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Axiom ax-mulass 9987
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 9963. Proofs should normally use mulass 10009 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 9919 . . . 4 class
31, 2wcel 1988 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1988 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 1988 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 1036 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 cmul 9926 . . . . 5 class ·
101, 4, 9co 6635 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 6635 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 6635 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 6635 . . 3 class (𝐴 · (𝐵 · 𝐶))
1411, 13wceq 1481 . 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  10009
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