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Mirrors > Home > MPE Home > Th. List > coskpi | Structured version Visualization version GIF version |
Description: The absolute value of the cosine of an integer multiple of π is 1. (Contributed by NM, 19-Aug-2008.) |
Ref | Expression |
---|---|
coskpi | ⊢ (𝐾 ∈ ℤ → (abs‘(cos‘(𝐾 · π))) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 12614 | . . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ) | |
2 | 2cn 12337 | . . . . . . . . . . 11 ⊢ 2 ∈ ℂ | |
3 | picn 26491 | . . . . . . . . . . 11 ⊢ π ∈ ℂ | |
4 | mul12 11422 | . . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℂ ∧ 2 ∈ ℂ ∧ π ∈ ℂ) → (𝐾 · (2 · π)) = (2 · (𝐾 · π))) | |
5 | 2, 3, 4 | mp3an23 1455 | . . . . . . . . . 10 ⊢ (𝐾 ∈ ℂ → (𝐾 · (2 · π)) = (2 · (𝐾 · π))) |
6 | 1, 5 | syl 17 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℤ → (𝐾 · (2 · π)) = (2 · (𝐾 · π))) |
7 | 6 | fveq2d 6908 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → (cos‘(𝐾 · (2 · π))) = (cos‘(2 · (𝐾 · π)))) |
8 | cos2kpi 26516 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → (cos‘(𝐾 · (2 · π))) = 1) | |
9 | zre 12613 | . . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
10 | pire 26490 | . . . . . . . . . . 11 ⊢ π ∈ ℝ | |
11 | remulcl 11236 | . . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℝ ∧ π ∈ ℝ) → (𝐾 · π) ∈ ℝ) | |
12 | 9, 10, 11 | sylancl 586 | . . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (𝐾 · π) ∈ ℝ) |
13 | 12 | recnd 11285 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℤ → (𝐾 · π) ∈ ℂ) |
14 | cos2t 16210 | . . . . . . . . 9 ⊢ ((𝐾 · π) ∈ ℂ → (cos‘(2 · (𝐾 · π))) = ((2 · ((cos‘(𝐾 · π))↑2)) − 1)) | |
15 | 13, 14 | syl 17 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → (cos‘(2 · (𝐾 · π))) = ((2 · ((cos‘(𝐾 · π))↑2)) − 1)) |
16 | 7, 8, 15 | 3eqtr3rd 2785 | . . . . . . 7 ⊢ (𝐾 ∈ ℤ → ((2 · ((cos‘(𝐾 · π))↑2)) − 1) = 1) |
17 | 12 | recoscld 16176 | . . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (cos‘(𝐾 · π)) ∈ ℝ) |
18 | 17 | recnd 11285 | . . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (cos‘(𝐾 · π)) ∈ ℂ) |
19 | 18 | sqcld 14180 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℤ → ((cos‘(𝐾 · π))↑2) ∈ ℂ) |
20 | mulcl 11235 | . . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ ((cos‘(𝐾 · π))↑2) ∈ ℂ) → (2 · ((cos‘(𝐾 · π))↑2)) ∈ ℂ) | |
21 | 2, 19, 20 | sylancr 587 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → (2 · ((cos‘(𝐾 · π))↑2)) ∈ ℂ) |
22 | ax-1cn 11209 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
23 | subadd 11507 | . . . . . . . . 9 ⊢ (((2 · ((cos‘(𝐾 · π))↑2)) ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · ((cos‘(𝐾 · π))↑2)) − 1) = 1 ↔ (1 + 1) = (2 · ((cos‘(𝐾 · π))↑2)))) | |
24 | 22, 22, 23 | mp3an23 1455 | . . . . . . . 8 ⊢ ((2 · ((cos‘(𝐾 · π))↑2)) ∈ ℂ → (((2 · ((cos‘(𝐾 · π))↑2)) − 1) = 1 ↔ (1 + 1) = (2 · ((cos‘(𝐾 · π))↑2)))) |
25 | 21, 24 | syl 17 | . . . . . . 7 ⊢ (𝐾 ∈ ℤ → (((2 · ((cos‘(𝐾 · π))↑2)) − 1) = 1 ↔ (1 + 1) = (2 · ((cos‘(𝐾 · π))↑2)))) |
26 | 16, 25 | mpbid 232 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → (1 + 1) = (2 · ((cos‘(𝐾 · π))↑2))) |
27 | 2t1e2 12425 | . . . . . . 7 ⊢ (2 · 1) = 2 | |
28 | df-2 12325 | . . . . . . 7 ⊢ 2 = (1 + 1) | |
29 | 27, 28 | eqtr2i 2765 | . . . . . 6 ⊢ (1 + 1) = (2 · 1) |
30 | 26, 29 | eqtr3di 2791 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (2 · ((cos‘(𝐾 · π))↑2)) = (2 · 1)) |
31 | 2cnne0 12472 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
32 | mulcan 11896 | . . . . . . 7 ⊢ ((((cos‘(𝐾 · π))↑2) ∈ ℂ ∧ 1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((2 · ((cos‘(𝐾 · π))↑2)) = (2 · 1) ↔ ((cos‘(𝐾 · π))↑2) = 1)) | |
33 | 22, 31, 32 | mp3an23 1455 | . . . . . 6 ⊢ (((cos‘(𝐾 · π))↑2) ∈ ℂ → ((2 · ((cos‘(𝐾 · π))↑2)) = (2 · 1) ↔ ((cos‘(𝐾 · π))↑2) = 1)) |
34 | 19, 33 | syl 17 | . . . . 5 ⊢ (𝐾 ∈ ℤ → ((2 · ((cos‘(𝐾 · π))↑2)) = (2 · 1) ↔ ((cos‘(𝐾 · π))↑2) = 1)) |
35 | 30, 34 | mpbid 232 | . . . 4 ⊢ (𝐾 ∈ ℤ → ((cos‘(𝐾 · π))↑2) = 1) |
36 | sq1 14230 | . . . 4 ⊢ (1↑2) = 1 | |
37 | 35, 36 | eqtr4di 2794 | . . 3 ⊢ (𝐾 ∈ ℤ → ((cos‘(𝐾 · π))↑2) = (1↑2)) |
38 | 1re 11257 | . . . 4 ⊢ 1 ∈ ℝ | |
39 | sqabs 15342 | . . . 4 ⊢ (((cos‘(𝐾 · π)) ∈ ℝ ∧ 1 ∈ ℝ) → (((cos‘(𝐾 · π))↑2) = (1↑2) ↔ (abs‘(cos‘(𝐾 · π))) = (abs‘1))) | |
40 | 17, 38, 39 | sylancl 586 | . . 3 ⊢ (𝐾 ∈ ℤ → (((cos‘(𝐾 · π))↑2) = (1↑2) ↔ (abs‘(cos‘(𝐾 · π))) = (abs‘1))) |
41 | 37, 40 | mpbid 232 | . 2 ⊢ (𝐾 ∈ ℤ → (abs‘(cos‘(𝐾 · π))) = (abs‘1)) |
42 | abs1 15332 | . 2 ⊢ (abs‘1) = 1 | |
43 | 41, 42 | eqtrdi 2792 | 1 ⊢ (𝐾 ∈ ℤ → (abs‘(cos‘(𝐾 · π))) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2939 ‘cfv 6559 (class class class)co 7429 ℂcc 11149 ℝcr 11150 0cc0 11151 1c1 11152 + caddc 11154 · cmul 11156 − cmin 11488 2c2 12317 ℤcz 12609 ↑cexp 14098 abscabs 15269 cosccos 16096 πcpi 16098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-inf2 9677 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-pre-sup 11229 ax-addf 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-se 5636 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-of 7694 df-om 7884 df-1st 8010 df-2nd 8011 df-supp 8182 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-2o 8503 df-er 8741 df-map 8864 df-pm 8865 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9398 df-fi 9447 df-sup 9478 df-inf 9479 df-oi 9546 df-card 9975 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-div 11917 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12875 df-q 12987 df-rp 13031 df-xneg 13150 df-xadd 13151 df-xmul 13152 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-starv 17308 df-sca 17309 df-vsca 17310 df-ip 17311 df-tset 17312 df-ple 17313 df-ds 17315 df-unif 17316 df-hom 17317 df-cco 17318 df-rest 17463 df-topn 17464 df-0g 17482 df-gsum 17483 df-topgen 17484 df-pt 17485 df-prds 17488 df-xrs 17543 df-qtop 17548 df-imas 17549 df-xps 17551 df-mre 17625 df-mrc 17626 df-acs 17628 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-submnd 18793 df-mulg 19082 df-cntz 19331 df-cmn 19796 df-psmet 21348 df-xmet 21349 df-met 21350 df-bl 21351 df-mopn 21352 df-fbas 21353 df-fg 21354 df-cnfld 21357 df-top 22890 df-topon 22907 df-topsp 22929 df-bases 22943 df-cld 23017 df-ntr 23018 df-cls 23019 df-nei 23096 df-lp 23134 df-perf 23135 df-cn 23225 df-cnp 23226 df-haus 23313 df-tx 23560 df-hmeo 23753 df-fil 23844 df-fm 23936 df-flim 23937 df-flf 23938 df-xms 24320 df-ms 24321 df-tms 24322 df-cncf 24894 df-limc 25891 df-dv 25892 |
This theorem is referenced by: (None) |
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