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| Mirrors > Home > MPE Home > Th. List > cosval | Structured version Visualization version GIF version | ||
| Description: Value of the cosine function. (Contributed by NM, 14-Mar-2005.) (Revised by Mario Carneiro, 10-Nov-2013.) | 
| Ref | Expression | 
|---|---|
| cosval | ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | oveq2 7440 | . . . . 5 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴)) | |
| 2 | 1 | fveq2d 6909 | . . . 4 ⊢ (𝑥 = 𝐴 → (exp‘(i · 𝑥)) = (exp‘(i · 𝐴))) | 
| 3 | oveq2 7440 | . . . . 5 ⊢ (𝑥 = 𝐴 → (-i · 𝑥) = (-i · 𝐴)) | |
| 4 | 3 | fveq2d 6909 | . . . 4 ⊢ (𝑥 = 𝐴 → (exp‘(-i · 𝑥)) = (exp‘(-i · 𝐴))) | 
| 5 | 2, 4 | oveq12d 7450 | . . 3 ⊢ (𝑥 = 𝐴 → ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) = ((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴)))) | 
| 6 | 5 | oveq1d 7447 | . 2 ⊢ (𝑥 = 𝐴 → (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2)) | 
| 7 | df-cos 16107 | . 2 ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)) | |
| 8 | ovex 7465 | . 2 ⊢ (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2) ∈ V | |
| 9 | 6, 7, 8 | fvmpt 7015 | 1 ⊢ (𝐴 ∈ ℂ → (cos‘𝐴) = (((exp‘(i · 𝐴)) + (exp‘(-i · 𝐴))) / 2)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 ici 11158 + caddc 11159 · cmul 11161 -cneg 11494 / cdiv 11921 2c2 12322 expce 16098 cosccos 16101 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-cos 16107 | 
| This theorem is referenced by: tanval2 16170 tanval3 16171 recosval 16173 cosneg 16184 efival 16189 coshval 16192 cosadd 16202 cosper 26525 pige3ALT 26563 cosargd 26651 asinsin 26936 cosasin 26948 cosatan 26965 | 
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