Theorem divdiv3d

Description: Division into a fraction. (Contributed by Glauco Siliprandi, 24-Dec-2020)

	Ref	Expression
Hypotheses	divdiv3d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	divdiv3d.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	divdiv3d.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	divdiv3d.4	⊢ ( 𝜑 → 𝐵 ≠ 0 )
	divdiv3d.5	⊢ ( 𝜑 → 𝐶 ≠ 0 )
Assertion	divdiv3d	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) / 𝐶 ) = ( 𝐴 / ( 𝐶 · 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		divdiv3d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		divdiv3d.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		divdiv3d.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		divdiv3d.4	⊢ ( 𝜑 → 𝐵 ≠ 0 )
5		divdiv3d.5	⊢ ( 𝜑 → 𝐶 ≠ 0 )
6	1 2 3 4 5	divdiv1d	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) / 𝐶 ) = ( 𝐴 / ( 𝐵 · 𝐶 ) ) )
7	2 3	mulcomd	⊢ ( 𝜑 → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
8	7	oveq2d	⊢ ( 𝜑 → ( 𝐴 / ( 𝐵 · 𝐶 ) ) = ( 𝐴 / ( 𝐶 · 𝐵 ) ) )
9	6 8	eqtrd	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) / 𝐶 ) = ( 𝐴 / ( 𝐶 · 𝐵 ) ) )