Metamath Proof Explorer


Theorem recgt1

Description: The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005)

Ref Expression
Assertion recgt1 A 0 < A 1 < A 1 A < 1

Proof

Step Hyp Ref Expression
1 1re 1
2 0lt1 0 < 1
3 ltrec 1 0 < 1 A 0 < A 1 < A 1 A < 1 1
4 1 2 3 mpanl12 A 0 < A 1 < A 1 A < 1 1
5 1div1e1 1 1 = 1
6 5 breq2i 1 A < 1 1 1 A < 1
7 4 6 bitrdi A 0 < A 1 < A 1 A < 1