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Metamath Proof Explorer

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 ) )