Axiom ax-mulass 11290

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11266. Proofs should normally use mulass 11312 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 11222	. . . 4 class ℂ
3	1, 2	wcel 2147	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2147	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2147	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1103	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 11229	. . . . 5 class ·
10	1, 4, 9	co 7440	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 7440	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 7440	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 7440	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1570	. 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 used by:  mulass  11312