Theorem tancld

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

	Ref	Expression
Hypotheses	sincld.1	|- ( ph -> A e. CC )
	tancld.2	|- ( ph -> ( cos ` A ) =/= 0 )
Assertion	tancld	|- ( ph -> ( tan ` A ) e. CC )

Proof

Step	Hyp	Ref	Expression
1		sincld.1	|- ( ph -> A e. CC )
2		tancld.2	|- ( ph -> ( cos ` A ) =/= 0 )
3		tancl	|- ( ( A e. CC /\ ( cos ` A ) =/= 0 ) -> ( tan ` A ) e. CC )
4	1 2 3	syl2anc	|- ( ph -> ( tan ` A ) e. CC )