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Metamath Proof Explorer

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 ) )