Theorem expdivd

Description: Nonnegative integer exponentiation of a quotient. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	expcld.1	|- ( ph -> A e. CC )
	mulexpd.2	|- ( ph -> B e. CC )
	sqdivd.3	|- ( ph -> B =/= 0 )
	expdivd.3	|- ( ph -> N e. NN0 )
Assertion	expdivd	|- ( ph -> ( ( A / B ) ^ N ) = ( ( A ^ N ) / ( B ^ N ) ) )

Proof

Step	Hyp	Ref	Expression
1		expcld.1	|- ( ph -> A e. CC )
2		mulexpd.2	|- ( ph -> B e. CC )
3		sqdivd.3	|- ( ph -> B =/= 0 )
4		expdivd.3	|- ( ph -> N e. NN0 )
5		expdiv	|- ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) /\ N e. NN0 ) -> ( ( A / B ) ^ N ) = ( ( A ^ N ) / ( B ^ N ) ) )
6	1 2 3 4 5	syl121anc	|- ( ph -> ( ( A / B ) ^ N ) = ( ( A ^ N ) / ( B ^ N ) ) )