Theorem nnnn0addcl 9572

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 9544	. 2 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
2		nnaddcl 9303	. . 3 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
3		oveq2 6083	. . . . 5 |- ( N = 0 -> ( M + N ) = ( M + 0 ) )
4		nncn 9291	. . . . . 6 |- ( M e. NN -> M e. CC )
5	4	addridd 8465	. . . . 5 |- ( M e. NN -> ( M + 0 ) = M )
6	3, 5	sylan9eqr 2293	. . . 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 2315	. . 3 |- ( ( M e. NN /\ N = 0 ) -> ( M + N ) e. NN )
9	2, 8	jaodan 809	. 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 720   = wceq 1402   e. wcel 2209  (class class class)co 6075   0cc0 8169   + caddc 8172   NNcn 9283   NN0cn0 9542

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4244  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addass 8271  ax-i2m1 8274  ax-0id 8277

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-iota 5332  df-fv 5380  df-ov 6078  df-inn 9284  df-n0 9543

This theorem is referenced by:  nn0nnaddcl  9573  elz2  9695  bcxmas  12234  dec2nprm  13172