Theorem divsubdird

Description: Distribution of division over subtraction. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	|- ( ph -> A e. CC )
	divcld.2	|- ( ph -> B e. CC )
	divmuld.3	|- ( ph -> C e. CC )
	divassd.4	|- ( ph -> C =/= 0 )
Assertion	divsubdird	|- ( ph -> ( ( A - B ) / C ) = ( ( A / C ) - ( B / C ) ) )

Proof

Step	Hyp	Ref	Expression
1		div1d.1	|- ( ph -> A e. CC )
2		divcld.2	|- ( ph -> B e. CC )
3		divmuld.3	|- ( ph -> C e. CC )
4		divassd.4	|- ( ph -> C =/= 0 )
5		divsubdir	|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A - B ) / C ) = ( ( A / C ) - ( B / C ) ) )
6	1 2 3 4 5	syl112anc	|- ( ph -> ( ( A - B ) / C ) = ( ( A / C ) - ( B / C ) ) )