Theorem expsubd

Description: Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	expcld.1	|- ( ph -> A e. CC )
	sqrecd.1	|- ( ph -> A =/= 0 )
	expclzd.3	|- ( ph -> N e. ZZ )
	expsubd.3	|- ( ph -> M e. ZZ )
Assertion	expsubd	|- ( ph -> ( A ^ ( M - N ) ) = ( ( A ^ M ) / ( A ^ N ) ) )

Proof

Step	Hyp	Ref	Expression
1		expcld.1	|- ( ph -> A e. CC )
2		sqrecd.1	|- ( ph -> A =/= 0 )
3		expclzd.3	|- ( ph -> N e. ZZ )
4		expsubd.3	|- ( ph -> M e. ZZ )
5		expsub	|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M - N ) ) = ( ( A ^ M ) / ( A ^ N ) ) )
6	1 2 4 3 5	syl22anc	|- ( ph -> ( A ^ ( M - N ) ) = ( ( A ^ M ) / ( A ^ N ) ) )