Theorem joinlmuladdmuld

Description: Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019)

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

Proof

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