Axiom ax-mulass 11141

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11117. Proofs should normally use mulass 11163 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 11073	. . . 4 class ℂ
3	1, 2	wcel 2144	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2144	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2144	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1099	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 11080	. . . . 5 class ·
10	1, 4, 9	co 7398	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 7398	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 7398	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 7398	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1562	. 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  11163