Axiom ax-mulass 10290

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 10266. Proofs should normally use mulass 10312 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 10222	. . . 4 class ℂ
3	1, 2	wcel 2157	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2157	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2157	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1100	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 10229	. . . . 5 class ·
10	1, 4, 9	co 6877	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 6877	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 6877	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 6877	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1637	. 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  10312