Axiom ax-mulass 11100

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11076. Proofs should normally use mulass 11122 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 11032	. . . 4 class ℂ
3	1, 2	wcel 2121	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2121	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2121	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1093	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 11039	. . . . 5 class ·
10	1, 4, 9	co 7359	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 7359	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 7359	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 7359	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1548	. 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  11122