This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem nn0pzuz

Description: The sum of a nonnegative integer and an integer is an integer greater than or equal to that integer. (Contributed by Alexander van der Vekens, 3-Oct-2018)

Ref Expression
Assertion nn0pzuz 
|- ( ( N e. NN0 /\ Z e. ZZ ) -> ( N + Z ) e. ( ZZ>= ` Z ) )

Proof

Step Hyp Ref Expression
1 simpr 
 |-  ( ( N e. NN0 /\ Z e. ZZ ) -> Z e. ZZ )
2 nn0z 
 |-  ( N e. NN0 -> N e. ZZ )
3 zaddcl 
 |-  ( ( N e. ZZ /\ Z e. ZZ ) -> ( N + Z ) e. ZZ )
4 2 3 sylan 
 |-  ( ( N e. NN0 /\ Z e. ZZ ) -> ( N + Z ) e. ZZ )
5 zre 
 |-  ( Z e. ZZ -> Z e. RR )
6 nn0addge2 
 |-  ( ( Z e. RR /\ N e. NN0 ) -> Z <_ ( N + Z ) )
7 5 6 sylan 
 |-  ( ( Z e. ZZ /\ N e. NN0 ) -> Z <_ ( N + Z ) )
8 7 ancoms 
 |-  ( ( N e. NN0 /\ Z e. ZZ ) -> Z <_ ( N + Z ) )
9 eluz2 
 |-  ( ( N + Z ) e. ( ZZ>= ` Z ) <-> ( Z e. ZZ /\ ( N + Z ) e. ZZ /\ Z <_ ( N + Z ) ) )
10 1 4 8 9 syl3anbrc 
 |-  ( ( N e. NN0 /\ Z e. ZZ ) -> ( N + Z ) e. ( ZZ>= ` Z ) )