Theorem adddirp1d

Description: Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	adddirp1d.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	adddirp1d.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	adddirp1d	⊢ ( 𝜑 → ( ( 𝐴 + 1 ) · 𝐵 ) = ( ( 𝐴 · 𝐵 ) + 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		adddirp1d.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		adddirp1d.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		1cnd	⊢ ( 𝜑 → 1 ∈ ℂ )
4	1 3 2	adddird	⊢ ( 𝜑 → ( ( 𝐴 + 1 ) · 𝐵 ) = ( ( 𝐴 · 𝐵 ) + ( 1 · 𝐵 ) ) )
5	2	mulid2d	⊢ ( 𝜑 → ( 1 · 𝐵 ) = 𝐵 )
6	5	oveq2d	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) + ( 1 · 𝐵 ) ) = ( ( 𝐴 · 𝐵 ) + 𝐵 ) )
7	4 6	eqtrd	⊢ ( 𝜑 → ( ( 𝐴 + 1 ) · 𝐵 ) = ( ( 𝐴 · 𝐵 ) + 𝐵 ) )