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	$$⊢\phi \to A\in ℂ$$
	divcld.2	$$⊢\phi \to B\in ℂ$$
	divmuld.3	$$⊢\phi \to C\in ℂ$$
	divmuldivd.4	$$⊢\phi \to D\in ℂ$$
	divmuldivd.5	$$⊢\phi \to B\ne 0$$
	divmuldivd.6	$$⊢\phi \to D\ne 0$$
	divdivdivd.7	$$⊢\phi \to C\ne 0$$
Assertion	divdivdivd	$$⊢\phi \to \frac{\frac{A}{B}}{\frac{C}{D}}=\frac{AD}{BC}$$

Proof

Step	Hyp	Ref	Expression
1		div1d.1	$$⊢\phi \to A\in ℂ$$
2		divcld.2	$$⊢\phi \to B\in ℂ$$
3		divmuld.3	$$⊢\phi \to C\in ℂ$$
4		divmuldivd.4	$$⊢\phi \to D\in ℂ$$
5		divmuldivd.5	$$⊢\phi \to B\ne 0$$
6		divmuldivd.6	$$⊢\phi \to D\ne 0$$
7		divdivdivd.7	$$⊢\phi \to C\ne 0$$
8	2 5	jca	$$⊢\phi \to \left(B\in ℂ\wedge B\ne 0\right)$$
9	3 7	jca	$$⊢\phi \to \left(C\in ℂ\wedge C\ne 0\right)$$
10	4 6	jca	$$⊢\phi \to \left(D\in ℂ\wedge D\ne 0\right)$$
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	syl22anc	$$⊢\phi \to \frac{\frac{A}{B}}{\frac{C}{D}}=\frac{AD}{BC}$$