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 ) )