Theorem dmdcan2d

Description: Cancellation law for division and multiplication. (Contributed by David Moews, 28-Feb-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	dmdcan2d	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) · ( 𝐵 / 𝐶 ) ) = ( 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	1 2 4	divcld	⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) ∈ ℂ )
7	2 3 5	divcld	⊢ ( 𝜑 → ( 𝐵 / 𝐶 ) ∈ ℂ )
8	6 7	mulcomd	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) · ( 𝐵 / 𝐶 ) ) = ( ( 𝐵 / 𝐶 ) · ( 𝐴 / 𝐵 ) ) )
9	1 2 3 4 5	dmdcand	⊢ ( 𝜑 → ( ( 𝐵 / 𝐶 ) · ( 𝐴 / 𝐵 ) ) = ( 𝐴 / 𝐶 ) )
10	8 9	eqtrd	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) · ( 𝐵 / 𝐶 ) ) = ( 𝐴 / 𝐶 ) )