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 A0<AA<11<1A

Proof

Step Hyp Ref Expression
1 1re 1
2 0lt1 0<1
3 ltrec A0<A10<1A<111<1A
4 1 2 3 mpanr12 A0<AA<111<1A
5 1div1e1 11=1
6 5 breq1i 11<1A1<1A
7 4 6 bitrdi A0<AA<11<1A