Theorem retancld

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

	Ref	Expression
Hypotheses	resincld.1	|- ( ph -> A e. RR )
	retancld.2	|- ( ph -> ( cos ` A ) =/= 0 )
Assertion	retancld	|- ( ph -> ( tan ` A ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		resincld.1	|- ( ph -> A e. RR )
2		retancld.2	|- ( ph -> ( cos ` A ) =/= 0 )
3		retancl	|- ( ( A e. RR /\ ( cos ` A ) =/= 0 ) -> ( tan ` A ) e. RR )
4	1 2 3	syl2anc	|- ( ph -> ( tan ` A ) e. RR )