Theorem rpcoshcl

Description: The hyperbolic cosine of a real number is a positive real. (Contributed by Mario Carneiro, 4-Apr-2015)

	Ref	Expression
Assertion	rpcoshcl	⊢ ( 𝐴 ∈ ℝ → ( cos ‘ ( i · 𝐴 ) ) ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
2		coshval	⊢ ( 𝐴 ∈ ℂ → ( cos ‘ ( i · 𝐴 ) ) = ( ( ( exp ‘ 𝐴 ) + ( exp ‘ - 𝐴 ) ) / 2 ) )
3	1 2	syl	⊢ ( 𝐴 ∈ ℝ → ( cos ‘ ( i · 𝐴 ) ) = ( ( ( exp ‘ 𝐴 ) + ( exp ‘ - 𝐴 ) ) / 2 ) )
4		rpefcl	⊢ ( 𝐴 ∈ ℝ → ( exp ‘ 𝐴 ) ∈ ℝ+ )
5		renegcl	⊢ ( 𝐴 ∈ ℝ → - 𝐴 ∈ ℝ )
6	5	rpefcld	⊢ ( 𝐴 ∈ ℝ → ( exp ‘ - 𝐴 ) ∈ ℝ+ )
7	4 6	rpaddcld	⊢ ( 𝐴 ∈ ℝ → ( ( exp ‘ 𝐴 ) + ( exp ‘ - 𝐴 ) ) ∈ ℝ+ )
8	7	rphalfcld	⊢ ( 𝐴 ∈ ℝ → ( ( ( exp ‘ 𝐴 ) + ( exp ‘ - 𝐴 ) ) / 2 ) ∈ ℝ+ )
9	3 8	eqeltrd	⊢ ( 𝐴 ∈ ℝ → ( cos ‘ ( i · 𝐴 ) ) ∈ ℝ+ )