Theorem nnnn0addcl 9399

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 9371	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		nnaddcl 9130	. . 3 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
3		oveq2 6009	. . . . 5 |- ( N = 0 -> ( M + N ) = ( M + 0 ) )
4		nncn 9118	. . . . . 6 |- ( M e. NN -> M e. CC )
5	4	addridd 8295	. . . . 5 |- ( M e. NN -> ( M + 0 ) = M )
6	3, 5	sylan9eqr 2284	. . . 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 2306	. . 3 |- ( ( M e. NN /\ N = 0 ) -> ( M + N ) e. NN )
9	2, 8	jaodan 802	. 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 713   = wceq 1395   e. wcel 2200  (class class class)co 6001   0cc0 7999   + caddc 8002   NNcn 9110   NN0cn0 9369

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-sep 4202  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addass 8101  ax-i2m1 8104  ax-0id 8107

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6004  df-inn 9111  df-n0 9370

This theorem is referenced by:  nn0nnaddcl  9400  elz2  9518  bcxmas  12000  dec2nprm  12938