Theorem dmmcand

Description: Cancellation law for division and multiplication. (Contributed by Glauco Siliprandi, 5-Apr-2020)

	Ref	Expression
Hypotheses	dmmcand.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	dmmcand.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	dmmcand.c	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	dmmcand.bn0	⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion	dmmcand	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) · ( 𝐵 · 𝐶 ) ) = ( 𝐴 · 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		dmmcand.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		dmmcand.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		dmmcand.c	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		dmmcand.bn0	⊢ ( 𝜑 → 𝐵 ≠ 0 )
5	2 3	mulcld	⊢ ( 𝜑 → ( 𝐵 · 𝐶 ) ∈ ℂ )
6	1 2 5 4	div32d	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) · ( 𝐵 · 𝐶 ) ) = ( 𝐴 · ( ( 𝐵 · 𝐶 ) / 𝐵 ) ) )
7	3 2 4	divcan3d	⊢ ( 𝜑 → ( ( 𝐵 · 𝐶 ) / 𝐵 ) = 𝐶 )
8	7	oveq2d	⊢ ( 𝜑 → ( 𝐴 · ( ( 𝐵 · 𝐶 ) / 𝐵 ) ) = ( 𝐴 · 𝐶 ) )
9		eqidd	⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) = ( 𝐴 · 𝐶 ) )
10	6 8 9	3eqtrd	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) · ( 𝐵 · 𝐶 ) ) = ( 𝐴 · 𝐶 ) )