Axiom ax-mulass 11172

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11148. Proofs should normally use mulass 11194 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 11104	. . . 4 class ℂ
3	1, 2	wcel 2142	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2142	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2142	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1102	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 11111	. . . . 5 class ·
10	1, 4, 9	co 7412	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 7412	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 7412	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 7412	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1569	. 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  11194