Theorem tancl

Description: The closure of the tangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014)

	Ref	Expression
Assertion	tancl	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( tan ‘ 𝐴 ) ∈ ℂ )

Proof

Step	Hyp	Ref	Expression
1		tanval	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( tan ‘ 𝐴 ) = ( ( sin ‘ 𝐴 ) / ( cos ‘ 𝐴 ) ) )
2		sincl	⊢ ( 𝐴 ∈ ℂ → ( sin ‘ 𝐴 ) ∈ ℂ )
3	2	adantr	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( sin ‘ 𝐴 ) ∈ ℂ )
4		coscl	⊢ ( 𝐴 ∈ ℂ → ( cos ‘ 𝐴 ) ∈ ℂ )
5	4	adantr	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( cos ‘ 𝐴 ) ∈ ℂ )
6		simpr	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( cos ‘ 𝐴 ) ≠ 0 )
7	3 5 6	divcld	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( ( sin ‘ 𝐴 ) / ( cos ‘ 𝐴 ) ) ∈ ℂ )
8	1 7	eqeltrd	⊢ ( ( 𝐴 ∈ ℂ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( tan ‘ 𝐴 ) ∈ ℂ )