Theorem divdivdivd

Description: Division of two ratios. Theorem I.15 of Apostol p. 18. (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 )
	divmuldivd.4	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
	divmuldivd.5	⊢ ( 𝜑 → 𝐵 ≠ 0 )
	divmuldivd.6	⊢ ( 𝜑 → 𝐷 ≠ 0 )
	divdivdivd.7	⊢ ( 𝜑 → 𝐶 ≠ 0 )
Assertion	divdivdivd	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) / ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐷 ) / ( 𝐵 · 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		divmuldivd.4	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
5		divmuldivd.5	⊢ ( 𝜑 → 𝐵 ≠ 0 )
6		divmuldivd.6	⊢ ( 𝜑 → 𝐷 ≠ 0 )
7		divdivdivd.7	⊢ ( 𝜑 → 𝐶 ≠ 0 )
8	2 5	jca	⊢ ( 𝜑 → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) )
9	3 7	jca	⊢ ( 𝜑 → ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) )
10	4 6	jca	⊢ ( 𝜑 → ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) )
11		divdivdiv	⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) ∧ ( ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ∧ ( 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0 ) ) ) → ( ( 𝐴 / 𝐵 ) / ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐷 ) / ( 𝐵 · 𝐶 ) ) )
12	1 8 9 10 11	syl22anc	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) / ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐷 ) / ( 𝐵 · 𝐶 ) ) )