Axiom ax-mulass 10938

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 10914. Proofs should normally use mulass 10960 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 10870	. . . 4 class ℂ
3	1, 2	wcel 2110	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2110	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2110	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1086	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 10877	. . . . 5 class ·
10	1, 4, 9	co 7271	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 7271	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 7271	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 7271	. . 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  10960