Theorem divcan5rd

Description: Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 1-Jan-2017)

	Ref	Expression
Hypotheses	div1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	divcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	divmuld.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	divmuld.4	⊢ ( 𝜑 → 𝐵 ≠ 0 )
	divdiv23d.5	⊢ ( 𝜑 → 𝐶 ≠ 0 )
Assertion	divcan5rd	⊢ ( 𝜑 → ( ( 𝐴 · 𝐶 ) / ( 𝐵 · 𝐶 ) ) = ( 𝐴 / 𝐵 ) )

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		divdiv23d.5	⊢ ( 𝜑 → 𝐶 ≠ 0 )
6	1 3	mulcomd	⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) )
7	2 3	mulcomd	⊢ ( 𝜑 → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
8	6 7	oveq12d	⊢ ( 𝜑 → ( ( 𝐴 · 𝐶 ) / ( 𝐵 · 𝐶 ) ) = ( ( 𝐶 · 𝐴 ) / ( 𝐶 · 𝐵 ) ) )
9	1 2 3 4 5	divcan5d	⊢ ( 𝜑 → ( ( 𝐶 · 𝐴 ) / ( 𝐶 · 𝐵 ) ) = ( 𝐴 / 𝐵 ) )
10	8 9	eqtrd	⊢ ( 𝜑 → ( ( 𝐴 · 𝐶 ) / ( 𝐵 · 𝐶 ) ) = ( 𝐴 / 𝐵 ) )