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Axiom ax-mulass 9858
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 9834. Proofs should normally use mulass 9880 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-mulass ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 9790 . . . 4 class
31, 2wcel 1976 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1976 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 1976 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 1030 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 cmul 9797 . . . . 5 class ·
101, 4, 9co 6526 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 6526 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 6526 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 6526 . . 3 class (𝐴 · (𝐵 · 𝐶))
1411, 13wceq 1474 . 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:  mulass  9880
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