Theorem divmuldivi

Description: Multiplication of two ratios. Theorem I.14 of Apostol p. 18. (Contributed by NM, 16-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	$$⊢D\ne 0$$
Assertion	divmuldivi	$$⊢\left(\frac{A}{B}\right)\left(\frac{C}{D}\right)=\frac{AC}{BD}$$

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	2 5	pm3.2i	$$⊢\left(B\in ℂ\wedge B\ne 0\right)$$
8	4 6	pm3.2i	$$⊢\left(D\in ℂ\wedge D\ne 0\right)$$
9		divmuldiv	$$⊢\left(\left(A\in ℂ\wedge C\in ℂ\right)\wedge \left(\left(B\in ℂ\wedge B\ne 0\right)\wedge \left(D\in ℂ\wedge D\ne 0\right)\right)\right)\to \left(\frac{A}{B}\right)\left(\frac{C}{D}\right)=\frac{AC}{BD}$$
10	1 3 7 8 9	mp4an	$$⊢\left(\frac{A}{B}\right)\left(\frac{C}{D}\right)=\frac{AC}{BD}$$