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Axiom ax-mulass 10592
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 10568. Proofs should normally use mulass 10614 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 10524 . . . 4 class
31, 2wcel 2111 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 2111 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 2111 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 1084 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 cmul 10531 . . . . 5 class ·
101, 4, 9co 7135 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 7135 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 7135 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 7135 . . 3 class (𝐴 · (𝐵 · 𝐶))
1411, 13wceq 1538 . 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  10614
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