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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 26322 | . 2 ⊢ (exp‘(i · (π / 2))) = i | |
2 | ax-icn 11164 | . . 3 ⊢ i ∈ ℂ | |
3 | ine0 11645 | . . 3 ⊢ i ≠ 0 | |
4 | halfpire 26315 | . . . . . 6 ⊢ (π / 2) ∈ ℝ | |
5 | 4 | recni 11224 | . . . . 5 ⊢ (π / 2) ∈ ℂ |
6 | 2, 5 | mulcli 11217 | . . . 4 ⊢ (i · (π / 2)) ∈ ℂ |
7 | pipos 26311 | . . . . . . 7 ⊢ 0 < π | |
8 | pire 26309 | . . . . . . . 8 ⊢ π ∈ ℝ | |
9 | lt0neg2 11717 | . . . . . . . 8 ⊢ (π ∈ ℝ → (0 < π ↔ -π < 0)) | |
10 | 8, 9 | ax-mp 5 | . . . . . . 7 ⊢ (0 < π ↔ -π < 0) |
11 | 7, 10 | mpbi 229 | . . . . . 6 ⊢ -π < 0 |
12 | halfpos2 12437 | . . . . . . . 8 ⊢ (π ∈ ℝ → (0 < π ↔ 0 < (π / 2))) | |
13 | 8, 12 | ax-mp 5 | . . . . . . 7 ⊢ (0 < π ↔ 0 < (π / 2)) |
14 | 7, 13 | mpbi 229 | . . . . . 6 ⊢ 0 < (π / 2) |
15 | 8 | renegcli 11517 | . . . . . . 7 ⊢ -π ∈ ℝ |
16 | 0re 11212 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
17 | 15, 16, 4 | lttri 11336 | . . . . . 6 ⊢ ((-π < 0 ∧ 0 < (π / 2)) → -π < (π / 2)) |
18 | 11, 14, 17 | mp2an 689 | . . . . 5 ⊢ -π < (π / 2) |
19 | reim 15052 | . . . . . . 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 15065 | . . . . . . 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 2754 | . . . . 5 ⊢ (ℑ‘(i · (π / 2))) = (π / 2) |
24 | 18, 23 | breqtrri 5165 | . . . 4 ⊢ -π < (ℑ‘(i · (π / 2))) |
25 | 8 | a1i 11 | . . . . . . . . . . 11 ⊢ (⊤ → π ∈ ℝ) |
26 | 25, 25 | ltaddposd 11794 | . . . . . . . . . 10 ⊢ (⊤ → (0 < π ↔ π < (π + π))) |
27 | 7, 26 | mpbii 232 | . . . . . . . . 9 ⊢ (⊤ → π < (π + π)) |
28 | picn 26310 | . . . . . . . . . 10 ⊢ π ∈ ℂ | |
29 | 28 | times2i 12347 | . . . . . . . . 9 ⊢ (π · 2) = (π + π) |
30 | 27, 29 | breqtrrdi 5180 | . . . . . . . 8 ⊢ (⊤ → π < (π · 2)) |
31 | 2rp 12975 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ+ | |
32 | 31 | a1i 11 | . . . . . . . . 9 ⊢ (⊤ → 2 ∈ ℝ+) |
33 | 25, 25, 32 | ltdivmul2d 13064 | . . . . . . . 8 ⊢ (⊤ → ((π / 2) < π ↔ π < (π · 2))) |
34 | 30, 33 | mpbird 257 | . . . . . . 7 ⊢ (⊤ → (π / 2) < π) |
35 | 34 | mptru 1540 | . . . . . 6 ⊢ (π / 2) < π |
36 | 4, 8, 35 | ltleii 11333 | . . . . 5 ⊢ (π / 2) ≤ π |
37 | 23, 36 | eqbrtri 5159 | . . . 4 ⊢ (ℑ‘(i · (π / 2))) ≤ π |
38 | ellogrn 26409 | . . . 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 1338 | . . 3 ⊢ (i · (π / 2)) ∈ ran log |
40 | logeftb 26433 | . . 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 1457 | . 2 ⊢ ((log‘i) = (i · (π / 2)) ↔ (exp‘(i · (π / 2))) = i) |
42 | 1, 41 | mpbir 230 | 1 ⊢ (log‘i) = (i · (π / 2)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ≠ wne 2932 class class class wbr 5138 ran crn 5667 ‘cfv 6533 (class class class)co 7401 ℂcc 11103 ℝcr 11104 0cc0 11105 ici 11107 + caddc 11108 · cmul 11110 < clt 11244 ≤ cle 11245 -cneg 11441 / cdiv 11867 2c2 12263 ℝ+crp 12970 ℜcre 15040 ℑcim 15041 expce 16001 πcpi 16006 logclog 26404 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-starv 17210 df-sca 17211 df-vsca 17212 df-ip 17213 df-tset 17214 df-ple 17215 df-ds 17217 df-unif 17218 df-hom 17219 df-cco 17220 df-rest 17366 df-topn 17367 df-0g 17385 df-gsum 17386 df-topgen 17387 df-pt 17388 df-prds 17391 df-xrs 17446 df-qtop 17451 df-imas 17452 df-xps 17454 df-mre 17528 df-mrc 17529 df-acs 17531 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-submnd 18703 df-mulg 18985 df-cntz 19222 df-cmn 19691 df-psmet 21219 df-xmet 21220 df-met 21221 df-bl 21222 df-mopn 21223 df-fbas 21224 df-fg 21225 df-cnfld 21228 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cld 22844 df-ntr 22845 df-cls 22846 df-nei 22923 df-lp 22961 df-perf 22962 df-cn 23052 df-cnp 23053 df-haus 23140 df-tx 23387 df-hmeo 23580 df-fil 23671 df-fm 23763 df-flim 23764 df-flf 23765 df-xms 24147 df-ms 24148 df-tms 24149 df-cncf 24719 df-limc 25716 df-dv 25717 df-log 26406 |
This theorem is referenced by: iexpire 35166 |
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