Theorem int-mulcomd

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

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

Proof

Step	Hyp	Ref	Expression
1		int-mulcomd.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
2		int-mulcomd.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
3		int-mulcomd.3	⊢ ( 𝜑 → 𝐴 = 𝐵 )
4	1	recnd	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
5	2	recnd	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
6	4 5	mulcomd	⊢ ( 𝜑 → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
7	3	eqcomd	⊢ ( 𝜑 → 𝐵 = 𝐴 )
8	7	oveq2d	⊢ ( 𝜑 → ( 𝐶 · 𝐵 ) = ( 𝐶 · 𝐴 ) )
9	6 8	eqtrd	⊢ ( 𝜑 → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐴 ) )