Theorem seccl

Description: The closure of the secant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014)

	Ref	Expression
Assertion	seccl	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( sec ‘ 𝐴 ) ∈ ℂ )

Proof

Step	Hyp	Ref	Expression
1		secval	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( sec ‘ 𝐴 ) = ( 1 / ( cos ‘ 𝐴 ) ) )
2		coscl	⊢ ( 𝐴 ∈ ℂ → ( cos ‘ 𝐴 ) ∈ ℂ )
3	2	adantr	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( cos ‘ 𝐴 ) ∈ ℂ )
4		simpr	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( cos ‘ 𝐴 ) ≠ 0 )
5	3 4	reccld	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( 1 / ( cos ‘ 𝐴 ) ) ∈ ℂ )
6	1 5	eqeltrd	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( sec ‘ 𝐴 ) ∈ ℂ )