![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 12594 | . . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ) | |
2 | 2cn 12318 | . . . . . . . . . . 11 ⊢ 2 ∈ ℂ | |
3 | picn 26407 | . . . . . . . . . . 11 ⊢ π ∈ ℂ | |
4 | mul12 11410 | . . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℂ ∧ 2 ∈ ℂ ∧ π ∈ ℂ) → (𝐾 · (2 · π)) = (2 · (𝐾 · π))) | |
5 | 2, 3, 4 | mp3an23 1450 | . . . . . . . . . 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 6901 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → (cos‘(𝐾 · (2 · π))) = (cos‘(2 · (𝐾 · π)))) |
8 | cos2kpi 26432 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → (cos‘(𝐾 · (2 · π))) = 1) | |
9 | zre 12593 | . . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
10 | pire 26406 | . . . . . . . . . . 11 ⊢ π ∈ ℝ | |
11 | remulcl 11224 | . . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℝ ∧ π ∈ ℝ) → (𝐾 · π) ∈ ℝ) | |
12 | 9, 10, 11 | sylancl 585 | . . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (𝐾 · π) ∈ ℝ) |
13 | 12 | recnd 11273 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℤ → (𝐾 · π) ∈ ℂ) |
14 | cos2t 16155 | . . . . . . . . 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 2777 | . . . . . . 7 ⊢ (𝐾 ∈ ℤ → ((2 · ((cos‘(𝐾 · π))↑2)) − 1) = 1) |
17 | 12 | recoscld 16121 | . . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (cos‘(𝐾 · π)) ∈ ℝ) |
18 | 17 | recnd 11273 | . . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (cos‘(𝐾 · π)) ∈ ℂ) |
19 | 18 | sqcld 14141 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℤ → ((cos‘(𝐾 · π))↑2) ∈ ℂ) |
20 | mulcl 11223 | . . . . . . . . 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 11197 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
23 | subadd 11494 | . . . . . . . . 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 1450 | . . . . . . . 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 231 | . . . . . 6 ⊢ (𝐾 ∈ ℤ → (1 + 1) = (2 · ((cos‘(𝐾 · π))↑2))) |
27 | 2t1e2 12406 | . . . . . . 7 ⊢ (2 · 1) = 2 | |
28 | df-2 12306 | . . . . . . 7 ⊢ 2 = (1 + 1) | |
29 | 27, 28 | eqtr2i 2757 | . . . . . 6 ⊢ (1 + 1) = (2 · 1) |
30 | 26, 29 | eqtr3di 2783 | . . . . 5 ⊢ (𝐾 ∈ ℤ → (2 · ((cos‘(𝐾 · π))↑2)) = (2 · 1)) |
31 | 2cnne0 12453 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
32 | mulcan 11882 | . . . . . . 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 1450 | . . . . . 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 231 | . . . 4 ⊢ (𝐾 ∈ ℤ → ((cos‘(𝐾 · π))↑2) = 1) |
36 | sq1 14191 | . . . 4 ⊢ (1↑2) = 1 | |
37 | 35, 36 | eqtr4di 2786 | . . 3 ⊢ (𝐾 ∈ ℤ → ((cos‘(𝐾 · π))↑2) = (1↑2)) |
38 | 1re 11245 | . . . 4 ⊢ 1 ∈ ℝ | |
39 | sqabs 15287 | . . . 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 231 | . 2 ⊢ (𝐾 ∈ ℤ → (abs‘(cos‘(𝐾 · π))) = (abs‘1)) |
42 | abs1 15277 | . 2 ⊢ (abs‘1) = 1 | |
43 | 41, 42 | eqtrdi 2784 | 1 ⊢ (𝐾 ∈ ℤ → (abs‘(cos‘(𝐾 · π))) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ‘cfv 6548 (class class class)co 7420 ℂcc 11137 ℝcr 11138 0cc0 11139 1c1 11140 + caddc 11142 · cmul 11144 − cmin 11475 2c2 12298 ℤcz 12589 ↑cexp 14059 abscabs 15214 cosccos 16041 πcpi 16043 |
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 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ioc 13362 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-seq 14000 df-exp 14060 df-fac 14266 df-bc 14295 df-hash 14323 df-shft 15047 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-limsup 15448 df-clim 15465 df-rlim 15466 df-sum 15666 df-ef 16044 df-sin 16046 df-cos 16047 df-pi 16049 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-mulg 19024 df-cntz 19268 df-cmn 19737 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cn 23144 df-cnp 23145 df-haus 23232 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-cncf 24811 df-limc 25808 df-dv 25809 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |