Theorem nn0nnaddcl

Description: A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005)

	Ref	Expression
Assertion	nn0nnaddcl	|- ( ( M e. NN0 /\ N e. NN ) -> ( M + N ) e. NN )

Proof

Step	Hyp	Ref	Expression
1		nncn	|- ( N e. NN -> N e. CC )
2		nn0cn	|- ( M e. NN0 -> M e. CC )
3		addcom	|- ( ( N e. CC /\ M e. CC ) -> ( N + M ) = ( M + N ) )
4	1 2 3	syl2an	|- ( ( N e. NN /\ M e. NN0 ) -> ( N + M ) = ( M + N ) )
5		nnnn0addcl	|- ( ( N e. NN /\ M e. NN0 ) -> ( N + M ) e. NN )
6	4 5	eqeltrrd	|- ( ( N e. NN /\ M e. NN0 ) -> ( M + N ) e. NN )
7	6	ancoms	|- ( ( M e. NN0 /\ N e. NN ) -> ( M + N ) e. NN )