lognegb - Metamath Proof Explorer

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Theorem lognegb 26554

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 26552	. . . . 5 ⊢ (-𝐴 ∈ ℝ⁺ → (log‘--𝐴) = ((log‘-𝐴) + (i · π)))
2	1	fveq2d 6844	. . . 4 ⊢ (-𝐴 ∈ ℝ⁺ → (ℑ‘(log‘--𝐴)) = (ℑ‘((log‘-𝐴) + (i · π))))
3		relogcl 26539	. . . . 5 ⊢ (-𝐴 ∈ ℝ⁺ → (log‘-𝐴) ∈ ℝ)
4		pire 26421	. . . . 5 ⊢ π ∈ ℝ
5		crim 15077	. . . . 5 ⊢ (((log‘-𝐴) ∈ ℝ ∧ π ∈ ℝ) → (ℑ‘((log‘-𝐴) + (i · π))) = π)
6	3, 4, 5	sylancl 587	. . . 4 ⊢ (-𝐴 ∈ ℝ⁺ → (ℑ‘((log‘-𝐴) + (i · π))) = π)
7	2, 6	eqtrd 2771	. . 3 ⊢ (-𝐴 ∈ ℝ⁺ → (ℑ‘(log‘--𝐴)) = π)
8		negneg 11444	. . . . . 6 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴)
9	8	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → --𝐴 = 𝐴)
10	9	fveq2d 6844	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘--𝐴) = (log‘𝐴))
11	10	fveqeq2d 6848	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((ℑ‘(log‘--𝐴)) = π ↔ (ℑ‘(log‘𝐴)) = π))
12	7, 11	imbitrid 244	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-𝐴 ∈ ℝ⁺ → (ℑ‘(log‘𝐴)) = π))
13		logcl 26532	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
14	13	replimd 15159	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) = ((ℜ‘(log‘𝐴)) + (i · (ℑ‘(log‘𝐴)))))
15	14	fveq2d 6844	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = (exp‘((ℜ‘(log‘𝐴)) + (i · (ℑ‘(log‘𝐴))))))
16		eflog 26540	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
17	13	recld 15156	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) ∈ ℝ)
18	17	recnd 11173	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) ∈ ℂ)
19		ax-icn 11097	. . . . . . 7 ⊢ i ∈ ℂ
20	13	imcld 15157	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℝ)
21	20	recnd 11173	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℂ)
22		mulcl 11122	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ (ℑ‘(log‘𝐴)) ∈ ℂ) → (i · (ℑ‘(log‘𝐴))) ∈ ℂ)
23	19, 21, 22	sylancr 588	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (i · (ℑ‘(log‘𝐴))) ∈ ℂ)
24		efadd 16059	. . . . . 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 585	. . . . 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 2779	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · (exp‘(i · (ℑ‘(log‘𝐴))))))
27		oveq2 7375	. . . . . . . 8 ⊢ ((ℑ‘(log‘𝐴)) = π → (i · (ℑ‘(log‘𝐴))) = (i · π))
28	27	fveq2d 6844	. . . . . . 7 ⊢ ((ℑ‘(log‘𝐴)) = π → (exp‘(i · (ℑ‘(log‘𝐴)))) = (exp‘(i · π)))
29		efipi 26437	. . . . . . 7 ⊢ (exp‘(i · π)) = -1
30	28, 29	eqtrdi 2787	. . . . . 6 ⊢ ((ℑ‘(log‘𝐴)) = π → (exp‘(i · (ℑ‘(log‘𝐴)))) = -1)
31	30	oveq2d 7383	. . . . 5 ⊢ ((ℑ‘(log‘𝐴)) = π → ((exp‘(ℜ‘(log‘𝐴))) · (exp‘(i · (ℑ‘(log‘𝐴))))) = ((exp‘(ℜ‘(log‘𝐴))) · -1))
32	31	eqeq2d 2747	. . . 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 245	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((ℑ‘(log‘𝐴)) = π → 𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · -1)))
34	17	rpefcld 16072	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘(log‘𝐴))) ∈ ℝ⁺)
35	34	rpcnd 12988	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(ℜ‘(log‘𝐴))) ∈ ℂ)
36		neg1cn 12144	. . . . . . . . 9 ⊢ -1 ∈ ℂ
37		mulcom 11124	. . . . . . . . 9 ⊢ (((exp‘(ℜ‘(log‘𝐴))) ∈ ℂ ∧ -1 ∈ ℂ) → ((exp‘(ℜ‘(log‘𝐴))) · -1) = (-1 · (exp‘(ℜ‘(log‘𝐴)))))
38	35, 36, 37	sylancl 587	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((exp‘(ℜ‘(log‘𝐴))) · -1) = (-1 · (exp‘(ℜ‘(log‘𝐴)))))
39	35	mulm1d 11602	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-1 · (exp‘(ℜ‘(log‘𝐴)))) = -(exp‘(ℜ‘(log‘𝐴))))
40	38, 39	eqtrd 2771	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((exp‘(ℜ‘(log‘𝐴))) · -1) = -(exp‘(ℜ‘(log‘𝐴))))
41	40	negeqd 11387	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -((exp‘(ℜ‘(log‘𝐴))) · -1) = --(exp‘(ℜ‘(log‘𝐴))))
42	35	negnegd 11496	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → --(exp‘(ℜ‘(log‘𝐴))) = (exp‘(ℜ‘(log‘𝐴))))
43	41, 42	eqtrd 2771	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -((exp‘(ℜ‘(log‘𝐴))) · -1) = (exp‘(ℜ‘(log‘𝐴))))
44	43, 34	eqeltrd 2836	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → -((exp‘(ℜ‘(log‘𝐴))) · -1) ∈ ℝ⁺)
45		negeq 11385	. . . . 5 ⊢ (𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · -1) → -𝐴 = -((exp‘(ℜ‘(log‘𝐴))) · -1))
46	45	eleq1d 2821	. . . 4 ⊢ (𝐴 = ((exp‘(ℜ‘(log‘𝐴))) · -1) → (-𝐴 ∈ ℝ⁺ ↔ -((exp‘(ℜ‘(log‘𝐴))) · -1) ∈ ℝ⁺))
47	44, 46	syl5ibrcom 247	. . 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 212	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-𝐴 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝐴)) = π))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 ici 11040 + caddc 11041 · cmul 11043 -cneg 11378 ℝ⁺crp 12942 ℜcre 15059 ℑcim 15060 expce 16026 πcpi 16031 logclog 26518

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520

This theorem is referenced by: logcj 26570 argimgt0 26576 dvloglem 26612 logf1o2 26614 logrec 26727 ang180lem2 26774 angpieqvdlem2 26793 asinneg 26850 arginv 32820

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