Metamath Proof Explorer


Theorem reclt1

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

Ref Expression
Assertion reclt1 A 0 < A A < 1 1 < 1 A

Proof

Step Hyp Ref Expression
1 1re 1
2 0lt1 0 < 1
3 ltrec A 0 < A 1 0 < 1 A < 1 1 1 < 1 A
4 1 2 3 mpanr12 A 0 < A A < 1 1 1 < 1 A
5 1div1e1 1 1 = 1
6 5 breq1i 1 1 < 1 A 1 < 1 A
7 4 6 bitrdi A 0 < A A < 1 1 < 1 A