Axiom ax-mulass

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass . Proofs should normally use mulass instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994)

	Ref	Expression
Assertion	ax-mulass	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐵 · 𝐶 ) ) )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cA	⊢ 𝐴
1		cc	⊢ ℂ
2	0 1	wcel	⊢ 𝐴 ∈ ℂ
3		cB	⊢ 𝐵
4	3 1	wcel	⊢ 𝐵 ∈ ℂ
5		cC	⊢ 𝐶
6	5 1	wcel	⊢ 𝐶 ∈ ℂ
7	2 4 6	w3a	⊢ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ )
8		cmul	⊢ ·
9	0 3 8	co	⊢ ( 𝐴 · 𝐵 )
10	9 5 8	co	⊢ ( ( 𝐴 · 𝐵 ) · 𝐶 )
11	3 5 8	co	⊢ ( 𝐵 · 𝐶 )
12	0 11 8	co	⊢ ( 𝐴 · ( 𝐵 · 𝐶 ) )
13	10 12	wceq	⊢ ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐵 · 𝐶 ) )
14	7 13	wi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐵 · 𝐶 ) ) )