Theorem retancld

Description: Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	resincld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	retancld.2	⊢ ( 𝜑 → ( cos ‘ 𝐴 ) ≠ 0 )
Assertion	retancld	⊢ ( 𝜑 → ( tan ‘ 𝐴 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		resincld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		retancld.2	⊢ ( 𝜑 → ( cos ‘ 𝐴 ) ≠ 0 )
3		retancl	⊢ ( ( 𝐴 ∈ ℝ ∧ ( cos ‘ 𝐴 ) ≠ 0 ) → ( tan ‘ 𝐴 ) ∈ ℝ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( tan ‘ 𝐴 ) ∈ ℝ )