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Metamath Proof Explorer

Theorem nnrecgt0

Description: The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999)

Ref Expression
Assertion nnrecgt0 
|- ( A e. NN -> 0 < ( 1 / A ) )

Proof

Step Hyp Ref Expression
1 nnge1 
 |-  ( A e. NN -> 1 <_ A )
2 0lt1 
 |-  0 < 1
3 nnre 
 |-  ( A e. NN -> A e. RR )
4 0re 
 |-  0 e. RR
5 1re 
 |-  1 e. RR
6 ltletr 
 |-  ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( ( 0 < 1 /\ 1 <_ A ) -> 0 < A ) )
7 4 5 6 mp3an12 
 |-  ( A e. RR -> ( ( 0 < 1 /\ 1 <_ A ) -> 0 < A ) )
8 recgt0 
 |-  ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )
9 8 ex 
 |-  ( A e. RR -> ( 0 < A -> 0 < ( 1 / A ) ) )
10 7 9 syld 
 |-  ( A e. RR -> ( ( 0 < 1 /\ 1 <_ A ) -> 0 < ( 1 / A ) ) )
11 3 10 syl 
 |-  ( A e. NN -> ( ( 0 < 1 /\ 1 <_ A ) -> 0 < ( 1 / A ) ) )
12 2 11 mpani 
 |-  ( A e. NN -> ( 1 <_ A -> 0 < ( 1 / A ) ) )
13 1 12 mpd 
 |-  ( A e. NN -> 0 < ( 1 / A ) )