Metamath Proof Explorer


Theorem numnncl

Description: Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014)

Ref Expression
Hypotheses numnncl.1
|- T e. NN0
numnncl.2
|- A e. NN0
numnncl.3
|- B e. NN
Assertion numnncl
|- ( ( T x. A ) + B ) e. NN

Proof

Step Hyp Ref Expression
1 numnncl.1
 |-  T e. NN0
2 numnncl.2
 |-  A e. NN0
3 numnncl.3
 |-  B e. NN
4 1 2 nn0mulcli
 |-  ( T x. A ) e. NN0
5 nn0nnaddcl
 |-  ( ( ( T x. A ) e. NN0 /\ B e. NN ) -> ( ( T x. A ) + B ) e. NN )
6 4 3 5 mp2an
 |-  ( ( T x. A ) + B ) e. NN