Theorem dmdcand

Description: Cancellation law for division and multiplication. (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 )
	divdiv23d.5	⊢ ( 𝜑 → 𝐶 ≠ 0 )
Assertion	dmdcand	⊢ ( 𝜑 → ( ( 𝐵 / 𝐶 ) · ( 𝐴 / 𝐵 ) ) = ( 𝐴 / 𝐶 ) )

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		dmdcan	⊢ ( ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ∧ 𝐴 ∈ ℂ ) → ( ( 𝐵 / 𝐶 ) · ( 𝐴 / 𝐵 ) ) = ( 𝐴 / 𝐶 ) )
7	2 4 3 5 1 6	syl221anc	⊢ ( 𝜑 → ( ( 𝐵 / 𝐶 ) · ( 𝐴 / 𝐵 ) ) = ( 𝐴 / 𝐶 ) )