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Axiom ax-distr 10205
 Description: Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 10181. Proofs should normally use adddi 10227 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-distr ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Detailed syntax breakdown of Axiom ax-distr
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 10136 . . . 4 class
31, 2wcel 2145 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 2145 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 2145 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 1071 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 caddc 10141 . . . . 5 class +
104, 6, 9co 6793 . . . 4 class (𝐵 + 𝐶)
11 cmul 10143 . . . 4 class ·
121, 10, 11co 6793 . . 3 class (𝐴 · (𝐵 + 𝐶))
131, 4, 11co 6793 . . . 4 class (𝐴 · 𝐵)
141, 6, 11co 6793 . . . 4 class (𝐴 · 𝐶)
1513, 14, 9co 6793 . . 3 class ((𝐴 · 𝐵) + (𝐴 · 𝐶))
1612, 15wceq 1631 . 2 wff (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))
178, 16wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
 Colors of variables: wff setvar class This axiom is referenced by:  adddi  10227
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