Axiom ax-mulass 9858

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 9834. Proofs should normally use mulass 9880 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

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 9790	. . . 4 class ℂ
3	1, 2	wcel 1976	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 1976	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 1976	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1030	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 9797	. . . . 5 class ·
10	1, 4, 9	co 6526	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 6526	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 6526	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 6526	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1474	. 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

This axiom is referenced by:  mulass  9880