Theorem mulassi

Description: Associative law for multiplication. (Contributed by NM, 23-Nov-1994)

	Ref	Expression
Hypotheses	axi.1	⊢ 𝐴 ∈ ℂ
	axi.2	⊢ 𝐵 ∈ ℂ
	axi.3	⊢ 𝐶 ∈ ℂ
Assertion	mulassi	⊢ ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐵 · 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		axi.1	⊢ 𝐴 ∈ ℂ
2		axi.2	⊢ 𝐵 ∈ ℂ
3		axi.3	⊢ 𝐶 ∈ ℂ
4		mulass	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐵 · 𝐶 ) ) )
5	1 2 3 4	mp3an	⊢ ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐵 · 𝐶 ) )