Axiom ax-mulass 11069

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11045. Proofs should normally use mulass 11091 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 11001	. . . 4 class ℂ
3	1, 2	wcel 2111	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2111	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2111	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1086	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 11008	. . . . 5 class ·
10	1, 4, 9	co 7346	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 7346	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 7346	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 7346	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1541	. 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  11091