Axiom ax-mulass 10040

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 10016. Proofs should normally use mulass 10062 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 9972	. . . 4 class ℂ
3	1, 2	wcel 2030	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2030	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2030	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1054	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 9979	. . . . 5 class ·
10	1, 4, 9	co 6690	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 6690	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 6690	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 6690	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1523	. 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  10062