Theorem nnnn0addcl 9526

Description: A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Assertion

Ref	Expression
nnnn0addcl	|- ( ( M e. NN /\ N e. NN0 ) -> ( M + N ) e. NN )

Proof of Theorem nnnn0addcl

Step	Hyp	Ref	Expression
1		elnn0 9498	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		nnaddcl 9257	. . 3 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
3		oveq2 6058	. . . . 5 |- ( N = 0 -> ( M + N ) = ( M + 0 ) )
4		nncn 9245	. . . . . 6 |- ( M e. NN -> M e. CC )
5	4	addridd 8422	. . . . 5 |- ( M e. NN -> ( M + 0 ) = M )
6	3, 5	sylan9eqr 2287	. . . 4 |- ( ( M e. NN /\ N = 0 ) -> ( M + N ) = M )
7		simpl 109	. . . 4 |- ( ( M e. NN /\ N = 0 ) -> M e. NN )
8	6, 7	eqeltrd 2309	. . 3 |- ( ( M e. NN /\ N = 0 ) -> ( M + N ) e. NN )
9	2, 8	jaodan 805	. 2 |- ( ( M e. NN /\ ( N e. NN \/ N = 0 ) ) -> ( M + N ) e. NN )
10	1, 9	sylan2b 287	1 |- ( ( M e. NN /\ N e. NN0 ) -> ( M + N ) e. NN )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2203  (class class class)co 6050   0cc0 8127   + caddc 8130   NNcn 9237   NN0cn0 9496

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 2214  ax-sep 4228  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addass 8229  ax-i2m1 8232  ax-0id 8235

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-iota 5312  df-fv 5360  df-ov 6053  df-inn 9238  df-n0 9497

This theorem is referenced by:  nn0nnaddcl  9527  elz2  9649  bcxmas  12175  dec2nprm  13113