Theorem nn0addcl

Description: Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002) (Proof shortened by Mario Carneiro, 17-Jul-2014)

	Ref	Expression
Assertion	nn0addcl	|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M + N ) e. NN0 )

Proof

Step	Hyp	Ref	Expression
1		nnsscn	|- NN C_ CC
2		id	|- ( NN C_ CC -> NN C_ CC )
3		df-n0	|- NN0 = ( NN u. { 0 } )
4		nnaddcl	|- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
5	4	adantl	|- ( ( NN C_ CC /\ ( M e. NN /\ N e. NN ) ) -> ( M + N ) e. NN )
6	2 3 5	un0addcl	|- ( ( NN C_ CC /\ ( M e. NN0 /\ N e. NN0 ) ) -> ( M + N ) e. NN0 )
7	1 6	mpan	|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M + N ) e. NN0 )