Metamath Proof Explorer


Theorem adddid

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

Ref Expression
Hypotheses addcld.1 ( 𝜑𝐴 ∈ ℂ )
addcld.2 ( 𝜑𝐵 ∈ ℂ )
addassd.3 ( 𝜑𝐶 ∈ ℂ )
Assertion adddid ( 𝜑 → ( 𝐴 · ( 𝐵 + 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) + ( 𝐴 · 𝐶 ) ) )

Proof

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