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 Description: Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 10178. Proofs should normally use addass 10224 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-addass ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Detailed syntax breakdown of Axiom ax-addass
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 10135 . . . 4 class
31, 2wcel 2144 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 2144 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 2144 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 1070 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 caddc 10140 . . . . 5 class +
101, 4, 9co 6792 . . . 4 class (𝐴 + 𝐵)
1110, 6, 9co 6792 . . 3 class ((𝐴 + 𝐵) + 𝐶)
124, 6, 9co 6792 . . . 4 class (𝐵 + 𝐶)
131, 12, 9co 6792 . . 3 class (𝐴 + (𝐵 + 𝐶))
1411, 13wceq 1630 . 2 wff ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
158, 14wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
 Colors of variables: wff setvar class This axiom is referenced by:  addass  10224
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