Theorem zexpcld

Description: Closure of exponentiation of integers, deductive form. (Contributed by SN, 15-Sep-2024)

	Ref	Expression
Hypotheses	zexpcld.1	|- ( ph -> A e. ZZ )
	zexpcld.2	|- ( ph -> N e. NN0 )
Assertion	zexpcld	|- ( ph -> ( A ^ N ) e. ZZ )

Proof

Step	Hyp	Ref	Expression
1		zexpcld.1	|- ( ph -> A e. ZZ )
2		zexpcld.2	|- ( ph -> N e. NN0 )
3		zexpcl	|- ( ( A e. ZZ /\ N e. NN0 ) -> ( A ^ N ) e. ZZ )
4	1 2 3	syl2anc	|- ( ph -> ( A ^ N ) e. ZZ )