Theorem int-mulassocd

Description: MultiplicationAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020)

	Ref	Expression
Hypotheses	int-mulassocd.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	int-mulassocd.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	int-mulassocd.3	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
	int-mulassocd.4	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	int-mulassocd	⊢ ( 𝜑 → ( 𝐵 · ( 𝐶 · 𝐷 ) ) = ( ( 𝐴 · 𝐶 ) · 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		int-mulassocd.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
2		int-mulassocd.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
3		int-mulassocd.3	⊢ ( 𝜑 → 𝐷 ∈ ℝ )
4		int-mulassocd.4	⊢ ( 𝜑 → 𝐴 = 𝐵 )
5	1	recnd	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
6	2	recnd	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
7	3	recnd	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
8	5 6 7	mulassd	⊢ ( 𝜑 → ( ( 𝐵 · 𝐶 ) · 𝐷 ) = ( 𝐵 · ( 𝐶 · 𝐷 ) ) )
9	4	eqcomd	⊢ ( 𝜑 → 𝐵 = 𝐴 )
10	9	oveq1d	⊢ ( 𝜑 → ( 𝐵 · 𝐶 ) = ( 𝐴 · 𝐶 ) )
11	10	oveq1d	⊢ ( 𝜑 → ( ( 𝐵 · 𝐶 ) · 𝐷 ) = ( ( 𝐴 · 𝐶 ) · 𝐷 ) )
12	8 11	eqtr3d	⊢ ( 𝜑 → ( 𝐵 · ( 𝐶 · 𝐷 ) ) = ( ( 𝐴 · 𝐶 ) · 𝐷 ) )