Axiom ax-mulass 11218

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11194. Proofs should normally use mulass 11240 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 11150	. . . 4 class ℂ
3	1, 2	wcel 2105	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2105	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2105	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1086	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 11157	. . . . 5 class ·
10	1, 4, 9	co 7430	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 7430	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 7430	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 7430	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1536	. 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  11240