Theorem nnaddcld 9055

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 9027	. 2 |- ( ( A e. NN /\ B e. NN ) -> ( A + B ) e. NN )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> ( A + B ) e. NN )

Syntax hints:   -> wi 4   e. wcel 2167  (class class class)co 5925   + caddc 7899   NNcn 9007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-sep 4152  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-addrcl 7993  ax-addass 7998

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-iota 5220  df-fv 5267  df-ov 5928  df-inn 9008

This theorem is referenced by:  pythagtriplem4  12462  pythagtriplem6  12464  pythagtriplem7  12465  pythagtriplem11  12468  pythagtriplem12  12469  pythagtriplem13  12470  pythagtriplem14  12471  pythagtriplem15  12472  pythagtriplem16  12473  mulgnndir  13357  perfectlem2  15320  lgseisenlem2  15396