Axiom ax-mulass 10002

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 9978. Proofs should normally use mulass 10024 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 9934	. . . 4 class ℂ
3	1, 2	wcel 1990	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 1990	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 1990	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1037	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 9941	. . . . 5 class ·
10	1, 4, 9	co 6650	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 6650	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 6650	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 6650	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1483	. 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  10024