Theorem divdirzi

Description: Distribution of division over addition. (Contributed by NM, 31-Jul-2004)

	Ref	Expression
Hypotheses	divclz.1	|- A e. CC
	divclz.2	|- B e. CC
	divmulz.3	|- C e. CC
Assertion	divdirzi	|- ( C =/= 0 -> ( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) )

Proof

Step	Hyp	Ref	Expression
1		divclz.1	|- A e. CC
2		divclz.2	|- B e. CC
3		divmulz.3	|- C e. CC
4		divdir	|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) )
5	1 2 4	mp3an12	|- ( ( C e. CC /\ C =/= 0 ) -> ( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) )
6	3 5	mpan	|- ( C =/= 0 -> ( ( A + B ) / C ) = ( ( A / C ) + ( B / C ) ) )