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Axiom ax-mulass 6985
Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 6945. Proofs should normally use mulass 7010 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 6885 . . . 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 6892 . . . . 5 class ·
101, 4, 9co 5512 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 5512 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 5512 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 5512 . . 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  7010
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