Theorem div13d

Description: A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	divcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	divmuld.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	divmuld.4	⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion	div13d	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) · 𝐶 ) = ( ( 𝐶 / 𝐵 ) · 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		div1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		divcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		divmuld.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		divmuld.4	⊢ ( 𝜑 → 𝐵 ≠ 0 )
5		div13	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 / 𝐵 ) · 𝐶 ) = ( ( 𝐶 / 𝐵 ) · 𝐴 ) )
6	1 2 4 3 5	syl121anc	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) · 𝐶 ) = ( ( 𝐶 / 𝐵 ) · 𝐴 ) )