This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )