MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-cos Structured version   Visualization version   GIF version

Definition df-cos 15021
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 15015 . 2 class cos
2 vx . . 3 setvar 𝑥
3 cc 10219 . . 3 class
4 ci 10223 . . . . . . 7 class i
52cv 1636 . . . . . . 7 class 𝑥
6 cmul 10226 . . . . . . 7 class ·
74, 5, 6co 6874 . . . . . 6 class (i · 𝑥)
8 ce 15012 . . . . . 6 class exp
97, 8cfv 6101 . . . . 5 class (exp‘(i · 𝑥))
104cneg 10552 . . . . . . 7 class -i
1110, 5, 6co 6874 . . . . . 6 class (-i · 𝑥)
1211, 8cfv 6101 . . . . 5 class (exp‘(-i · 𝑥))
13 caddc 10224 . . . . 5 class +
149, 12, 13co 6874 . . . 4 class ((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥)))
15 c2 11356 . . . 4 class 2
16 cdiv 10969 . . . 4 class /
1714, 15, 16co 6874 . . 3 class (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2)
182, 3, 17cmpt 4923 . 2 class (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
191, 18wceq 1637 1 wff cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
Colors of variables: wff setvar class
This definition is referenced by:  cosval  15073  cosf  15075  dvsincos  23958  coscn  24413
  Copyright terms: Public domain W3C validator