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Metamath Proof Explorer

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 ) ) = 1_Q )

Proof

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