logimul - Metamath Proof Explorer

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Theorem logimul 26568

Description: Multiplying a number by i increases the logarithm of the number by iπ / 2. (Contributed by Mario Carneiro, 4-Apr-2015.)

**Assertion**
Ref	Expression
logimul	⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (log‘(i · 𝐴)) = ((log‘𝐴) + (i · (π / 2))))

**Proof of Theorem logimul**
Step	Hyp	Ref	Expression
1		logcl 26522	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
2	1	3adant3 1129	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (log‘𝐴) ∈ ℂ)
3		ax-icn 11205	. . . . . 6 ⊢ i ∈ ℂ
4		halfpire 26419	. . . . . . 7 ⊢ (π / 2) ∈ ℝ
5	4	recni 11266	. . . . . 6 ⊢ (π / 2) ∈ ℂ
6	3, 5	mulcli 11259	. . . . 5 ⊢ (i · (π / 2)) ∈ ℂ
7		efadd 16078	. . . . 5 ⊢ (((log‘𝐴) ∈ ℂ ∧ (i · (π / 2)) ∈ ℂ) → (exp‘((log‘𝐴) + (i · (π / 2)))) = ((exp‘(log‘𝐴)) · (exp‘(i · (π / 2)))))
8	2, 6, 7	sylancl 584	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (exp‘((log‘𝐴) + (i · (π / 2)))) = ((exp‘(log‘𝐴)) · (exp‘(i · (π / 2)))))
9		eflog 26530	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
10	9	3adant3 1129	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (exp‘(log‘𝐴)) = 𝐴)
11		efhalfpi 26426	. . . . . 6 ⊢ (exp‘(i · (π / 2))) = i
12	11	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (exp‘(i · (π / 2))) = i)
13	10, 12	oveq12d 7444	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((exp‘(log‘𝐴)) · (exp‘(i · (π / 2)))) = (𝐴 · i))
14		simp1 1133	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → 𝐴 ∈ ℂ)
15		mulcom 11232	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ i ∈ ℂ) → (𝐴 · i) = (i · 𝐴))
16	14, 3, 15	sylancl 584	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (𝐴 · i) = (i · 𝐴))
17	8, 13, 16	3eqtrd 2772	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (exp‘((log‘𝐴) + (i · (π / 2)))) = (i · 𝐴))
18	17	fveq2d 6906	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (log‘(exp‘((log‘𝐴) + (i · (π / 2))))) = (log‘(i · 𝐴)))
19		addcl 11228	. . . . 5 ⊢ (((log‘𝐴) ∈ ℂ ∧ (i · (π / 2)) ∈ ℂ) → ((log‘𝐴) + (i · (π / 2))) ∈ ℂ)
20	2, 6, 19	sylancl 584	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((log‘𝐴) + (i · (π / 2))) ∈ ℂ)
21		pire 26413	. . . . . . . 8 ⊢ π ∈ ℝ
22	21	renegcli 11559	. . . . . . 7 ⊢ -π ∈ ℝ
23	22	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → -π ∈ ℝ)
24	2	imcld 15182	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ ℝ)
25		readdcl 11229	. . . . . . 7 ⊢ (((ℑ‘(log‘𝐴)) ∈ ℝ ∧ (π / 2) ∈ ℝ) → ((ℑ‘(log‘𝐴)) + (π / 2)) ∈ ℝ)
26	24, 4, 25	sylancl 584	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((ℑ‘(log‘𝐴)) + (π / 2)) ∈ ℝ)
27		logimcl 26523	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
28	27	3adant3 1129	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
29	28	simpld 493	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → -π < (ℑ‘(log‘𝐴)))
30		pirp 26416	. . . . . . . 8 ⊢ π ∈ ℝ⁺
31		rphalfcl 13041	. . . . . . . 8 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
32	30, 31	ax-mp 5	. . . . . . 7 ⊢ (π / 2) ∈ ℝ⁺
33		ltaddrp 13051	. . . . . . 7 ⊢ (((ℑ‘(log‘𝐴)) ∈ ℝ ∧ (π / 2) ∈ ℝ⁺) → (ℑ‘(log‘𝐴)) < ((ℑ‘(log‘𝐴)) + (π / 2)))
34	24, 32, 33	sylancl 584	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) < ((ℑ‘(log‘𝐴)) + (π / 2)))
35	23, 24, 26, 29, 34	lttrd 11413	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → -π < ((ℑ‘(log‘𝐴)) + (π / 2)))
36		imadd 15121	. . . . . . 7 ⊢ (((log‘𝐴) ∈ ℂ ∧ (i · (π / 2)) ∈ ℂ) → (ℑ‘((log‘𝐴) + (i · (π / 2)))) = ((ℑ‘(log‘𝐴)) + (ℑ‘(i · (π / 2)))))
37	2, 6, 36	sylancl 584	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘((log‘𝐴) + (i · (π / 2)))) = ((ℑ‘(log‘𝐴)) + (ℑ‘(i · (π / 2)))))
38		reim 15096	. . . . . . . . 9 ⊢ ((π / 2) ∈ ℂ → (ℜ‘(π / 2)) = (ℑ‘(i · (π / 2))))
39	5, 38	ax-mp 5	. . . . . . . 8 ⊢ (ℜ‘(π / 2)) = (ℑ‘(i · (π / 2)))
40		rere 15109	. . . . . . . . 9 ⊢ ((π / 2) ∈ ℝ → (ℜ‘(π / 2)) = (π / 2))
41	4, 40	ax-mp 5	. . . . . . . 8 ⊢ (ℜ‘(π / 2)) = (π / 2)
42	39, 41	eqtr3i 2758	. . . . . . 7 ⊢ (ℑ‘(i · (π / 2))) = (π / 2)
43	42	oveq2i 7437	. . . . . 6 ⊢ ((ℑ‘(log‘𝐴)) + (ℑ‘(i · (π / 2)))) = ((ℑ‘(log‘𝐴)) + (π / 2))
44	37, 43	eqtrdi 2784	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘((log‘𝐴) + (i · (π / 2)))) = ((ℑ‘(log‘𝐴)) + (π / 2)))
45	35, 44	breqtrrd 5180	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → -π < (ℑ‘((log‘𝐴) + (i · (π / 2)))))
46		argrege0 26565	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (-(π / 2)[,](π / 2)))
47	4	renegcli 11559	. . . . . . . . . 10 ⊢ -(π / 2) ∈ ℝ
48	47, 4	elicc2i 13430	. . . . . . . . 9 ⊢ ((ℑ‘(log‘𝐴)) ∈ (-(π / 2)[,](π / 2)) ↔ ((ℑ‘(log‘𝐴)) ∈ ℝ ∧ -(π / 2) ≤ (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ (π / 2)))
49	48	simp3bi 1144	. . . . . . . 8 ⊢ ((ℑ‘(log‘𝐴)) ∈ (-(π / 2)[,](π / 2)) → (ℑ‘(log‘𝐴)) ≤ (π / 2))
50	46, 49	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ≤ (π / 2))
51	21	recni 11266	. . . . . . . 8 ⊢ π ∈ ℂ
52		pidiv2halves 26422	. . . . . . . 8 ⊢ ((π / 2) + (π / 2)) = π
53	51, 5, 5, 52	subaddrii 11587	. . . . . . 7 ⊢ (π − (π / 2)) = (π / 2)
54	50, 53	breqtrrdi 5194	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ≤ (π − (π / 2)))
55	4	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (π / 2) ∈ ℝ)
56	21	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → π ∈ ℝ)
57		leaddsub 11728	. . . . . . 7 ⊢ (((ℑ‘(log‘𝐴)) ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ π ∈ ℝ) → (((ℑ‘(log‘𝐴)) + (π / 2)) ≤ π ↔ (ℑ‘(log‘𝐴)) ≤ (π − (π / 2))))
58	24, 55, 56, 57	syl3anc 1368	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (((ℑ‘(log‘𝐴)) + (π / 2)) ≤ π ↔ (ℑ‘(log‘𝐴)) ≤ (π − (π / 2))))
59	54, 58	mpbird 256	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((ℑ‘(log‘𝐴)) + (π / 2)) ≤ π)
60	44, 59	eqbrtrd 5174	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘((log‘𝐴) + (i · (π / 2)))) ≤ π)
61		ellogrn 26513	. . . 4 ⊢ (((log‘𝐴) + (i · (π / 2))) ∈ ran log ↔ (((log‘𝐴) + (i · (π / 2))) ∈ ℂ ∧ -π < (ℑ‘((log‘𝐴) + (i · (π / 2)))) ∧ (ℑ‘((log‘𝐴) + (i · (π / 2)))) ≤ π))
62	20, 45, 60, 61	syl3anbrc 1340	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → ((log‘𝐴) + (i · (π / 2))) ∈ ran log)
63		logef 26535	. . 3 ⊢ (((log‘𝐴) + (i · (π / 2))) ∈ ran log → (log‘(exp‘((log‘𝐴) + (i · (π / 2))))) = ((log‘𝐴) + (i · (π / 2))))
64	62, 63	syl 17	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (log‘(exp‘((log‘𝐴) + (i · (π / 2))))) = ((log‘𝐴) + (i · (π / 2))))
65	18, 64	eqtr3d 2770	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (log‘(i · 𝐴)) = ((log‘𝐴) + (i · (π / 2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 class class class wbr 5152 ran crn 5683 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 ℝcr 11145 0cc0 11146 ici 11148 + caddc 11149 · cmul 11151 < clt 11286 ≤ cle 11287 − cmin 11482 -cneg 11483 / cdiv 11909 2c2 12305 ℝ⁺crp 13014 [,]cicc 13367 ℜcre 15084 ℑcim 15085 expce 16045 πcpi 16050 logclog 26508

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 df-sin 16053 df-cos 16054 df-pi 16056 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cn 23151 df-cnp 23152 df-haus 23239 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24818 df-limc 25815 df-dv 25816 df-log 26510

This theorem is referenced by: atanlogsublem 26867

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