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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 25712 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ) | |
2 | 1 | recnd 10987 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℂ) |
3 | ax-icn 10914 | . . . . . 6 ⊢ i ∈ ℂ | |
4 | picn 25597 | . . . . . 6 ⊢ π ∈ ℂ | |
5 | 3, 4 | mulcli 10966 | . . . . 5 ⊢ (i · π) ∈ ℂ |
6 | efadd 15784 | . . . . 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 25611 | . . . . . 6 ⊢ (exp‘(i · π)) = -1 | |
9 | 8 | oveq2i 7279 | . . . . 5 ⊢ ((exp‘(log‘𝐴)) · (exp‘(i · π))) = ((exp‘(log‘𝐴)) · -1) |
10 | reeflog 25717 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → (exp‘(log‘𝐴)) = 𝐴) | |
11 | 10 | oveq1d 7283 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → ((exp‘(log‘𝐴)) · -1) = (𝐴 · -1)) |
12 | 9, 11 | eqtrid 2791 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → ((exp‘(log‘𝐴)) · (exp‘(i · π))) = (𝐴 · -1)) |
13 | rpcn 12722 | . . . . . 6 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
14 | neg1cn 12070 | . . . . . 6 ⊢ -1 ∈ ℂ | |
15 | mulcom 10941 | . . . . . 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 11410 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → (-1 · 𝐴) = -𝐴) |
18 | 16, 17 | eqtrd 2779 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (𝐴 · -1) = -𝐴) |
19 | 7, 12, 18 | 3eqtrd 2783 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (exp‘((log‘𝐴) + (i · π))) = -𝐴) |
20 | 19 | fveq2d 6772 | . 2 ⊢ (𝐴 ∈ ℝ+ → (log‘(exp‘((log‘𝐴) + (i · π)))) = (log‘-𝐴)) |
21 | addcl 10937 | . . . . 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 25598 | . . . . . . 7 ⊢ 0 < π | |
24 | pire 25596 | . . . . . . . 8 ⊢ π ∈ ℝ | |
25 | lt0neg2 11465 | . . . . . . . 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 11265 | . . . . . . 7 ⊢ -π ∈ ℝ |
29 | 0re 10961 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
30 | 28, 29, 24 | lttri 11084 | . . . . . 6 ⊢ ((-π < 0 ∧ 0 < π) → -π < π) |
31 | 27, 23, 30 | mp2an 688 | . . . . 5 ⊢ -π < π |
32 | crim 14807 | . . . . . 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 5116 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → -π < (ℑ‘((log‘𝐴) + (i · π)))) |
35 | 24 | leidi 11492 | . . . . 5 ⊢ π ≤ π |
36 | 33, 35 | eqbrtrdi 5117 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (ℑ‘((log‘𝐴) + (i · π))) ≤ π) |
37 | ellogrn 25696 | . . . 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 25718 | . . 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 2781 | 1 ⊢ (𝐴 ∈ ℝ+ → (log‘-𝐴) = ((log‘𝐴) + (i · π))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2109 class class class wbr 5078 ran crn 5589 ‘cfv 6430 (class class class)co 7268 ℂcc 10853 ℝcr 10854 0cc0 10855 1c1 10856 ici 10857 + caddc 10858 · cmul 10860 < clt 10993 ≤ cle 10994 -cneg 11189 ℝ+crp 12712 ℑcim 14790 expce 15752 πcpi 15757 logclog 25691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-inf2 9360 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 ax-addf 10934 ax-mulf 10935 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-supp 7962 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-2o 8282 df-er 8472 df-map 8591 df-pm 8592 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fsupp 9090 df-fi 9131 df-sup 9162 df-inf 9163 df-oi 9230 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-q 12671 df-rp 12713 df-xneg 12830 df-xadd 12831 df-xmul 12832 df-ioo 13065 df-ioc 13066 df-ico 13067 df-icc 13068 df-fz 13222 df-fzo 13365 df-fl 13493 df-mod 13571 df-seq 13703 df-exp 13764 df-fac 13969 df-bc 13998 df-hash 14026 df-shft 14759 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-limsup 15161 df-clim 15178 df-rlim 15179 df-sum 15379 df-ef 15758 df-sin 15760 df-cos 15761 df-pi 15763 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-starv 16958 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-unif 16966 df-hom 16967 df-cco 16968 df-rest 17114 df-topn 17115 df-0g 17133 df-gsum 17134 df-topgen 17135 df-pt 17136 df-prds 17139 df-xrs 17194 df-qtop 17199 df-imas 17200 df-xps 17202 df-mre 17276 df-mrc 17277 df-acs 17279 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-submnd 18412 df-mulg 18682 df-cntz 18904 df-cmn 19369 df-psmet 20570 df-xmet 20571 df-met 20572 df-bl 20573 df-mopn 20574 df-fbas 20575 df-fg 20576 df-cnfld 20579 df-top 22024 df-topon 22041 df-topsp 22063 df-bases 22077 df-cld 22151 df-ntr 22152 df-cls 22153 df-nei 22230 df-lp 22268 df-perf 22269 df-cn 22359 df-cnp 22360 df-haus 22447 df-tx 22694 df-hmeo 22887 df-fil 22978 df-fm 23070 df-flim 23071 df-flf 23072 df-xms 23454 df-ms 23455 df-tms 23456 df-cncf 24022 df-limc 25011 df-dv 25012 df-log 25693 |
This theorem is referenced by: logm1 25725 lognegb 25726 cxpsqrt 25839 |
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