ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-cos GIF version

Definition df-cos 11691
Description: Define the cosine function. (Contributed by NM, 14-Mar-2005.)
Assertion
Ref Expression
df-cos cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))

Detailed syntax breakdown of Definition df-cos
StepHypRef Expression
1 ccos 11685 . 2 class cos
2 vx . . 3 setvar 𝑥
3 cc 7839 . . 3 class
4 ci 7843 . . . . . . 7 class i
52cv 1363 . . . . . . 7 class 𝑥
6 cmul 7846 . . . . . . 7 class ·
74, 5, 6co 5896 . . . . . 6 class (i · 𝑥)
8 ce 11682 . . . . . 6 class exp
97, 8cfv 5235 . . . . 5 class (exp‘(i · 𝑥))
104cneg 8159 . . . . . . 7 class -i
1110, 5, 6co 5896 . . . . . 6 class (-i · 𝑥)
1211, 8cfv 5235 . . . . 5 class (exp‘(-i · 𝑥))
13 caddc 7844 . . . . 5 class +
149, 12, 13co 5896 . . . 4 class ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))
15 c2 9000 . . . 4 class 2
16 cdiv 8659 . . . 4 class /
1714, 15, 16co 5896 . . 3 class (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)
182, 3, 17cmpt 4079 . 2 class (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
191, 18wceq 1364 1 wff cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
Colors of variables: wff set class
This definition is referenced by:  cosval  11743  cosf  11745  coscn  14648
  Copyright terms: Public domain W3C validator