Axiom ax-mulass 11157

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11133. Proofs should normally use mulass 11179 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 11089	. . . 4 class ℂ
3	1, 2	wcel 2106	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2106	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2106	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1087	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 11096	. . . . 5 class ·
10	1, 4, 9	co 7392	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 7392	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 7392	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 7392	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1541	. 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  11179