Theorem nnnn0addcl

Description: A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005) (Proof shortened by Mario Carneiro, 16-May-2014)

	Ref	Expression
Assertion	nnnn0addcl	|- ( ( M e. NN /\ N e. NN0 ) -> ( M + N ) e. NN )

Proof

Step	Hyp	Ref	Expression
1		elnn0	|- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		nnaddcl	|- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
3		oveq2	|- ( N = 0 -> ( M + N ) = ( M + 0 ) )
4		nncn	|- ( M e. NN -> M e. CC )
5	4	addid1d	|- ( M e. NN -> ( M + 0 ) = M )
6	3 5	sylan9eqr	|- ( ( M e. NN /\ N = 0 ) -> ( M + N ) = M )
7		simpl	|- ( ( M e. NN /\ N = 0 ) -> M e. NN )
8	6 7	eqeltrd	|- ( ( M e. NN /\ N = 0 ) -> ( M + N ) e. NN )
9	2 8	jaodan	|- ( ( M e. NN /\ ( N e. NN \/ N = 0 ) ) -> ( M + N ) e. NN )
10	1 9	sylan2b	|- ( ( M e. NN /\ N e. NN0 ) -> ( M + N ) e. NN )