Axiom ax-mulass 11165

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11141. Proofs should normally use mulass 11187 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 11097	. . . 4 class ℂ
3	1, 2	wcel 2141	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2141	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2141	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1101	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 11104	. . . . 5 class ·
10	1, 4, 9	co 7410	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 7410	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 7410	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 7410	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1568	. 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  11187