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Metamath Proof Explorer

Theorem adddir

Description: Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004)

Ref Expression
Assertion adddir ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 adddi ⊢ ( ( 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐶 · ( 𝐴 + 𝐵 ) ) = ( ( 𝐶 · 𝐴 ) + ( 𝐶 · 𝐵 ) ) )
2 1 3coml ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐶 · ( 𝐴 + 𝐵 ) ) = ( ( 𝐶 · 𝐴 ) + ( 𝐶 · 𝐵 ) ) )
3 addcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) ∈ ℂ )
4 mulcom ⊢ ( ( ( 𝐴 + 𝐵 ) ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( 𝐶 · ( 𝐴 + 𝐵 ) ) )
5 3 4 stoic3 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( 𝐶 · ( 𝐴 + 𝐵 ) ) )
6 mulcom ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
7 6 3adant2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
8 mulcom ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
9 8 3adant1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
10 7 9 oveq12d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) = ( ( 𝐶 · 𝐴 ) + ( 𝐶 · 𝐵 ) ) )
11 2 5 10 3eqtr4d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) + ( 𝐵 · 𝐶 ) ) )