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Axiom ax-mulass 10203
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 10179. Proofs should normally use mulass 10225 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 10135 . . . 4 class
31, 2wcel 2144 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 2144 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 2144 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 1070 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 cmul 10142 . . . . 5 class ·
101, 4, 9co 6792 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 6792 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 6792 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 6792 . . 3 class (𝐴 · (𝐵 · 𝐶))
1411, 13wceq 1630 . 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  10225
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