Theorem divdivdivi

Description: Division of two ratios. Theorem I.15 of Apostol p. 18. (Contributed by NM, 22-Feb-1995)

	Ref	Expression
Hypotheses	divclz.1	⊢ 𝐴 ∈ ℂ
	divclz.2	⊢ 𝐵 ∈ ℂ
	divmulz.3	⊢ 𝐶 ∈ ℂ
	divmuldiv.4	⊢ 𝐷 ∈ ℂ
	divmuldiv.5	⊢ 𝐵 ≠ 0
	divmuldiv.6	⊢ 𝐷 ≠ 0
	divdivdiv.7	⊢ 𝐶 ≠ 0
Assertion	divdivdivi	⊢ ( ( 𝐴 / 𝐵 ) / ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐷 ) / ( 𝐵 · 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		divclz.1	⊢ 𝐴 ∈ ℂ
2		divclz.2	⊢ 𝐵 ∈ ℂ
3		divmulz.3	⊢ 𝐶 ∈ ℂ
4		divmuldiv.4	⊢ 𝐷 ∈ ℂ
5		divmuldiv.5	⊢ 𝐵 ≠ 0
6		divmuldiv.6	⊢ 𝐷 ≠ 0
7		divdivdiv.7	⊢ 𝐶 ≠ 0
8	2 5	pm3.2i	⊢ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 )
9	3 7	pm3.2i	⊢ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 )
10	4 6	pm3.2i	⊢ ( 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	mp4an	⊢ ( ( 𝐴 / 𝐵 ) / ( 𝐶 / 𝐷 ) ) = ( ( 𝐴 · 𝐷 ) / ( 𝐵 · 𝐶 ) )