Axiom ax-mulass 11193

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11169. Proofs should normally use mulass 11215 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 11125	. . . 4 class ℂ
3	1, 2	wcel 2145	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2145	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2145	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1103	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 11132	. . . . 5 class ·
10	1, 4, 9	co 7416	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 7416	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 7416	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 7416	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1570	. 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  11215