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Mirrors > Home > ILE Home > Th. List > cosper | GIF version |
Description: The cosine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.) |
Ref | Expression |
---|---|
cosper | ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℤ) → (cos‘(𝐴 + (𝐾 · (2 · π)))) = (cos‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cosval 11417 | . 2 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2)) | |
2 | cosval 11417 | . 2 ⊢ ((𝐴 + (𝐾 · (2 · π))) ∈ ℂ → (cos‘(𝐴 + (𝐾 · (2 · π)))) = (((exp‘(i · (𝐴 + (𝐾 · (2 · π))))) + (exp‘(-i · (𝐴 + (𝐾 · (2 · π)))))) / 2)) | |
3 | 1, 2 | sinperlem 12899 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℤ) → (cos‘(𝐴 + (𝐾 · (2 · π)))) = (cos‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ‘cfv 5123 (class class class)co 5774 ℂcc 7625 + caddc 7630 · cmul 7632 2c2 8778 ℤcz 9061 cosccos 11358 πcpi 11360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-mulrcl 7726 ax-addcom 7727 ax-mulcom 7728 ax-addass 7729 ax-mulass 7730 ax-distr 7731 ax-i2m1 7732 ax-0lt1 7733 ax-1rid 7734 ax-0id 7735 ax-rnegex 7736 ax-precex 7737 ax-cnre 7738 ax-pre-ltirr 7739 ax-pre-ltwlin 7740 ax-pre-lttrn 7741 ax-pre-apti 7742 ax-pre-ltadd 7743 ax-pre-mulgt0 7744 ax-pre-mulext 7745 ax-arch 7746 ax-caucvg 7747 ax-pre-suploc 7748 ax-addf 7749 ax-mulf 7750 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-disj 3907 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-of 5982 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-map 6544 df-pm 6545 df-en 6635 df-dom 6636 df-fin 6637 df-sup 6871 df-inf 6872 df-pnf 7809 df-mnf 7810 df-xr 7811 df-ltxr 7812 df-le 7813 df-sub 7942 df-neg 7943 df-reap 8344 df-ap 8351 df-div 8440 df-inn 8728 df-2 8786 df-3 8787 df-4 8788 df-5 8789 df-6 8790 df-7 8791 df-8 8792 df-9 8793 df-n0 8985 df-z 9062 df-uz 9334 df-q 9419 df-rp 9449 df-xneg 9566 df-xadd 9567 df-ioo 9682 df-ioc 9683 df-ico 9684 df-icc 9685 df-fz 9798 df-fzo 9927 df-seqfrec 10226 df-exp 10300 df-fac 10479 df-bc 10501 df-ihash 10529 df-shft 10594 df-cj 10621 df-re 10622 df-im 10623 df-rsqrt 10777 df-abs 10778 df-clim 11055 df-sumdc 11130 df-ef 11361 df-sin 11363 df-cos 11364 df-pi 11366 df-rest 12132 df-topgen 12151 df-psmet 12166 df-xmet 12167 df-met 12168 df-bl 12169 df-mopn 12170 df-top 12175 df-topon 12188 df-bases 12220 df-ntr 12275 df-cn 12367 df-cnp 12368 df-tx 12432 df-cncf 12737 df-limced 12804 df-dvap 12805 |
This theorem is referenced by: cos2kpi 12903 cos2pim 12905 cosq34lt1 12941 |
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