Metamath Proof Explorer


Theorem nnaddcld

Description: Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses nnge1d.1
|- ( ph -> A e. NN )
nnmulcld.2
|- ( ph -> B e. NN )
Assertion nnaddcld
|- ( ph -> ( A + B ) e. NN )

Proof

Step Hyp Ref Expression
1 nnge1d.1
 |-  ( ph -> A e. NN )
2 nnmulcld.2
 |-  ( ph -> B e. NN )
3 nnaddcl
 |-  ( ( A e. NN /\ B e. NN ) -> ( A + B ) e. NN )
4 1 2 3 syl2anc
 |-  ( ph -> ( A + B ) e. NN )