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Axiom ax-mulass 10040
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 10016. Proofs should normally use mulass 10062 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 9972 . . . 4 class
31, 2wcel 2030 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 2030 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 2030 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 1054 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 cmul 9979 . . . . 5 class ·
101, 4, 9co 6690 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 6690 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 6690 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 6690 . . 3 class (𝐴 · (𝐵 · 𝐶))
1411, 13wceq 1523 . 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  10062
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