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Mirrors > Home > MPE Home > Th. List > lognegb | Structured version Visualization version GIF version |
Description: If a number has imaginary part equal to π, then it is on the negative real axis and vice-versa. (Contributed by Mario Carneiro, 23-Sep-2014.) |
Ref | Expression |
---|---|
lognegb | ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-𝐴 ∈ ℝ+ ↔ (ℑ‘(log‘𝐴)) = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | logneg 25753 | . . . . 5 ⊢ (-𝐴 ∈ ℝ+ → (log‘--𝐴) = ((log‘-𝐴) + (i · π))) | |
2 | 1 | fveq2d 6770 | . . . 4 ⊢ (-𝐴 ∈ ℝ+ → (ℑ‘(log‘--𝐴)) = (ℑ‘((log‘-𝐴) + (i · π)))) |
3 | relogcl 25741 | . . . . 5 ⊢ (-𝐴 ∈ ℝ+ → (log‘-𝐴) ∈ ℝ) | |
4 | pire 25625 | . . . . 5 ⊢ π ∈ ℝ | |
5 | crim 14836 | . . . . 5 ⊢ (((log‘-𝐴) ∈ ℝ ∧ π ∈ ℝ) → (ℑ‘((log‘-𝐴) + (i · π))) = π) | |
6 | 3, 4, 5 | sylancl 586 | . . . 4 ⊢ (-𝐴 ∈ ℝ+ → (ℑ‘((log‘-𝐴) + (i · π))) = π) |
7 | 2, 6 | eqtrd 2778 | . . 3 ⊢ (-𝐴 ∈ ℝ+ → (ℑ‘(log‘--𝐴)) = π) |
8 | negneg 11281 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴) | |
9 | 8 | adantr 481 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → --𝐴 = 𝐴) |
10 | 9 | fveq2d 6770 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘--𝐴) = (log‘𝐴)) |
11 | 10 | fveqeq2d 6774 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((ℑ‘(log‘--𝐴)) = π ↔ (ℑ‘(log‘𝐴)) = π)) |
12 | 7, 11 | syl5ib 243 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-𝐴 ∈ ℝ+ → (ℑ‘(log‘𝐴)) = π)) |
13 | logcl 25734 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ) | |
14 | 13 | replimd 14918 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) = ((ℜ‘(log‘𝐴)) + (i · (ℑ‘(log‘𝐴))))) |
15 | 14 | fveq2d 6770 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = (exp‘((ℜ‘(log‘𝐴)) + (i · (ℑ‘(log‘𝐴)))))) |
16 | eflog 25742 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴) | |
17 | 13 | recld 14915 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) ∈ ℝ) |
18 | 17 | recnd 11013 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) ∈ ℂ) |
19 | ax-icn 10940 | . . . . . . 7 ⊢ i ∈ ℂ | |
20 | 13 | imcld 14916 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℝ) |
21 | 20 | recnd 11013 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℂ) |
22 | mulcl 10965 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ (ℑ‘(log‘𝐴)) ∈ ℂ) → (i · (ℑ‘(log‘𝐴))) ∈ ℂ) | |
23 | 19, 21, 22 | sylancr 587 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (i · (ℑ‘(log‘𝐴))) ∈ ℂ) |
24 | efadd 15813 | . . . . . 6 ⊢ (((ℜ‘(log‘𝐴)) ∈ ℂ ∧ (i · (ℑ‘(log‘𝐴))) ∈ ℂ) → (exp‘((ℜ‘(log‘𝐴)) + (i · (ℑ‘(log‘𝐴))))) = ((exp‘(ℜ‘(log‘𝐴))) · (exp‘(i · (ℑ‘(log‘𝐴)))))) | |
25 | 18, 23, 24 | syl2anc 584 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘((ℜ‘(log‘𝐴)) + (i · (ℑ‘(log‘𝐴))))) = ((exp‘(ℜ‘(log‘𝐴))) · (exp‘(i · (ℑ‘(log‘𝐴)))))) |
26 | 15, 16, 25 | 3eqtr3d 2786 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · (exp‘(i · (ℑ‘(log‘𝐴)))))) |
27 | oveq2 7275 | . . . . . . . 8 ⊢ ((ℑ‘(log‘𝐴)) = π → (i · (ℑ‘(log‘𝐴))) = (i · π)) | |
28 | 27 | fveq2d 6770 | . . . . . . 7 ⊢ ((ℑ‘(log‘𝐴)) = π → (exp‘(i · (ℑ‘(log‘𝐴)))) = (exp‘(i · π))) |
29 | efipi 25640 | . . . . . . 7 ⊢ (exp‘(i · π)) = -1 | |
30 | 28, 29 | eqtrdi 2794 | . . . . . 6 ⊢ ((ℑ‘(log‘𝐴)) = π → (exp‘(i · (ℑ‘(log‘𝐴)))) = -1) |
31 | 30 | oveq2d 7283 | . . . . 5 ⊢ ((ℑ‘(log‘𝐴)) = π → ((exp‘(ℜ‘(log‘𝐴))) · (exp‘(i · (ℑ‘(log‘𝐴))))) = ((exp‘(ℜ‘(log‘𝐴))) · -1)) |
32 | 31 | eqeq2d 2749 | . . . 4 ⊢ ((ℑ‘(log‘𝐴)) = π → (𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · (exp‘(i · (ℑ‘(log‘𝐴))))) ↔ 𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · -1))) |
33 | 26, 32 | syl5ibcom 244 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((ℑ‘(log‘𝐴)) = π → 𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · -1))) |
34 | 17 | rpefcld 15824 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘(log‘𝐴))) ∈ ℝ+) |
35 | 34 | rpcnd 12784 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘(log‘𝐴))) ∈ ℂ) |
36 | neg1cn 12097 | . . . . . . . . 9 ⊢ -1 ∈ ℂ | |
37 | mulcom 10967 | . . . . . . . . 9 ⊢ (((exp‘(ℜ‘(log‘𝐴))) ∈ ℂ ∧ -1 ∈ ℂ) → ((exp‘(ℜ‘(log‘𝐴))) · -1) = (-1 · (exp‘(ℜ‘(log‘𝐴))))) | |
38 | 35, 36, 37 | sylancl 586 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((exp‘(ℜ‘(log‘𝐴))) · -1) = (-1 · (exp‘(ℜ‘(log‘𝐴))))) |
39 | 35 | mulm1d 11437 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-1 · (exp‘(ℜ‘(log‘𝐴)))) = -(exp‘(ℜ‘(log‘𝐴)))) |
40 | 38, 39 | eqtrd 2778 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((exp‘(ℜ‘(log‘𝐴))) · -1) = -(exp‘(ℜ‘(log‘𝐴)))) |
41 | 40 | negeqd 11225 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -((exp‘(ℜ‘(log‘𝐴))) · -1) = --(exp‘(ℜ‘(log‘𝐴)))) |
42 | 35 | negnegd 11333 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → --(exp‘(ℜ‘(log‘𝐴))) = (exp‘(ℜ‘(log‘𝐴)))) |
43 | 41, 42 | eqtrd 2778 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -((exp‘(ℜ‘(log‘𝐴))) · -1) = (exp‘(ℜ‘(log‘𝐴)))) |
44 | 43, 34 | eqeltrd 2839 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -((exp‘(ℜ‘(log‘𝐴))) · -1) ∈ ℝ+) |
45 | negeq 11223 | . . . . 5 ⊢ (𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · -1) → -𝐴 = -((exp‘(ℜ‘(log‘𝐴))) · -1)) | |
46 | 45 | eleq1d 2823 | . . . 4 ⊢ (𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · -1) → (-𝐴 ∈ ℝ+ ↔ -((exp‘(ℜ‘(log‘𝐴))) · -1) ∈ ℝ+)) |
47 | 44, 46 | syl5ibrcom 246 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · -1) → -𝐴 ∈ ℝ+)) |
48 | 33, 47 | syld 47 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((ℑ‘(log‘𝐴)) = π → -𝐴 ∈ ℝ+)) |
49 | 12, 48 | impbid 211 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-𝐴 ∈ ℝ+ ↔ (ℑ‘(log‘𝐴)) = π)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ‘cfv 6426 (class class class)co 7267 ℂcc 10879 ℝcr 10880 0cc0 10881 1c1 10882 ici 10883 + caddc 10884 · cmul 10886 -cneg 11216 ℝ+crp 12740 ℜcre 14818 ℑcim 14819 expce 15781 πcpi 15786 logclog 25720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-inf2 9386 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959 ax-addf 10960 ax-mulf 10961 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-se 5540 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-isom 6435 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-om 7703 df-1st 7820 df-2nd 7821 df-supp 7965 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-2o 8285 df-er 8485 df-map 8604 df-pm 8605 df-ixp 8673 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fsupp 9116 df-fi 9157 df-sup 9188 df-inf 9189 df-oi 9256 df-card 9707 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-q 12699 df-rp 12741 df-xneg 12858 df-xadd 12859 df-xmul 12860 df-ioo 13093 df-ioc 13094 df-ico 13095 df-icc 13096 df-fz 13250 df-fzo 13393 df-fl 13522 df-mod 13600 df-seq 13732 df-exp 13793 df-fac 13998 df-bc 14027 df-hash 14055 df-shft 14788 df-cj 14820 df-re 14821 df-im 14822 df-sqrt 14956 df-abs 14957 df-limsup 15190 df-clim 15207 df-rlim 15208 df-sum 15408 df-ef 15787 df-sin 15789 df-cos 15790 df-pi 15792 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-starv 16987 df-sca 16988 df-vsca 16989 df-ip 16990 df-tset 16991 df-ple 16992 df-ds 16994 df-unif 16995 df-hom 16996 df-cco 16997 df-rest 17143 df-topn 17144 df-0g 17162 df-gsum 17163 df-topgen 17164 df-pt 17165 df-prds 17168 df-xrs 17223 df-qtop 17228 df-imas 17229 df-xps 17231 df-mre 17305 df-mrc 17306 df-acs 17308 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-submnd 18441 df-mulg 18711 df-cntz 18933 df-cmn 19398 df-psmet 20599 df-xmet 20600 df-met 20601 df-bl 20602 df-mopn 20603 df-fbas 20604 df-fg 20605 df-cnfld 20608 df-top 22053 df-topon 22070 df-topsp 22092 df-bases 22106 df-cld 22180 df-ntr 22181 df-cls 22182 df-nei 22259 df-lp 22297 df-perf 22298 df-cn 22388 df-cnp 22389 df-haus 22476 df-tx 22723 df-hmeo 22916 df-fil 23007 df-fm 23099 df-flim 23100 df-flf 23101 df-xms 23483 df-ms 23484 df-tms 23485 df-cncf 24051 df-limc 25040 df-dv 25041 df-log 25722 |
This theorem is referenced by: logcj 25771 argimgt0 25777 dvloglem 25813 logf1o2 25815 logrec 25923 ang180lem2 25970 angpieqvdlem2 25989 asinneg 26046 |
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