Theorem nnnn0addcl 9431

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 9403	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		nnaddcl 9162	. . 3 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
3		oveq2 6025	. . . . 5 |- ( N = 0 -> ( M + N ) = ( M + 0 ) )
4		nncn 9150	. . . . . 6 |- ( M e. NN -> M e. CC )
5	4	addridd 8327	. . . . 5 |- ( M e. NN -> ( M + 0 ) = M )
6	3, 5	sylan9eqr 2286	. . . 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 2308	. . 3 |- ( ( M e. NN /\ N = 0 ) -> ( M + N ) e. NN )
9	2, 8	jaodan 804	. 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 715   = wceq 1397   e. wcel 2202  (class class class)co 6017   0cc0 8031   + caddc 8034   NNcn 9142   NN0cn0 9401

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-sep 4207  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addass 8133  ax-i2m1 8136  ax-0id 8139

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6020  df-inn 9143  df-n0 9402

This theorem is referenced by:  nn0nnaddcl  9432  elz2  9550  bcxmas  12049  dec2nprm  12987