Theorem adddiri

Description: Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995)

	Ref	Expression
Hypotheses	axi.1	⊢ 𝐴 ∈ ℂ
	axi.2	⊢ 𝐵 ∈ ℂ
	axi.3	⊢ 𝐶 ∈ ℂ
Assertion	adddiri	⊢ ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		axi.1	⊢ 𝐴 ∈ ℂ
2		axi.2	⊢ 𝐵 ∈ ℂ
3		axi.3	⊢ 𝐶 ∈ ℂ
4		adddir	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) )
5	1 2 3 4	mp3an	⊢ ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) )