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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > logi | Structured version Visualization version GIF version |
Description: Calculate the logarithm of i. (Contributed by Scott Fenton, 13-Apr-2020.) |
Ref | Expression |
---|---|
logi | ⊢ (log‘i) = (i · (π / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efhalfpi 24622 | . 2 ⊢ (exp‘(i · (π / 2))) = i | |
2 | ax-icn 10310 | . . 3 ⊢ i ∈ ℂ | |
3 | ine0 10788 | . . 3 ⊢ i ≠ 0 | |
4 | halfpire 24615 | . . . . . 6 ⊢ (π / 2) ∈ ℝ | |
5 | 4 | recni 10370 | . . . . 5 ⊢ (π / 2) ∈ ℂ |
6 | 2, 5 | mulcli 10363 | . . . 4 ⊢ (i · (π / 2)) ∈ ℂ |
7 | pipos 24611 | . . . . . . 7 ⊢ 0 < π | |
8 | pire 24609 | . . . . . . . 8 ⊢ π ∈ ℝ | |
9 | lt0neg2 10858 | . . . . . . . 8 ⊢ (π ∈ ℝ → (0 < π ↔ -π < 0)) | |
10 | 8, 9 | ax-mp 5 | . . . . . . 7 ⊢ (0 < π ↔ -π < 0) |
11 | 7, 10 | mpbi 222 | . . . . . 6 ⊢ -π < 0 |
12 | halfpos2 11586 | . . . . . . . 8 ⊢ (π ∈ ℝ → (0 < π ↔ 0 < (π / 2))) | |
13 | 8, 12 | ax-mp 5 | . . . . . . 7 ⊢ (0 < π ↔ 0 < (π / 2)) |
14 | 7, 13 | mpbi 222 | . . . . . 6 ⊢ 0 < (π / 2) |
15 | 8 | renegcli 10662 | . . . . . . 7 ⊢ -π ∈ ℝ |
16 | 0re 10357 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
17 | 15, 16, 4 | lttri 10481 | . . . . . 6 ⊢ ((-π < 0 ∧ 0 < (π / 2)) → -π < (π / 2)) |
18 | 11, 14, 17 | mp2an 685 | . . . . 5 ⊢ -π < (π / 2) |
19 | reim 14225 | . . . . . . 7 ⊢ ((π / 2) ∈ ℂ → (ℜ‘(π / 2)) = (ℑ‘(i · (π / 2)))) | |
20 | 5, 19 | ax-mp 5 | . . . . . 6 ⊢ (ℜ‘(π / 2)) = (ℑ‘(i · (π / 2))) |
21 | rere 14238 | . . . . . . 7 ⊢ ((π / 2) ∈ ℝ → (ℜ‘(π / 2)) = (π / 2)) | |
22 | 4, 21 | ax-mp 5 | . . . . . 6 ⊢ (ℜ‘(π / 2)) = (π / 2) |
23 | 20, 22 | eqtr3i 2850 | . . . . 5 ⊢ (ℑ‘(i · (π / 2))) = (π / 2) |
24 | 18, 23 | breqtrri 4899 | . . . 4 ⊢ -π < (ℑ‘(i · (π / 2))) |
25 | 8 | a1i 11 | . . . . . . . . . . 11 ⊢ (⊤ → π ∈ ℝ) |
26 | 25, 25 | ltaddposd 10935 | . . . . . . . . . 10 ⊢ (⊤ → (0 < π ↔ π < (π + π))) |
27 | 7, 26 | mpbii 225 | . . . . . . . . 9 ⊢ (⊤ → π < (π + π)) |
28 | picn 24610 | . . . . . . . . . 10 ⊢ π ∈ ℂ | |
29 | 28 | times2i 11496 | . . . . . . . . 9 ⊢ (π · 2) = (π + π) |
30 | 27, 29 | syl6breqr 4914 | . . . . . . . 8 ⊢ (⊤ → π < (π · 2)) |
31 | 2rp 12116 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ+ | |
32 | 31 | a1i 11 | . . . . . . . . 9 ⊢ (⊤ → 2 ∈ ℝ+) |
33 | 25, 25, 32 | ltdivmul2d 12207 | . . . . . . . 8 ⊢ (⊤ → ((π / 2) < π ↔ π < (π · 2))) |
34 | 30, 33 | mpbird 249 | . . . . . . 7 ⊢ (⊤ → (π / 2) < π) |
35 | 34 | mptru 1666 | . . . . . 6 ⊢ (π / 2) < π |
36 | 4, 8, 35 | ltleii 10478 | . . . . 5 ⊢ (π / 2) ≤ π |
37 | 23, 36 | eqbrtri 4893 | . . . 4 ⊢ (ℑ‘(i · (π / 2))) ≤ π |
38 | ellogrn 24704 | . . . 4 ⊢ ((i · (π / 2)) ∈ ran log ↔ ((i · (π / 2)) ∈ ℂ ∧ -π < (ℑ‘(i · (π / 2))) ∧ (ℑ‘(i · (π / 2))) ≤ π)) | |
39 | 6, 24, 37, 38 | mpbir3an 1447 | . . 3 ⊢ (i · (π / 2)) ∈ ran log |
40 | logeftb 24728 | . . 3 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ (i · (π / 2)) ∈ ran log) → ((log‘i) = (i · (π / 2)) ↔ (exp‘(i · (π / 2))) = i)) | |
41 | 2, 3, 39, 40 | mp3an 1591 | . 2 ⊢ ((log‘i) = (i · (π / 2)) ↔ (exp‘(i · (π / 2))) = i) |
42 | 1, 41 | mpbir 223 | 1 ⊢ (log‘i) = (i · (π / 2)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1658 ⊤wtru 1659 ∈ wcel 2166 ≠ wne 2998 class class class wbr 4872 ran crn 5342 ‘cfv 6122 (class class class)co 6904 ℂcc 10249 ℝcr 10250 0cc0 10251 ici 10253 + caddc 10254 · cmul 10256 < clt 10390 ≤ cle 10391 -cneg 10585 / cdiv 11008 2c2 11405 ℝ+crp 12111 ℜcre 14213 ℑcim 14214 expce 15163 πcpi 15168 logclog 24699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 ax-addf 10330 ax-mulf 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-iin 4742 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-of 7156 df-om 7326 df-1st 7427 df-2nd 7428 df-supp 7559 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-2o 7826 df-oadd 7829 df-er 8008 df-map 8123 df-pm 8124 df-ixp 8175 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-fsupp 8544 df-fi 8585 df-sup 8616 df-inf 8617 df-oi 8683 df-card 9077 df-cda 9304 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-5 11416 df-6 11417 df-7 11418 df-8 11419 df-9 11420 df-n0 11618 df-z 11704 df-dec 11821 df-uz 11968 df-q 12071 df-rp 12112 df-xneg 12231 df-xadd 12232 df-xmul 12233 df-ioo 12466 df-ioc 12467 df-ico 12468 df-icc 12469 df-fz 12619 df-fzo 12760 df-fl 12887 df-mod 12963 df-seq 13095 df-exp 13154 df-fac 13353 df-bc 13382 df-hash 13410 df-shft 14183 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 df-abs 14352 df-limsup 14578 df-clim 14595 df-rlim 14596 df-sum 14793 df-ef 15169 df-sin 15171 df-cos 15172 df-pi 15174 df-struct 16223 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-mulr 16318 df-starv 16319 df-sca 16320 df-vsca 16321 df-ip 16322 df-tset 16323 df-ple 16324 df-ds 16326 df-unif 16327 df-hom 16328 df-cco 16329 df-rest 16435 df-topn 16436 df-0g 16454 df-gsum 16455 df-topgen 16456 df-pt 16457 df-prds 16460 df-xrs 16514 df-qtop 16519 df-imas 16520 df-xps 16522 df-mre 16598 df-mrc 16599 df-acs 16601 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-submnd 17688 df-mulg 17894 df-cntz 18099 df-cmn 18547 df-psmet 20097 df-xmet 20098 df-met 20099 df-bl 20100 df-mopn 20101 df-fbas 20102 df-fg 20103 df-cnfld 20106 df-top 21068 df-topon 21085 df-topsp 21107 df-bases 21120 df-cld 21193 df-ntr 21194 df-cls 21195 df-nei 21272 df-lp 21310 df-perf 21311 df-cn 21401 df-cnp 21402 df-haus 21489 df-tx 21735 df-hmeo 21928 df-fil 22019 df-fm 22111 df-flim 22112 df-flf 22113 df-xms 22494 df-ms 22495 df-tms 22496 df-cncf 23050 df-limc 24028 df-dv 24029 df-log 24701 |
This theorem is referenced by: iexpire 32162 |
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