Theorem mulassd

Description: Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	addcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	addcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	addassd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion	mulassd	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐵 · 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		addcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		addcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		addassd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		mulass	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐵 · 𝐶 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐵 · 𝐶 ) ) )