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Metamath Proof Explorer

Theorem adddird

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

Ref Expression
Hypotheses addcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
addcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
addassd.3 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion adddird ( 𝜑 → ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) )

Proof

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