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Mirrors > Home > MPE Home > Th. List > argimlt0 | Structured version Visualization version GIF version |
Description: Closure of the argument of a complex number with negative imaginary part. (Contributed by Mario Carneiro, 25-Feb-2015.) |
Ref | Expression |
---|---|
argimlt0 | ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐴)) ∈ (-π(,)0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 478 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐴) < 0) | |
2 | 1 | lt0ne0d 10884 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐴) ≠ 0) |
3 | fveq2 6410 | . . . . . . 7 ⊢ (𝐴 = 0 → (ℑ‘𝐴) = (ℑ‘0)) | |
4 | im0 14231 | . . . . . . 7 ⊢ (ℑ‘0) = 0 | |
5 | 3, 4 | syl6eq 2848 | . . . . . 6 ⊢ (𝐴 = 0 → (ℑ‘𝐴) = 0) |
6 | 5 | necon3i 3002 | . . . . 5 ⊢ ((ℑ‘𝐴) ≠ 0 → 𝐴 ≠ 0) |
7 | 2, 6 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → 𝐴 ≠ 0) |
8 | logcl 24653 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ) | |
9 | 7, 8 | syldan 586 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (log‘𝐴) ∈ ℂ) |
10 | 9 | imcld 14273 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐴)) ∈ ℝ) |
11 | logcj 24690 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) ≠ 0) → (log‘(∗‘𝐴)) = (∗‘(log‘𝐴))) | |
12 | 2, 11 | syldan 586 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (log‘(∗‘𝐴)) = (∗‘(log‘𝐴))) |
13 | 12 | fveq2d 6414 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘(∗‘𝐴))) = (ℑ‘(∗‘(log‘𝐴)))) |
14 | 9 | imcjd 14283 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘(∗‘(log‘𝐴))) = -(ℑ‘(log‘𝐴))) |
15 | 13, 14 | eqtrd 2832 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘(∗‘𝐴))) = -(ℑ‘(log‘𝐴))) |
16 | cjcl 14183 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
17 | 16 | adantr 473 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (∗‘𝐴) ∈ ℂ) |
18 | imcl 14189 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
19 | 18 | adantr 473 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘𝐴) ∈ ℝ) |
20 | 19 | lt0neg1d 10888 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → ((ℑ‘𝐴) < 0 ↔ 0 < -(ℑ‘𝐴))) |
21 | 1, 20 | mpbid 224 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → 0 < -(ℑ‘𝐴)) |
22 | imcj 14210 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴)) | |
23 | 22 | adantr 473 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘(∗‘𝐴)) = -(ℑ‘𝐴)) |
24 | 21, 23 | breqtrrd 4870 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → 0 < (ℑ‘(∗‘𝐴))) |
25 | argimgt0 24696 | . . . . . . 7 ⊢ (((∗‘𝐴) ∈ ℂ ∧ 0 < (ℑ‘(∗‘𝐴))) → (ℑ‘(log‘(∗‘𝐴))) ∈ (0(,)π)) | |
26 | 17, 24, 25 | syl2anc 580 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘(∗‘𝐴))) ∈ (0(,)π)) |
27 | eliooord 12479 | . . . . . 6 ⊢ ((ℑ‘(log‘(∗‘𝐴))) ∈ (0(,)π) → (0 < (ℑ‘(log‘(∗‘𝐴))) ∧ (ℑ‘(log‘(∗‘𝐴))) < π)) | |
28 | 26, 27 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (0 < (ℑ‘(log‘(∗‘𝐴))) ∧ (ℑ‘(log‘(∗‘𝐴))) < π)) |
29 | 28 | simprd 490 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘(∗‘𝐴))) < π) |
30 | 15, 29 | eqbrtrrd 4866 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → -(ℑ‘(log‘𝐴)) < π) |
31 | pire 24549 | . . . 4 ⊢ π ∈ ℝ | |
32 | ltnegcon1 10820 | . . . 4 ⊢ (((ℑ‘(log‘𝐴)) ∈ ℝ ∧ π ∈ ℝ) → (-(ℑ‘(log‘𝐴)) < π ↔ -π < (ℑ‘(log‘𝐴)))) | |
33 | 10, 31, 32 | sylancl 581 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (-(ℑ‘(log‘𝐴)) < π ↔ -π < (ℑ‘(log‘𝐴)))) |
34 | 30, 33 | mpbid 224 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → -π < (ℑ‘(log‘𝐴))) |
35 | 28 | simpld 489 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → 0 < (ℑ‘(log‘(∗‘𝐴)))) |
36 | 35, 15 | breqtrd 4868 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → 0 < -(ℑ‘(log‘𝐴))) |
37 | 10 | lt0neg1d 10888 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → ((ℑ‘(log‘𝐴)) < 0 ↔ 0 < -(ℑ‘(log‘𝐴)))) |
38 | 36, 37 | mpbird 249 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐴)) < 0) |
39 | 31 | renegcli 10633 | . . . 4 ⊢ -π ∈ ℝ |
40 | 39 | rexri 10386 | . . 3 ⊢ -π ∈ ℝ* |
41 | 0xr 10374 | . . 3 ⊢ 0 ∈ ℝ* | |
42 | elioo2 12462 | . . 3 ⊢ ((-π ∈ ℝ* ∧ 0 ∈ ℝ*) → ((ℑ‘(log‘𝐴)) ∈ (-π(,)0) ↔ ((ℑ‘(log‘𝐴)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) < 0))) | |
43 | 40, 41, 42 | mp2an 684 | . 2 ⊢ ((ℑ‘(log‘𝐴)) ∈ (-π(,)0) ↔ ((ℑ‘(log‘𝐴)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) < 0)) |
44 | 10, 34, 38, 43 | syl3anbrc 1444 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐴)) ∈ (-π(,)0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ≠ wne 2970 class class class wbr 4842 ‘cfv 6100 (class class class)co 6877 ℂcc 10221 ℝcr 10222 0cc0 10223 ℝ*cxr 10361 < clt 10362 -cneg 10556 (,)cioo 12421 ∗ccj 14174 ℑcim 14176 πcpi 15130 logclog 24639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-rep 4963 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-inf2 8787 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 ax-pre-sup 10301 ax-addf 10302 ax-mulf 10303 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-int 4667 df-iun 4711 df-iin 4712 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-se 5271 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-isom 6109 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-of 7130 df-om 7299 df-1st 7400 df-2nd 7401 df-supp 7532 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-1o 7798 df-2o 7799 df-oadd 7802 df-er 7981 df-map 8096 df-pm 8097 df-ixp 8148 df-en 8195 df-dom 8196 df-sdom 8197 df-fin 8198 df-fsupp 8517 df-fi 8558 df-sup 8589 df-inf 8590 df-oi 8656 df-card 9050 df-cda 9277 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-div 10976 df-nn 11312 df-2 11373 df-3 11374 df-4 11375 df-5 11376 df-6 11377 df-7 11378 df-8 11379 df-9 11380 df-n0 11578 df-z 11664 df-dec 11781 df-uz 11928 df-q 12031 df-rp 12072 df-xneg 12190 df-xadd 12191 df-xmul 12192 df-ioo 12425 df-ioc 12426 df-ico 12427 df-icc 12428 df-fz 12578 df-fzo 12718 df-fl 12845 df-mod 12921 df-seq 13053 df-exp 13112 df-fac 13311 df-bc 13340 df-hash 13368 df-shft 14145 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-limsup 14540 df-clim 14557 df-rlim 14558 df-sum 14755 df-ef 15131 df-sin 15133 df-cos 15134 df-pi 15136 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-starv 16279 df-sca 16280 df-vsca 16281 df-ip 16282 df-tset 16283 df-ple 16284 df-ds 16286 df-unif 16287 df-hom 16288 df-cco 16289 df-rest 16395 df-topn 16396 df-0g 16414 df-gsum 16415 df-topgen 16416 df-pt 16417 df-prds 16420 df-xrs 16474 df-qtop 16479 df-imas 16480 df-xps 16482 df-mre 16558 df-mrc 16559 df-acs 16561 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-submnd 17648 df-mulg 17854 df-cntz 18059 df-cmn 18507 df-psmet 20057 df-xmet 20058 df-met 20059 df-bl 20060 df-mopn 20061 df-fbas 20062 df-fg 20063 df-cnfld 20066 df-top 21024 df-topon 21041 df-topsp 21063 df-bases 21076 df-cld 21149 df-ntr 21150 df-cls 21151 df-nei 21228 df-lp 21266 df-perf 21267 df-cn 21357 df-cnp 21358 df-haus 21445 df-tx 21691 df-hmeo 21884 df-fil 21975 df-fm 22067 df-flim 22068 df-flf 22069 df-xms 22450 df-ms 22451 df-tms 22452 df-cncf 23006 df-limc 23968 df-dv 23969 df-log 24641 |
This theorem is referenced by: logcnlem3 24728 atanlogaddlem 24989 |
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