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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 12609 | . . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ) | |
2 | 2cn 12332 | . . . . . . . . . . 11 ⊢ 2 ∈ ℂ | |
3 | picn 26497 | . . . . . . . . . . 11 ⊢ π ∈ ℂ | |
4 | mul12 11417 | . . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℂ ∧ 2 ∈ ℂ ∧ π ∈ ℂ) → (𝐾 · (2 · π)) = (2 · (𝐾 · π))) | |
5 | 2, 3, 4 | mp3an23 1451 | . . . . . . . . . 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 6905 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → (cos‘(𝐾 · (2 · π))) = (cos‘(2 · (𝐾 · π)))) |
8 | cos2kpi 26522 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → (cos‘(𝐾 · (2 · π))) = 1) | |
9 | zre 12608 | . . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
10 | pire 26496 | . . . . . . . . . . 11 ⊢ π ∈ ℝ | |
11 | remulcl 11231 | . . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℝ ∧ π ∈ ℝ) → (𝐾 · π) ∈ ℝ) | |
12 | 9, 10, 11 | sylancl 585 | . . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (𝐾 · π) ∈ ℝ) |
13 | 12 | recnd 11280 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℤ → (𝐾 · π) ∈ ℂ) |
14 | cos2t 16200 | . . . . . . . . 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 2782 | . . . . . . 7 ⊢ (𝐾 ∈ ℤ → ((2 · ((cos‘(𝐾 · π))↑2)) − 1) = 1) |
17 | 12 | recoscld 16166 | . . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (cos‘(𝐾 · π)) ∈ ℝ) |
18 | 17 | recnd 11280 | . . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (cos‘(𝐾 · π)) ∈ ℂ) |
19 | 18 | sqcld 14170 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℤ → ((cos‘(𝐾 · π))↑2) ∈ ℂ) |
20 | mulcl 11230 | . . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ ((cos‘(𝐾 · π))↑2) ∈ ℂ) → (2 · ((cos‘(𝐾 · π))↑2)) ∈ ℂ) | |
21 | 2, 19, 20 | sylancr 586 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → (2 · ((cos‘(𝐾 · π))↑2)) ∈ ℂ) |
22 | ax-1cn 11204 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
23 | subadd 11502 | . . . . . . . . 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 1451 | . . . . . . . 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 12420 | . . . . . . 7 ⊢ (2 · 1) = 2 | |
28 | df-2 12320 | . . . . . . 7 ⊢ 2 = (1 + 1) | |
29 | 27, 28 | eqtr2i 2762 | . . . . . 6 ⊢ (1 + 1) = (2 · 1) |
30 | 26, 29 | eqtr3di 2788 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (2 · ((cos‘(𝐾 · π))↑2)) = (2 · 1)) |
31 | 2cnne0 12467 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
32 | mulcan 11891 | . . . . . . 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 1451 | . . . . . 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 14220 | . . . 4 ⊢ (1↑2) = 1 | |
37 | 35, 36 | eqtr4di 2791 | . . 3 ⊢ (𝐾 ∈ ℤ → ((cos‘(𝐾 · π))↑2) = (1↑2)) |
38 | 1re 11252 | . . . 4 ⊢ 1 ∈ ℝ | |
39 | sqabs 15332 | . . . 4 ⊢ (((cos‘(𝐾 · π)) ∈ ℝ ∧ 1 ∈ ℝ) → (((cos‘(𝐾 · π))↑2) = (1↑2) ↔ (abs‘(cos‘(𝐾 · π))) = (abs‘1))) | |
40 | 17, 38, 39 | sylancl 585 | . . 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 15322 | . 2 ⊢ (abs‘1) = 1 | |
43 | 41, 42 | eqtrdi 2789 | 1 ⊢ (𝐾 ∈ ℤ → (abs‘(cos‘(𝐾 · π))) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1535 ∈ wcel 2104 ≠ wne 2936 ‘cfv 6558 (class class class)co 7425 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 − cmin 11483 2c2 12312 ℤcz 12604 ↑cexp 14088 abscabs 15259 cosccos 16086 πcpi 16088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4915 df-int 4954 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-of 7691 df-om 7881 df-1st 8007 df-2nd 8008 df-supp 8179 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-1o 8499 df-2o 8500 df-er 8738 df-map 8861 df-pm 8862 df-ixp 8931 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11485 df-neg 11486 df-div 11912 df-nn 12258 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12518 df-z 12605 df-dec 12725 df-uz 12870 df-q 12982 df-rp 13026 df-xneg 13145 df-xadd 13146 df-xmul 13147 df-ioo 13381 df-ioc 13382 df-ico 13383 df-icc 13384 df-fz 13538 df-fzo 13682 df-fl 13818 df-seq 14029 df-exp 14089 df-fac 14299 df-bc 14328 df-hash 14356 df-shft 15092 df-cj 15124 df-re 15125 df-im 15126 df-sqrt 15260 df-abs 15261 df-limsup 15493 df-clim 15510 df-rlim 15511 df-sum 15709 df-ef 16089 df-sin 16091 df-cos 16092 df-pi 16094 df-struct 17170 df-sets 17187 df-slot 17205 df-ndx 17217 df-base 17235 df-ress 17264 df-plusg 17300 df-mulr 17301 df-starv 17302 df-sca 17303 df-vsca 17304 df-ip 17305 df-tset 17306 df-ple 17307 df-ds 17309 df-unif 17310 df-hom 17311 df-cco 17312 df-rest 17458 df-topn 17459 df-0g 17477 df-gsum 17478 df-topgen 17479 df-pt 17480 df-prds 17483 df-xrs 17538 df-qtop 17543 df-imas 17544 df-xps 17546 df-mre 17620 df-mrc 17621 df-acs 17623 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18795 df-mulg 19084 df-cntz 19333 df-cmn 19800 df-psmet 21355 df-xmet 21356 df-met 21357 df-bl 21358 df-mopn 21359 df-fbas 21360 df-fg 21361 df-cnfld 21364 df-top 22897 df-topon 22914 df-topsp 22936 df-bases 22950 df-cld 23024 df-ntr 23025 df-cls 23026 df-nei 23103 df-lp 23141 df-perf 23142 df-cn 23232 df-cnp 23233 df-haus 23320 df-tx 23567 df-hmeo 23760 df-fil 23851 df-fm 23943 df-flim 23944 df-flf 23945 df-xms 24327 df-ms 24328 df-tms 24329 df-cncf 24899 df-limc 25897 df-dv 25898 |
This theorem is referenced by: (None) |
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