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 ) )