Theorem joinlmulsubmuli

Description: Join AB-CB into (A-C) on LHS. (Contributed by David A. Wheeler, 11-Oct-2018)

	Ref	Expression
Hypotheses	joinlmulsubmuli.1	$$⊢A\in ℂ$$
	joinlmulsubmuli.2	$$⊢B\in ℂ$$
	joinlmulsubmuli.3	$$⊢C\in ℂ$$
	joinlmulsubmuli.4	$$⊢AB-CB=D$$
Assertion	joinlmulsubmuli	$$⊢\left(A-C\right)B=D$$

Proof

Step	Hyp	Ref	Expression
1		joinlmulsubmuli.1	$$⊢A\in ℂ$$
2		joinlmulsubmuli.2	$$⊢B\in ℂ$$
3		joinlmulsubmuli.3	$$⊢C\in ℂ$$
4		joinlmulsubmuli.4	$$⊢AB-CB=D$$
5	1 3 2	subdiri	$$⊢\left(A-C\right)B=AB-CB$$
6	5 4	eqtri	$$⊢\left(A-C\right)B=D$$