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Axiom ax-mulass 6968
Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 6928. Proofs should normally use mulass 6993 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 6868 . . . 4 class
31, 2wcel 1393 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1393 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 1393 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 885 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 cmul 6875 . . . . 5 class ·
101, 4, 9co 5499 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 5499 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 5499 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 5499 . . 3 class (𝐴 · (𝐵 · 𝐶))
1411, 13wceq 1243 . 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 set class
This axiom is referenced by:  mulass  6993
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