Metamath Proof Explorer


Theorem elfzoext

Description: Membership of an integer in an extended open range of integers, extension added to the right. (Contributed by AV, 30-Apr-2020) (Proof shortened by AV, 23-Sep-2025)

Ref Expression
Assertion elfzoext Z M ..^ N I 0 Z M ..^ N + I

Proof

Step Hyp Ref Expression
1 elfzoextl Z M ..^ N I 0 Z M ..^ I + N
2 elfzoel2 Z M ..^ N N
3 2 zcnd Z M ..^ N N
4 3 adantr Z M ..^ N I 0 N
5 nn0cn I 0 I
6 5 adantl Z M ..^ N I 0 I
7 4 6 addcomd Z M ..^ N I 0 N + I = I + N
8 7 oveq2d Z M ..^ N I 0 M ..^ N + I = M ..^ I + N
9 1 8 eleqtrrd Z M ..^ N I 0 Z M ..^ N + I