Theorem cosval

Description: Value of the cosine function. (Contributed by NM, 14-Mar-2005) (Revised by Mario Carneiro, 10-Nov-2013)

	Ref	Expression
Assertion	cosval	⊢ ( 𝐴 ∈ ℂ → ( cos ‘ 𝐴 ) = ( ( ( exp ‘ ( i · 𝐴 ) ) + ( exp ‘ ( - i · 𝐴 ) ) ) / 2 ) )

Proof

Step	Hyp	Ref	Expression
1		oveq2	⊢ ( 𝑥 = 𝐴 → ( i · 𝑥 ) = ( i · 𝐴 ) )
2	1	fveq2d	⊢ ( 𝑥 = 𝐴 → ( exp ‘ ( i · 𝑥 ) ) = ( exp ‘ ( i · 𝐴 ) ) )
3		oveq2	⊢ ( 𝑥 = 𝐴 → ( - i · 𝑥 ) = ( - i · 𝐴 ) )
4	3	fveq2d	⊢ ( 𝑥 = 𝐴 → ( exp ‘ ( - i · 𝑥 ) ) = ( exp ‘ ( - i · 𝐴 ) ) )
5	2 4	oveq12d	⊢ ( 𝑥 = 𝐴 → ( ( exp ‘ ( i · 𝑥 ) ) + ( exp ‘ ( - i · 𝑥 ) ) ) = ( ( exp ‘ ( i · 𝐴 ) ) + ( exp ‘ ( - i · 𝐴 ) ) ) )
6	5	oveq1d	⊢ ( 𝑥 = 𝐴 → ( ( ( exp ‘ ( i · 𝑥 ) ) + ( exp ‘ ( - i · 𝑥 ) ) ) / 2 ) = ( ( ( exp ‘ ( i · 𝐴 ) ) + ( exp ‘ ( - i · 𝐴 ) ) ) / 2 ) )
7		df-cos	⊢ cos = ( 𝑥 ∈ ℂ ↦ ( ( ( exp ‘ ( i · 𝑥 ) ) + ( exp ‘ ( - i · 𝑥 ) ) ) / 2 ) )
8		ovex	⊢ ( ( ( exp ‘ ( i · 𝐴 ) ) + ( exp ‘ ( - i · 𝐴 ) ) ) / 2 ) ∈ V
9	6 7 8	fvmpt	⊢ ( 𝐴 ∈ ℂ → ( cos ‘ 𝐴 ) = ( ( ( exp ‘ ( i · 𝐴 ) ) + ( exp ‘ ( - i · 𝐴 ) ) ) / 2 ) )