Theorem csccl

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

	Ref	Expression
Assertion	csccl	⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) ≠ 0 ) → ( csc ‘ 𝐴 ) ∈ ℂ )

Proof

Step	Hyp	Ref	Expression
1		cscval	⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) ≠ 0 ) → ( csc ‘ 𝐴 ) = ( 1 / ( sin ‘ 𝐴 ) ) )
2		sincl	⊢ ( 𝐴 ∈ ℂ → ( sin ‘ 𝐴 ) ∈ ℂ )
3	2	adantr	⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) ≠ 0 ) → ( sin ‘ 𝐴 ) ∈ ℂ )
4		simpr	⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) ≠ 0 ) → ( sin ‘ 𝐴 ) ≠ 0 )
5	3 4	reccld	⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) ≠ 0 ) → ( 1 / ( sin ‘ 𝐴 ) ) ∈ ℂ )
6	1 5	eqeltrd	⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) ≠ 0 ) → ( csc ‘ 𝐴 ) ∈ ℂ )