∈ ℂ | 
| 6 | efadd 16131 | . . . . 5 ⊢ (((log‘𝐴) ∈ ℂ ∧ (i · π) ∈ ℂ) → (exp‘((log‘𝐴) + (i · π))) = ((exp‘(log‘𝐴)) · (exp‘(i · π)))) | |
| 7 | 2, 5, 6 | sylancl 586 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (exp‘((log‘𝐴) + (i · π))) = ((exp‘(log‘𝐴)) · (exp‘(i · π)))) | 
| 8 | efipi 26516 | . . . . . 6 ⊢ (exp‘(i · π)) = -1 | |
| 9 | 8 | oveq2i 7443 | . . . . 5 ⊢ ((exp‘(log‘𝐴)) · (exp‘(i · π))) = ((exp‘(log‘𝐴)) · -1) | 
| 10 | reeflog 26623 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (exp‘(log‘𝐴)) = 𝐴) | |
| 11 | 10 | oveq1d 7447 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → ((exp‘(log‘𝐴)) · -1) = (𝐴 · -1)) | 
| 12 | 9, 11 | eqtrid 2788 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → ((exp‘(log‘𝐴)) · (exp‘(i · π))) = (𝐴 · -1)) | 
| 13 | rpcn 13046 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
| 14 | neg1cn 12381 | . . . . . 6 ⊢ -1 ∈ ℂ | |
| 15 | mulcom 11242 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ -1 ∈ ℂ) → (𝐴 · -1) = (-1 · 𝐴)) | |
| 16 | 13, 14, 15 | sylancl 586 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (𝐴 · -1) = (-1 · 𝐴)) | 
| 17 | 13 | mulm1d 11716 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (-1 · 𝐴) = -𝐴) | 
| 18 | 16, 17 | eqtrd 2776 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (𝐴 · -1) = -𝐴) | 
| 19 | 7, 12, 18 | 3eqtrd 2780 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (exp‘((log‘𝐴) + (i · π))) = -𝐴) | 
| 20 | 19 | fveq2d 6909 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘(exp‘((log‘𝐴) + (i · π)))) = (log‘-𝐴)) | 
| 21 | addcl 11238 | . . . . 5 ⊢ (((log‘𝐴) ∈ ℂ ∧ (i · π) ∈ ℂ) → ((log‘𝐴) + (i · π)) ∈ ℂ) | |
| 22 | 2, 5, 21 | sylancl 586 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → ((log‘𝐴) + (i · π)) ∈ ℂ) | 
| 23 | pipos 26503 | . . . . . . 7 ⊢ 0 < π | |
| 24 | pire 26501 | . . . . . . . 8 ⊢ π ∈ ℝ | |
| 25 | lt0neg2 11771 | . . . . . . . 8 ⊢ (π ∈ ℝ → (0 < π ↔ -π < 0)) | |
| 26 | 24, 25 | ax-mp 5 | . . . . . . 7 ⊢ (0 < π ↔ -π < 0) | 
| 27 | 23, 26 | mpbi 230 | . . . . . 6 ⊢ -π < 0 | 
| 28 | 24 | renegcli 11571 | . . . . . . 7 ⊢ -π ∈ ℝ | 
| 29 | 0re 11264 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
| 30 | 28, 29, 24 | lttri 11388 | . . . . . 6 ⊢ ((-π < 0 ∧ 0 < π) → -π < π) | 
| 31 | 27, 23, 30 | mp2an 692 | . . . . 5 ⊢ -π < π | 
| 32 | crim 15155 | . . . . . 6 ⊢ (((log‘𝐴) ∈ ℝ ∧ π ∈ ℝ) → (ℑ‘((log‘𝐴) + (i · π))) = π) | |
| 33 | 1, 24, 32 | sylancl 586 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (ℑ‘((log‘𝐴) + (i · π))) = π) | 
| 34 | 31, 33 | breqtrrid 5180 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → -π < (ℑ‘((log‘𝐴) + (i · π)))) | 
| 35 | 24 | leidi 11798 | . . . . 5 ⊢ π ≤ π | 
| 36 | 33, 35 | eqbrtrdi 5181 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (ℑ‘((log‘𝐴) + (i · π))) ≤ π) | 
| 37 | ellogrn 26602 | . . . 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 1343 | . . 3 ⊢ (𝐴 ∈ ℝ+ → ((log‘𝐴) + (i · π)) ∈ ran log) | 
| 39 | logef 26624 | . . 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 2778 | 1 ⊢ (𝐴 ∈ ℝ+ → (log‘-𝐴) = ((log‘𝐴) + (i · π))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 ran crn 5685 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 ℝcr 11155 0cc0 11156 1c1 11157 ici 11158 + caddc 11159 · cmul 11161 < clt 11296 ≤ cle 11297 -cneg 11494 ℝ+crp 13035 ℑcim 15138 expce 16098 πcpi 16103 logclog 26597 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 df-sin 16106 df-cos 16107 df-pi 16109 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-limc 25902 df-dv 25903 df-log 26599 | 
| This theorem is referenced by: logm1 26632 lognegb 26633 cxpsqrt 26746 | 
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