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Axiom ax-mulass 10290
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 10266. Proofs should normally use mulass 10312 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 10222 . . . 4 class
31, 2wcel 2157 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 2157 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 2157 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 1100 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 cmul 10229 . . . . 5 class ·
101, 4, 9co 6877 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 6877 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 6877 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 6877 . . 3 class (𝐴 · (𝐵 · 𝐶))
1411, 13wceq 1637 . 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  10312
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