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 ∈ Q → ( 𝐴 ·Q ( *Q ‘ 𝐴 ) ) = 1Q )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( *Q ‘ 𝐴 ) = ( *Q ‘ 𝐴 )
2		recmulnq	⊢ ( 𝐴 ∈ Q → ( ( *Q ‘ 𝐴 ) = ( *Q ‘ 𝐴 ) ↔ ( 𝐴 ·Q ( *Q ‘ 𝐴 ) ) = 1Q ) )
3	1 2	mpbii	⊢ ( 𝐴 ∈ Q → ( 𝐴 ·Q ( *Q ‘ 𝐴 ) ) = 1Q )