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 ∈ ℂ
	joinlmulsubmuli.2	⊢ 𝐵 ∈ ℂ
	joinlmulsubmuli.3	⊢ 𝐶 ∈ ℂ
	joinlmulsubmuli.4	⊢ ( ( 𝐴 · 𝐵 ) − ( 𝐶 · 𝐵 ) ) = 𝐷
Assertion	joinlmulsubmuli	⊢ ( ( 𝐴 − 𝐶 ) · 𝐵 ) = 𝐷

Proof

Step	Hyp	Ref	Expression
1		joinlmulsubmuli.1	⊢ 𝐴 ∈ ℂ
2		joinlmulsubmuli.2	⊢ 𝐵 ∈ ℂ
3		joinlmulsubmuli.3	⊢ 𝐶 ∈ ℂ
4		joinlmulsubmuli.4	⊢ ( ( 𝐴 · 𝐵 ) − ( 𝐶 · 𝐵 ) ) = 𝐷
5	1 3 2	subdiri	⊢ ( ( 𝐴 − 𝐶 ) · 𝐵 ) = ( ( 𝐴 · 𝐵 ) − ( 𝐶 · 𝐵 ) )
6	5 4	eqtri	⊢ ( ( 𝐴 − 𝐶 ) · 𝐵 ) = 𝐷