Axiom ax-mulass 11106

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11082. Proofs should normally use mulass 11128 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 11038	. . . 4 class ℂ
3	1, 2	wcel 2114	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2114	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2114	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1087	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 11045	. . . . 5 class ·
10	1, 4, 9	co 7370	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 7370	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 7370	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 7370	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1542	. 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  11128