Theorem nnaddcld 9287

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 9259	. 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 2205  (class class class)co 6052   + caddc 8132   NNcn 9239

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-sep 4230  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-addrcl 8226  ax-addass 8231

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-iota 5314  df-fv 5362  df-ov 6055  df-inn 9240

This theorem is referenced by:  pythagtriplem4  12970  pythagtriplem6  12972  pythagtriplem7  12973  pythagtriplem11  12976  pythagtriplem12  12977  pythagtriplem13  12978  pythagtriplem14  12979  pythagtriplem15  12980  pythagtriplem16  12981  mulgnndir  13885  perfectlem2  15885  lgseisenlem2  15961