Axiom ax-mulass 11178

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11154. Proofs should normally use mulass 11200 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 11110	. . . 4 class ℂ
3	1, 2	wcel 2098	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2098	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2098	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1084	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 11117	. . . . 5 class ·
10	1, 4, 9	co 7405	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 7405	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 7405	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 7405	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1533	. 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  11200