Theorem divmul13i

Description: Swap denominators of two ratios. (Contributed by NM, 6-Aug-1999)

	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	divmul13i	$$⊢\left(\frac{A}{B}\right)\left(\frac{C}{D}\right)=\left(\frac{C}{B}\right)\left(\frac{A}{D}\right)$$

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	3 1	mulcomi	$$⊢CA=AC$$
8	7	oveq1i	$$⊢\frac{CA}{BD}=\frac{AC}{BD}$$
9	3 2 1 4 5 6	divmuldivi	$$⊢\left(\frac{C}{B}\right)\left(\frac{A}{D}\right)=\frac{CA}{BD}$$
10	1 2 3 4 5 6	divmuldivi	$$⊢\left(\frac{A}{B}\right)\left(\frac{C}{D}\right)=\frac{AC}{BD}$$
11	8 9 10	3eqtr4ri	$$⊢\left(\frac{A}{B}\right)\left(\frac{C}{D}\right)=\left(\frac{C}{B}\right)\left(\frac{A}{D}\right)$$