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	$$⊢A\in ℂ$$
	divclz.2	$$⊢B\in ℂ$$
	divmulz.3	$$⊢C\in ℂ$$
	divmuldiv.4	$$⊢D\in ℂ$$
	divmuldiv.5	$$⊢B\ne 0$$
	divmuldiv.6	$$⊢D\ne 0$$
	divdivdiv.7	$$⊢C\ne 0$$
Assertion	divdivdivi	$$⊢\frac{\frac{A}{B}}{\frac{C}{D}}=\frac{AD}{BC}$$

Proof

Step	Hyp	Ref	Expression
1		divclz.1	$$⊢A\in ℂ$$
2		divclz.2	$$⊢B\in ℂ$$
3		divmulz.3	$$⊢C\in ℂ$$
4		divmuldiv.4	$$⊢D\in ℂ$$
5		divmuldiv.5	$$⊢B\ne 0$$
6		divmuldiv.6	$$⊢D\ne 0$$
7		divdivdiv.7	$$⊢C\ne 0$$
8	2 5	pm3.2i	$$⊢\left(B\in ℂ\wedge B\ne 0\right)$$
9	3 7	pm3.2i	$$⊢\left(C\in ℂ\wedge C\ne 0\right)$$
10	4 6	pm3.2i	$$⊢\left(D\in ℂ\wedge D\ne 0\right)$$
11		divdivdiv	$$⊢\left(\left(A\in ℂ\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\wedge \left(\left(C\in ℂ\wedge C\ne 0\right)\wedge \left(D\in ℂ\wedge D\ne 0\right)\right)\right)\to \frac{\frac{A}{B}}{\frac{C}{D}}=\frac{AD}{BC}$$
12	1 8 9 10 11	mp4an	$$⊢\frac{\frac{A}{B}}{\frac{C}{D}}=\frac{AD}{BC}$$