Theorem adddird

Description: Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	addcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	addcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	addassd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion	adddird	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		addcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		addcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		addassd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		adddir	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) )