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Metamath Proof Explorer

Theorem adddii

Description: Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994)

Ref Expression
Hypotheses axi.1 ⊢ A ∈ ℂ
axi.2 ⊢ B ∈ ℂ
axi.3 ⊢ C ∈ ℂ
Assertion adddii ⊢ A ⁢ B + C = A ⁢ B + A ⁢ C

Proof

Step Hyp Ref Expression
1 axi.1 ⊢ A ∈ ℂ
2 axi.2 ⊢ B ∈ ℂ
3 axi.3 ⊢ C ∈ ℂ
4 adddi ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ → A ⁢ B + C = A ⁢ B + A ⁢ C
5 1 2 3 4 mp3an ⊢ A ⁢ B + C = A ⁢ B + A ⁢ C