Theorem reexpclzd

Description: Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpexpclzd.1	|- ( ph -> A e. RR )
	rpexpclzd.2	|- ( ph -> A =/= 0 )
	rpexpclzd.3	|- ( ph -> N e. ZZ )
Assertion	reexpclzd	|- ( ph -> ( A ^ N ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		rpexpclzd.1	|- ( ph -> A e. RR )
2		rpexpclzd.2	|- ( ph -> A =/= 0 )
3		rpexpclzd.3	|- ( ph -> N e. ZZ )
4		reexpclz	|- ( ( A e. RR /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ N ) e. RR )
5	1 2 3 4	syl3anc	|- ( ph -> ( A ^ N ) e. RR )