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Mirrors > Home > MPE Home > Th. List > abslogle | Structured version Visualization version GIF version |
Description: Bound on the magnitude of the complex logarithm function. (Contributed by Mario Carneiro, 3-Jul-2017.) |
Ref | Expression |
---|---|
abslogle | ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(log‘𝐴)) ≤ ((abs‘(log‘(abs‘𝐴))) + π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | logcl 26520 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ) | |
2 | 1 | abscld 15415 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(log‘𝐴)) ∈ ℝ) |
3 | absrpcl 15267 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ+) | |
4 | relogcl 26527 | . . . . . 6 ⊢ ((abs‘𝐴) ∈ ℝ+ → (log‘(abs‘𝐴)) ∈ ℝ) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘(abs‘𝐴)) ∈ ℝ) |
6 | 5 | recnd 11272 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘(abs‘𝐴)) ∈ ℂ) |
7 | 6 | abscld 15415 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(log‘(abs‘𝐴))) ∈ ℝ) |
8 | 1 | imcld 15174 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℝ) |
9 | 8 | recnd 11272 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℑ‘(log‘𝐴)) ∈ ℂ) |
10 | 9 | abscld 15415 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘(log‘𝐴))) ∈ ℝ) |
11 | 7, 10 | readdcld 11273 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘(log‘(abs‘𝐴))) + (abs‘(ℑ‘(log‘𝐴)))) ∈ ℝ) |
12 | pire 26411 | . . . 4 ⊢ π ∈ ℝ | |
13 | 12 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → π ∈ ℝ) |
14 | 7, 13 | readdcld 11273 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘(log‘(abs‘𝐴))) + π) ∈ ℝ) |
15 | 1 | recld 15173 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) ∈ ℝ) |
16 | 15 | recnd 11272 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) ∈ ℂ) |
17 | ax-icn 11197 | . . . . . 6 ⊢ i ∈ ℂ | |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → i ∈ ℂ) |
19 | 18, 9 | mulcld 11264 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (i · (ℑ‘(log‘𝐴))) ∈ ℂ) |
20 | 16, 19 | abstrid 15435 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘((ℜ‘(log‘𝐴)) + (i · (ℑ‘(log‘𝐴))))) ≤ ((abs‘(ℜ‘(log‘𝐴))) + (abs‘(i · (ℑ‘(log‘𝐴)))))) |
21 | 1 | replimd 15176 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) = ((ℜ‘(log‘𝐴)) + (i · (ℑ‘(log‘𝐴))))) |
22 | 21 | fveq2d 6896 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(log‘𝐴)) = (abs‘((ℜ‘(log‘𝐴)) + (i · (ℑ‘(log‘𝐴)))))) |
23 | relog 26549 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴))) | |
24 | 23 | eqcomd 2731 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘(abs‘𝐴)) = (ℜ‘(log‘𝐴))) |
25 | 24 | fveq2d 6896 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(log‘(abs‘𝐴))) = (abs‘(ℜ‘(log‘𝐴)))) |
26 | 18, 9 | absmuld 15433 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(i · (ℑ‘(log‘𝐴)))) = ((abs‘i) · (abs‘(ℑ‘(log‘𝐴))))) |
27 | absi 15265 | . . . . . . 7 ⊢ (abs‘i) = 1 | |
28 | 27 | oveq1i 7426 | . . . . . 6 ⊢ ((abs‘i) · (abs‘(ℑ‘(log‘𝐴)))) = (1 · (abs‘(ℑ‘(log‘𝐴)))) |
29 | 10 | recnd 11272 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘(log‘𝐴))) ∈ ℂ) |
30 | 29 | mullidd 11262 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 · (abs‘(ℑ‘(log‘𝐴)))) = (abs‘(ℑ‘(log‘𝐴)))) |
31 | 28, 30 | eqtrid 2777 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘i) · (abs‘(ℑ‘(log‘𝐴)))) = (abs‘(ℑ‘(log‘𝐴)))) |
32 | 26, 31 | eqtr2d 2766 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘(log‘𝐴))) = (abs‘(i · (ℑ‘(log‘𝐴))))) |
33 | 25, 32 | oveq12d 7434 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘(log‘(abs‘𝐴))) + (abs‘(ℑ‘(log‘𝐴)))) = ((abs‘(ℜ‘(log‘𝐴))) + (abs‘(i · (ℑ‘(log‘𝐴)))))) |
34 | 20, 22, 33 | 3brtr4d 5175 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(log‘𝐴)) ≤ ((abs‘(log‘(abs‘𝐴))) + (abs‘(ℑ‘(log‘𝐴))))) |
35 | abslogimle 26525 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘(log‘𝐴))) ≤ π) | |
36 | 10, 13, 7, 35 | leadd2dd 11859 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘(log‘(abs‘𝐴))) + (abs‘(ℑ‘(log‘𝐴)))) ≤ ((abs‘(log‘(abs‘𝐴))) + π)) |
37 | 2, 11, 14, 34, 36 | letrd 11401 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(log‘𝐴)) ≤ ((abs‘(log‘(abs‘𝐴))) + π)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ≠ wne 2930 class class class wbr 5143 ‘cfv 6543 (class class class)co 7416 ℂcc 11136 ℝcr 11137 0cc0 11138 1c1 11139 ici 11140 + caddc 11141 · cmul 11143 ≤ cle 11279 ℝ+crp 13006 ℜcre 15076 ℑcim 15077 abscabs 15213 πcpi 16042 logclog 26506 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-fac 14265 df-bc 14294 df-hash 14322 df-shft 15046 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-sum 15665 df-ef 16043 df-sin 16045 df-cos 16046 df-pi 16048 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-mulg 19028 df-cntz 19272 df-cmn 19741 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-fbas 21280 df-fg 21281 df-cnfld 21284 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-lp 23058 df-perf 23059 df-cn 23149 df-cnp 23150 df-haus 23237 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-xms 24244 df-ms 24245 df-tms 24246 df-cncf 24816 df-limc 25813 df-dv 25814 df-log 26508 |
This theorem is referenced by: lgamgulmlem5 26983 lgambdd 26987 |
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