Theorem divcan6

Description: Cancellation of inverted fractions. (Contributed by NM, 28-Dec-2007)

	Ref	Expression
Assertion	divcan6	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \left(\frac{A}{B}\right)\left(\frac{B}{A}\right)=1$$

Proof

Step	Hyp	Ref	Expression
1		recdiv	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \frac{1}{\frac{A}{B}}=\frac{B}{A}$$
2	1	oveq2d	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \left(\frac{A}{B}\right)\left(\frac{1}{\frac{A}{B}}\right)=\left(\frac{A}{B}\right)\left(\frac{B}{A}\right)$$
3		divcl	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \frac{A}{B}\in ℂ$$
4	3	3expb	$$⊢\left(A\in ℂ\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \frac{A}{B}\in ℂ$$
5	4	adantlr	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \frac{A}{B}\in ℂ$$
6		divne0	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \frac{A}{B}\ne 0$$
7		recid	$$⊢\left(\frac{A}{B}\in ℂ\wedge \frac{A}{B}\ne 0\right)\to \left(\frac{A}{B}\right)\left(\frac{1}{\frac{A}{B}}\right)=1$$
8	5 6 7	syl2anc	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \left(\frac{A}{B}\right)\left(\frac{1}{\frac{A}{B}}\right)=1$$
9	2 8	eqtr3d	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge \left(B\in ℂ\wedge B\ne 0\right)\right)\to \left(\frac{A}{B}\right)\left(\frac{B}{A}\right)=1$$