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Axiom ax-mulass 9881
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 9857. Proofs should normally use mulass 9903 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 9813 . . . 4 class
31, 2wcel 1977 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1977 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 1977 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 1031 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 cmul 9820 . . . . 5 class ·
101, 4, 9co 6549 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 6549 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 6549 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 6549 . . 3 class (𝐴 · (𝐵 · 𝐶))
1411, 13wceq 1475 . 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  9903
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