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Metamath Proof Explorer

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 mullidd ⊢ ( 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 ) )