Theorem rerecrecd

Description: A number is equal to the reciprocal of its reciprocal. (Contributed by SN, 2-Apr-2026)

	Ref	Expression
Hypotheses	sn-rereccld.a	|- ( ph -> A e. RR )
	sn-rereccld.z	|- ( ph -> A =/= 0 )
Assertion	rerecrecd	|- ( ph -> ( 1 /R ( 1 /R A ) ) = A )

Proof

Step	Hyp	Ref	Expression
1		sn-rereccld.a	|- ( ph -> A e. RR )
2		sn-rereccld.z	|- ( ph -> A =/= 0 )
3	1 2	rerecid2d	|- ( ph -> ( ( 1 /R A ) x. A ) = 1 )
4		1red	|- ( ph -> 1 e. RR )
5	1 2	sn-rereccld	|- ( ph -> ( 1 /R A ) e. RR )
6	1 2	rerecne0d	|- ( ph -> ( 1 /R A ) =/= 0 )
7	4 1 5 6	redivmuld	|- ( ph -> ( ( 1 /R ( 1 /R A ) ) = A <-> ( ( 1 /R A ) x. A ) = 1 ) )
8	3 7	mpbird	|- ( ph -> ( 1 /R ( 1 /R A ) ) = A )