Theorem recgt1i

Description: The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005)

	Ref	Expression
Assertion	recgt1i	|- ( ( A e. RR /\ 1 < A ) -> ( 0 < ( 1 / A ) /\ ( 1 / A ) < 1 ) )

Proof

Step	Hyp	Ref	Expression
1		0lt1	|- 0 < 1
2		0re	|- 0 e. RR
3		1re	|- 1 e. RR
4		lttr	|- ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
5	2 3 4	mp3an12	|- ( A e. RR -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
6	1 5	mpani	|- ( A e. RR -> ( 1 < A -> 0 < A ) )
7	6	imdistani	|- ( ( A e. RR /\ 1 < A ) -> ( A e. RR /\ 0 < A ) )
8		recgt0	|- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )
9	7 8	syl	|- ( ( A e. RR /\ 1 < A ) -> 0 < ( 1 / A ) )
10		recgt1	|- ( ( A e. RR /\ 0 < A ) -> ( 1 < A <-> ( 1 / A ) < 1 ) )
11	10	biimpa	|- ( ( ( A e. RR /\ 0 < A ) /\ 1 < A ) -> ( 1 / A ) < 1 )
12	7 11	sylancom	|- ( ( A e. RR /\ 1 < A ) -> ( 1 / A ) < 1 )
13	9 12	jca	|- ( ( A e. RR /\ 1 < A ) -> ( 0 < ( 1 / A ) /\ ( 1 / A ) < 1 ) )