Theorem nnaddcld 8905

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

Hypotheses

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

Assertion

Ref	Expression
nnaddcld	|- ( ph -> ( A + B ) e. NN )

Proof of Theorem nnaddcld

Step	Hyp	Ref	Expression
1		nnge1d.1	. 2 |- ( ph -> A e. NN )
2		nnmulcld.2	. 2 |- ( ph -> B e. NN )
3		nnaddcl 8877	. 2 |- ( ( A e. NN /\ B e. NN ) -> ( A + B ) e. NN )
4	1, 2, 3	syl2anc 409	1 |- ( ph -> ( A + B ) e. NN )

Syntax hints:   -> wi 4   e. wcel 2136  (class class class)co 5842   + caddc 7756   NNcn 8857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-sep 4100  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-addrcl 7850  ax-addass 7855

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845  df-inn 8858

This theorem is referenced by:  pythagtriplem4  12200  pythagtriplem6  12202  pythagtriplem7  12203  pythagtriplem11  12206  pythagtriplem12  12207  pythagtriplem13  12208  pythagtriplem14  12209  pythagtriplem15  12210  pythagtriplem16  12211