Theorem cosf

Description: Domain and codomain of the cosine function. (Contributed by Paul Chapman, 22-Oct-2007) (Revised by Mario Carneiro, 30-Apr-2014)

	Ref	Expression
Assertion	cosf	⊢ cos : ℂ ⟶ ℂ

Proof

Step	Hyp	Ref	Expression
1		df-cos	⊢ cos = ( 𝑥 ∈ ℂ ↦ ( ( ( exp ‘ ( i · 𝑥 ) ) + ( exp ‘ ( - i · 𝑥 ) ) ) / 2 ) )
2		ax-icn	⊢ i ∈ ℂ
3		mulcl	⊢ ( ( i ∈ ℂ ∧ 𝑥 ∈ ℂ ) → ( i · 𝑥 ) ∈ ℂ )
4	2 3	mpan	⊢ ( 𝑥 ∈ ℂ → ( i · 𝑥 ) ∈ ℂ )
5		efcl	⊢ ( ( i · 𝑥 ) ∈ ℂ → ( exp ‘ ( i · 𝑥 ) ) ∈ ℂ )
6	4 5	syl	⊢ ( 𝑥 ∈ ℂ → ( exp ‘ ( i · 𝑥 ) ) ∈ ℂ )
7		negicn	⊢ - i ∈ ℂ
8		mulcl	⊢ ( ( - i ∈ ℂ ∧ 𝑥 ∈ ℂ ) → ( - i · 𝑥 ) ∈ ℂ )
9	7 8	mpan	⊢ ( 𝑥 ∈ ℂ → ( - i · 𝑥 ) ∈ ℂ )
10		efcl	⊢ ( ( - i · 𝑥 ) ∈ ℂ → ( exp ‘ ( - i · 𝑥 ) ) ∈ ℂ )
11	9 10	syl	⊢ ( 𝑥 ∈ ℂ → ( exp ‘ ( - i · 𝑥 ) ) ∈ ℂ )
12	6 11	addcld	⊢ ( 𝑥 ∈ ℂ → ( ( exp ‘ ( i · 𝑥 ) ) + ( exp ‘ ( - i · 𝑥 ) ) ) ∈ ℂ )
13	12	halfcld	⊢ ( 𝑥 ∈ ℂ → ( ( ( exp ‘ ( i · 𝑥 ) ) + ( exp ‘ ( - i · 𝑥 ) ) ) / 2 ) ∈ ℂ )
14	1 13	fmpti	⊢ cos : ℂ ⟶ ℂ