logneg2 - Metamath Proof Explorer

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Theorem logneg2 26973

Description: The logarithm of the negative of a number with positive imaginary part is i · π less than the original. (Compare logneg 26946.) (Contributed by Mario Carneiro, 3-Apr-2015.)

**Assertion**
Ref	Expression
logneg2	⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (log‘ -𝐴) = ((log‘𝐴) − (i · π)))

**Proof of Theorem logneg2**
Step	Hyp	Ref	Expression
1		imcl 15307	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ)
2		gt0ne0 11808	. . . . . . . 8 ⊢ (((ℑ‘𝐴) ∈ ℝ ∧ 0 < (ℑ‘𝐴)) → (ℑ‘𝐴) ≠ 0)
3	1, 2	sylan 592	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (ℑ‘𝐴) ≠ 0)
4		fveq2 6908	. . . . . . . . 9 ⊢ (𝐴 = 0 → (ℑ‘𝐴) = (ℑ‘0))
5		im0 15349	. . . . . . . . 9 ⊢ (ℑ‘0) = 0
6	4, 5	eqtrdi 2814	. . . . . . . 8 ⊢ (𝐴 = 0 → (ℑ‘𝐴) = 0)
7	6	necon3i 2991	. . . . . . 7 ⊢ ((ℑ‘𝐴) ≠ 0 → 𝐴 ≠ 0)
8	3, 7	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → 𝐴 ≠ 0)
9		logcl 26926	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
10	8, 9	syldan 603	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (log‘𝐴) ∈ ℂ)
11		ax-icn 11286	. . . . . 6 ⊢ i ∈ ℂ
12		picn 26815	. . . . . 6 ⊢ π ∈ ℂ
13	11, 12	mulcli 11343	. . . . 5 ⊢ (i · π) ∈ ℂ
14		efsub 16297	. . . . 5 ⊢ (((log‘𝐴) ∈ ℂ ∧ (i · π) ∈ ℂ) → (exp‘((log‘𝐴) − (i · π))) = ((exp‘(log‘𝐴)) / (exp‘(i · π))))
15	10, 13, 14	sylancl 598	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (exp‘((log‘𝐴) − (i · π))) = ((exp‘(log‘𝐴)) / (exp‘(i · π))))
16		eflog 26934	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
17	8, 16	syldan 603	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (exp‘(log‘𝐴)) = 𝐴)
18		efipi 26832	. . . . . 6 ⊢ (exp‘(i · π)) = -1
19	18	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (exp‘(i · π)) = -1)
20	17, 19	oveq12d 7461	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → ((exp‘(log‘𝐴)) / (exp‘(i · π))) = (𝐴 / -1))
21		ax-1cn 11285	. . . . . . 7 ⊢ 1 ∈ ℂ
22		ax-1ne0 11296	. . . . . . 7 ⊢ 1 ≠ 0
23		divneg2 12068	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → -(𝐴 / 1) = (𝐴 / -1))
24	21, 22, 23	mp3an23 1483	. . . . . 6 ⊢ (𝐴 ∈ ℂ → -(𝐴 / 1) = (𝐴 / -1))
25		div1 12033	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴)
26	25	negeqd 11578	. . . . . 6 ⊢ (𝐴 ∈ ℂ → -(𝐴 / 1) = -𝐴)
27	24, 26	eqtr3d 2800	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 / -1) = -𝐴)
28	27	adantr 486	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (𝐴 / -1) = -𝐴)
29	15, 20, 28	3eqtrd 2802	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (exp‘((log‘𝐴) − (i · π))) = -𝐴)
30	29	fveq2d 6912	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (log‘(exp‘((log‘𝐴) − (i · π)))) = (log‘ -𝐴))
31		subcl 11583	. . . . 5 ⊢ (((log‘𝐴) ∈ ℂ ∧ (i · π) ∈ ℂ) → ((log‘𝐴) − (i · π)) ∈ ℂ)
32	10, 13, 31	sylancl 598	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → ((log‘𝐴) − (i · π)) ∈ ℂ)
33		argimgt0 26970	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (0(,)π))
34		eliooord 13563	. . . . . . . . 9 ⊢ ((ℑ‘(log‘𝐴)) ∈ (0(,)π) → (0 < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) < π))
35	33, 34	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (0 < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) < π))
36	35	simpld 500	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → 0 < (ℑ‘(log‘𝐴)))
37		imcl 15307	. . . . . . . . 9 ⊢ ((log‘𝐴) ∈ ℂ → (ℑ‘(log‘𝐴)) ∈ ℝ)
38	10, 37	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ ℝ)
39		pire 26813	. . . . . . . . 9 ⊢ π ∈ ℝ
40	39	renegcli 11646	. . . . . . . 8 ⊢ -π ∈ ℝ
41		ltaddpos2 11834	. . . . . . . 8 ⊢ (((ℑ‘(log‘𝐴)) ∈ ℝ ∧ -π ∈ ℝ) → (0 < (ℑ‘(log‘𝐴)) ↔ -π < ((ℑ‘(log‘𝐴)) + -π)))
42	38, 40, 41	sylancl 598	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (0 < (ℑ‘(log‘𝐴)) ↔ -π < ((ℑ‘(log‘𝐴)) + -π)))
43	36, 42	mpbid 235	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → -π < ((ℑ‘(log‘𝐴)) + -π))
44	38	recnd 11364	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ ℂ)
45		negsub 11633	. . . . . . 7 ⊢ (((ℑ‘(log‘𝐴)) ∈ ℂ ∧ π ∈ ℂ) → ((ℑ‘(log‘𝐴)) + -π) = ((ℑ‘(log‘𝐴)) − π))
46	44, 12, 45	sylancl 598	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐴)) + -π) = ((ℑ‘(log‘𝐴)) − π))
47	43, 46	breqtrd 5134	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → -π < ((ℑ‘(log‘𝐴)) − π))
48		imsub 15331	. . . . . . 7 ⊢ (((log‘𝐴) ∈ ℂ ∧ (i · π) ∈ ℂ) → (ℑ‘((log‘𝐴) − (i · π))) = ((ℑ‘(log‘𝐴)) − (ℑ‘(i · π))))
49	10, 13, 48	sylancl 598	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (ℑ‘((log‘𝐴) − (i · π))) = ((ℑ‘(log‘𝐴)) − (ℑ‘(i · π))))
50		reim 15305	. . . . . . . . 9 ⊢ (π ∈ ℂ → (ℜ‘π) = (ℑ‘(i · π)))
51	12, 50	ax-mp 5	. . . . . . . 8 ⊢ (ℜ‘π) = (ℑ‘(i · π))
52		rere 15318	. . . . . . . . 9 ⊢ (π ∈ ℝ → (ℜ‘π) = π)
53	39, 52	ax-mp 5	. . . . . . . 8 ⊢ (ℜ‘π) = π
54	51, 53	eqtr3i 2788	. . . . . . 7 ⊢ (ℑ‘(i · π)) = π
55	54	oveq2i 7454	. . . . . 6 ⊢ ((ℑ‘(log‘𝐴)) − (ℑ‘(i · π))) = ((ℑ‘(log‘𝐴)) − π)
56	49, 55	eqtrdi 2814	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (ℑ‘((log‘𝐴) − (i · π))) = ((ℑ‘(log‘𝐴)) − π))
57	47, 56	breqtrrd 5136	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → -π < (ℑ‘((log‘𝐴) − (i · π))))
58		resubcl 11649	. . . . . . 7 ⊢ (((ℑ‘(log‘𝐴)) ∈ ℝ ∧ π ∈ ℝ) → ((ℑ‘(log‘𝐴)) − π) ∈ ℝ)
59	38, 39, 58	sylancl 598	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐴)) − π) ∈ ℝ)
60	39	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → π ∈ ℝ)
61		0re 11337	. . . . . . . 8 ⊢ 0 ∈ ℝ
62		pipos 26817	. . . . . . . 8 ⊢ 0 < π
63	61, 39, 62	ltleii 11460	. . . . . . 7 ⊢ 0 ≤ π
64		subge02 11859	. . . . . . . 8 ⊢ (((ℑ‘(log‘𝐴)) ∈ ℝ ∧ π ∈ ℝ) → (0 ≤ π ↔ ((ℑ‘(log‘𝐴)) − π) ≤ (ℑ‘(log‘𝐴))))
65	38, 39, 64	sylancl 598	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (0 ≤ π ↔ ((ℑ‘(log‘𝐴)) − π) ≤ (ℑ‘(log‘𝐴))))
66	63, 65	mpbii 236	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐴)) − π) ≤ (ℑ‘(log‘𝐴)))
67		logimcl 26927	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ( -π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
68	8, 67	syldan 603	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → ( -π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
69	68	simprd 501	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐴)) ≤ π)
70	59, 38, 60, 66, 69	letrd 11494	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → ((ℑ‘(log‘𝐴)) − π) ≤ π)
71	56, 70	eqbrtrd 5130	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (ℑ‘((log‘𝐴) − (i · π))) ≤ π)
72		ellogrn 26917	. . . 4 ⊢ (((log‘𝐴) − (i · π)) ∈ ran log ↔ (((log‘𝐴) − (i · π)) ∈ ℂ ∧ -π < (ℑ‘((log‘𝐴) − (i · π))) ∧ (ℑ‘((log‘𝐴) − (i · π))) ≤ π))
73	32, 57, 71, 72	syl3anbrc 1362	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → ((log‘𝐴) − (i · π)) ∈ ran log)
74		logef 26939	. . 3 ⊢ (((log‘𝐴) − (i · π)) ∈ ran log → (log‘(exp‘((log‘𝐴) − (i · π)))) = ((log‘𝐴) − (i · π)))
75	73, 74	syl 18	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (log‘(exp‘((log‘𝐴) − (i · π)))) = ((log‘𝐴) − (i · π)))
76	30, 75	eqtr3d 2800	1 ⊢ ((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (log‘ -𝐴) = ((log‘𝐴) − (i · π)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 class class class wbr 5106 ran crn 5656 ‘cfv 6558 (class class class)co 7443 ℂcc 11225 ℝcr 11226 0cc0 11227 1c1 11228 ici 11229 + caddc 11230 · cmul 11232 < clt 11370 ≤ cle 11371 − cmin 11568 -cneg 11569 / cdiv 12000 (,)cioo 13503 ℜcre 15293 ℑcim 15294 expce 16256 πcpi 16261 logclog 26912

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-inf2 9669 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305 ax-addf 11306

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-om 7902 df-1st 8025 df-2nd 8026 df-supp 8205 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-er 8745 df-map 8877 df-pm 8878 df-ixp 8954 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fsupp 9382 df-fi 9430 df-sup 9461 df-inf 9462 df-oi 9531 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-q 13103 df-rp 13148 df-xneg 13268 df-xadd 13269 df-xmul 13270 df-ioo 13507 df-ioc 13508 df-ico 13509 df-icc 13510 df-fz 13667 df-fzo 13816 df-fl 13959 df-mod 14037 df-seq 14172 df-exp 14232 df-fac 14445 df-bc 14474 df-hash 14502 df-shft 15249 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-limsup 15667 df-clim 15684 df-rlim 15685 df-sum 15883 df-ef 16262 df-sin 16264 df-cos 16265 df-pi 16267 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-starv 17472 df-sca 17473 df-vsca 17474 df-ip 17475 df-tset 17476 df-ple 17477 df-ds 17479 df-unif 17480 df-hom 17481 df-cco 17482 df-rest 17622 df-topn 17623 df-0g 17641 df-gsum 17642 df-topgen 17643 df-pt 17644 df-prds 17647 df-xrs 17703 df-qtop 17708 df-imas 17709 df-xps 17711 df-mre 17785 df-mrc 17786 df-acs 17788 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-submnd 19008 df-mulg 19307 df-cntz 19561 df-cmn 20026 df-psmet 21700 df-xmet 21701 df-met 21702 df-bl 21703 df-mopn 21704 df-fbas 21705 df-fg 21706 df-cnfld 21709 df-top 23242 df-topon 23259 df-topsp 23281 df-bases 23294 df-cld 23367 df-ntr 23368 df-cls 23369 df-nei 23446 df-lp 23484 df-perf 23485 df-cn 23575 df-cnp 23576 df-haus 23663 df-tx 23911 df-hmeo 24104 df-fil 24195 df-fm 24287 df-flim 24288 df-flf 24289 df-xms 24669 df-ms 24670 df-tms 24671 df-cncf 25229 df-limc 26216 df-dv 26217 df-log 26914

This theorem is used by: atanlogsublem 27273

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