Axiom ax-mulass 10456

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 10432. Proofs should normally use mulass 10478 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 10388	. . . 4 class ℂ
3	1, 2	wcel 2083	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2083	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2083	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1080	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 10395	. . . . 5 class ·
10	1, 4, 9	co 7023	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 7023	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 7023	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 7023	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1525	. 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  10478