Theorem cxpaddd

Description: Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of Gleason p. 135. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	cxp0d.1	|- ( ph -> A e. CC )
	cxpefd.2	|- ( ph -> A =/= 0 )
	cxpefd.3	|- ( ph -> B e. CC )
	cxpaddd.4	|- ( ph -> C e. CC )
Assertion	cxpaddd	|- ( ph -> ( A ^c ( B + C ) ) = ( ( A ^c B ) x. ( A ^c C ) ) )

Proof

Step	Hyp	Ref	Expression
1		cxp0d.1	|- ( ph -> A e. CC )
2		cxpefd.2	|- ( ph -> A =/= 0 )
3		cxpefd.3	|- ( ph -> B e. CC )
4		cxpaddd.4	|- ( ph -> C e. CC )
5		cxpadd	|- ( ( ( A e. CC /\ A =/= 0 ) /\ B e. CC /\ C e. CC ) -> ( A ^c ( B + C ) ) = ( ( A ^c B ) x. ( A ^c C ) ) )
6	1 2 3 4 5	syl211anc	|- ( ph -> ( A ^c ( B + C ) ) = ( ( A ^c B ) x. ( A ^c C ) ) )