Theorem muldivdid

Description: Distribution of division over addition with a multiplication. (Contributed by Thierry Arnoux, 6-Jul-2025)

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

Proof

Step	Hyp	Ref	Expression
1		muldivdid.1	|- ( ph -> A e. CC )
2		muldivdid.2	|- ( ph -> B e. CC )
3		muldivdid.3	|- ( ph -> C e. CC )
4		muldivdid.4	|- ( ph -> B =/= 0 )
5	1 2	mulcomd	|- ( ph -> ( A x. B ) = ( B x. A ) )
6	5	oveq1d	|- ( ph -> ( ( A x. B ) + C ) = ( ( B x. A ) + C ) )
7	6	oveq1d	|- ( ph -> ( ( ( A x. B ) + C ) / B ) = ( ( ( B x. A ) + C ) / B ) )
8		muldivdir	|- ( ( A e. CC /\ C e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( ( B x. A ) + C ) / B ) = ( A + ( C / B ) ) )
9	1 3 2 4 8	syl112anc	|- ( ph -> ( ( ( B x. A ) + C ) / B ) = ( A + ( C / B ) ) )
10	7 9	eqtrd	|- ( ph -> ( ( ( A x. B ) + C ) / B ) = ( A + ( C / B ) ) )