Theorem recidnq

Description: A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996) (Revised by Mario Carneiro, 8-May-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	recidnq	$$⊢A\in 𝑸\to A{\cdot }_{𝑸}{*}_{𝑸}\left(A\right)=1_{𝑸}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢{*}_{𝑸}\left(A\right)={*}_{𝑸}\left(A\right)$$
2		recmulnq	$$⊢A\in 𝑸\to \left({*}_{𝑸}\left(A\right)={*}_{𝑸}\left(A\right)\leftrightarrow A{\cdot }_{𝑸}{*}_{𝑸}\left(A\right)=1_{𝑸}\right)$$
3	1 2	mpbii	$$⊢A\in 𝑸\to A{\cdot }_{𝑸}{*}_{𝑸}\left(A\right)=1_{𝑸}$$