Axiom ax-mulass 10868

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 10844. Proofs should normally use mulass 10890 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 10800	. . . 4 class ℂ
3	1, 2	wcel 2108	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2108	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2108	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1085	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 10807	. . . . 5 class ·
10	1, 4, 9	co 7255	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 7255	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 7255	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 7255	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1539	. 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  10890