Axiom ax-mulass 11134

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11110. Proofs should normally use mulass 11156 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 11066	. . . 4 class ℂ
3	1, 2	wcel 2109	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2109	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2109	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1086	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 11073	. . . . 5 class ·
10	1, 4, 9	co 7387	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 7387	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 7387	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 7387	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1540	. 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  11156