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Theorem lognegb 24381

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.)

**Assertion**
Ref	Expression
lognegb	⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-𝐴 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝐴)) = π))

**Proof of Theorem lognegb**
Step	Hyp	Ref	Expression
1		logneg 24379	. . . . 5 ⊢ (-𝐴 ∈ ℝ⁺ → (log‘--𝐴) = ((log‘-𝐴) + (i · π)))
2	1	fveq2d 6233	. . . 4 ⊢ (-𝐴 ∈ ℝ⁺ → (ℑ‘(log‘--𝐴)) = (ℑ‘((log‘-𝐴) + (i · π))))
3		relogcl 24367	. . . . 5 ⊢ (-𝐴 ∈ ℝ⁺ → (log‘-𝐴) ∈ ℝ)
4		pire 24255	. . . . 5 ⊢ π ∈ ℝ
5		crim 13899	. . . . 5 ⊢ (((log‘-𝐴) ∈ ℝ ∧ π ∈ ℝ) → (ℑ‘((log‘-𝐴) + (i · π))) = π)
6	3, 4, 5	sylancl 695	. . . 4 ⊢ (-𝐴 ∈ ℝ⁺ → (ℑ‘((log‘-𝐴) + (i · π))) = π)
7	2, 6	eqtrd 2685	. . 3 ⊢ (-𝐴 ∈ ℝ⁺ → (ℑ‘(log‘--𝐴)) = π)
8		negneg 10369	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴)
9	8	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → --𝐴 = 𝐴)
10	9	fveq2d 6233	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘--𝐴) = (log‘𝐴))
11	10	fveq2d 6233	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘--𝐴)) = (ℑ‘(log‘𝐴)))
12	11	eqeq1d 2653	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((ℑ‘(log‘--𝐴)) = π ↔ (ℑ‘(log‘𝐴)) = π))
13	7, 12	syl5ib 234	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-𝐴 ∈ ℝ⁺ → (ℑ‘(log‘𝐴)) = π))
14		logcl 24360	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
15	14	replimd 13981	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) = ((ℜ‘(log‘𝐴)) + (i · (ℑ‘(log‘𝐴)))))
16	15	fveq2d 6233	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = (exp‘((ℜ‘(log‘𝐴)) + (i · (ℑ‘(log‘𝐴))))))
17		eflog 24368	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
18	14	recld 13978	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) ∈ ℝ)
19	18	recnd 10106	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) ∈ ℂ)
20		ax-icn 10033	. . . . . . 7 ⊢ i ∈ ℂ
21	14	imcld 13979	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℝ)
22	21	recnd 10106	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℂ)
23		mulcl 10058	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ (ℑ‘(log‘𝐴)) ∈ ℂ) → (i · (ℑ‘(log‘𝐴))) ∈ ℂ)
24	20, 22, 23	sylancr 696	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (i · (ℑ‘(log‘𝐴))) ∈ ℂ)
25		efadd 14868	. . . . . 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))))))
26	19, 24, 25	syl2anc 694	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘((ℜ‘(log‘𝐴)) + (i · (ℑ‘(log‘𝐴))))) = ((exp‘(ℜ‘(log‘𝐴))) · (exp‘(i · (ℑ‘(log‘𝐴))))))
27	16, 17, 26	3eqtr3d 2693	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · (exp‘(i · (ℑ‘(log‘𝐴))))))
28		oveq2 6698	. . . . . . . 8 ⊢ ((ℑ‘(log‘𝐴)) = π → (i · (ℑ‘(log‘𝐴))) = (i · π))
29	28	fveq2d 6233	. . . . . . 7 ⊢ ((ℑ‘(log‘𝐴)) = π → (exp‘(i · (ℑ‘(log‘𝐴)))) = (exp‘(i · π)))
30		efipi 24270	. . . . . . 7 ⊢ (exp‘(i · π)) = -1
31	29, 30	syl6eq 2701	. . . . . 6 ⊢ ((ℑ‘(log‘𝐴)) = π → (exp‘(i · (ℑ‘(log‘𝐴)))) = -1)
32	31	oveq2d 6706	. . . . 5 ⊢ ((ℑ‘(log‘𝐴)) = π → ((exp‘(ℜ‘(log‘𝐴))) · (exp‘(i · (ℑ‘(log‘𝐴))))) = ((exp‘(ℜ‘(log‘𝐴))) · -1))
33	32	eqeq2d 2661	. . . 4 ⊢ ((ℑ‘(log‘𝐴)) = π → (𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · (exp‘(i · (ℑ‘(log‘𝐴))))) ↔ 𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · -1)))
34	27, 33	syl5ibcom 235	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((ℑ‘(log‘𝐴)) = π → 𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · -1)))
35	18	rpefcld 14879	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘(log‘𝐴))) ∈ ℝ⁺)
36	35	rpcnd 11912	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘(log‘𝐴))) ∈ ℂ)
37		neg1cn 11162	. . . . . . . . 9 ⊢ -1 ∈ ℂ
38		mulcom 10060	. . . . . . . . 9 ⊢ (((exp‘(ℜ‘(log‘𝐴))) ∈ ℂ ∧ -1 ∈ ℂ) → ((exp‘(ℜ‘(log‘𝐴))) · -1) = (-1 · (exp‘(ℜ‘(log‘𝐴)))))
39	36, 37, 38	sylancl 695	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((exp‘(ℜ‘(log‘𝐴))) · -1) = (-1 · (exp‘(ℜ‘(log‘𝐴)))))
40	36	mulm1d 10520	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-1 · (exp‘(ℜ‘(log‘𝐴)))) = -(exp‘(ℜ‘(log‘𝐴))))
41	39, 40	eqtrd 2685	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((exp‘(ℜ‘(log‘𝐴))) · -1) = -(exp‘(ℜ‘(log‘𝐴))))
42	41	negeqd 10313	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -((exp‘(ℜ‘(log‘𝐴))) · -1) = --(exp‘(ℜ‘(log‘𝐴))))
43	36	negnegd 10421	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → --(exp‘(ℜ‘(log‘𝐴))) = (exp‘(ℜ‘(log‘𝐴))))
44	42, 43	eqtrd 2685	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -((exp‘(ℜ‘(log‘𝐴))) · -1) = (exp‘(ℜ‘(log‘𝐴))))
45	44, 35	eqeltrd 2730	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -((exp‘(ℜ‘(log‘𝐴))) · -1) ∈ ℝ⁺)
46		negeq 10311	. . . . 5 ⊢ (𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · -1) → -𝐴 = -((exp‘(ℜ‘(log‘𝐴))) · -1))
47	46	eleq1d 2715	. . . 4 ⊢ (𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · -1) → (-𝐴 ∈ ℝ⁺ ↔ -((exp‘(ℜ‘(log‘𝐴))) · -1) ∈ ℝ⁺))
48	45, 47	syl5ibrcom 237	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · -1) → -𝐴 ∈ ℝ⁺))
49	34, 48	syld 47	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((ℑ‘(log‘𝐴)) = π → -𝐴 ∈ ℝ⁺))
50	13, 49	impbid 202	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-𝐴 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝐴)) = π))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 ℝcr 9973 0cc0 9974 1c1 9975 ici 9976 + caddc 9977 · cmul 9979 -cneg 10305 ℝ⁺crp 11870 ℜcre 13881 ℑcim 13882 expce 14836 πcpi 14841 logclog 24346

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ioc 12218 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-seq 12842 df-exp 12901 df-fac 13101 df-bc 13130 df-hash 13158 df-shft 13851 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-ef 14842 df-sin 14844 df-cos 14845 df-pi 14847 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-mulg 17588 df-cntz 17796 df-cmn 18241 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-fbas 19791 df-fg 19792 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cld 20871 df-ntr 20872 df-cls 20873 df-nei 20950 df-lp 20988 df-perf 20989 df-cn 21079 df-cnp 21080 df-haus 21167 df-tx 21413 df-hmeo 21606 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-xms 22172 df-ms 22173 df-tms 22174 df-cncf 22728 df-limc 23675 df-dv 23676 df-log 24348

This theorem is referenced by: logcj 24397 argimgt0 24403 dvloglem 24439 logf1o2 24441 logrec 24546 ang180lem2 24585 angpieqvdlem2 24601 asinneg 24658

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