Theorem joinlmulsubmuld

Description: Join AB-CB into (A-C) on LHS. (Contributed by David A. Wheeler, 15-Oct-2018)

	Ref	Expression
Hypotheses	joinlmulsubmuld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	joinlmulsubmuld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	joinlmulsubmuld.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	joinlmulsubmuld.4	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) − ( 𝐶 · 𝐵 ) ) = 𝐷 )
Assertion	joinlmulsubmuld	⊢ ( 𝜑 → ( ( 𝐴 − 𝐶 ) · 𝐵 ) = 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		joinlmulsubmuld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		joinlmulsubmuld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		joinlmulsubmuld.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		joinlmulsubmuld.4	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) − ( 𝐶 · 𝐵 ) ) = 𝐷 )
5	1 3 2	subdird	⊢ ( 𝜑 → ( ( 𝐴 − 𝐶 ) · 𝐵 ) = ( ( 𝐴 · 𝐵 ) − ( 𝐶 · 𝐵 ) ) )
6	5 4	eqtrd	⊢ ( 𝜑 → ( ( 𝐴 − 𝐶 ) · 𝐵 ) = 𝐷 )