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Axiom ax-mulass 7045
Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 7005. Proofs should normally use mulass 7070 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 6945 . . . 4 class
31, 2wcel 1409 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1409 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 1409 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 896 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 cmul 6952 . . . . 5 class ·
101, 4, 9co 5540 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 5540 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 5540 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 5540 . . 3 class (𝐴 · (𝐵 · 𝐶))
1411, 13wceq 1259 . 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  7070
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