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Theorem nnnn0addcl 8958
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
StepHypRef Expression
1 elnn0 8930 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnaddcl 8697 . . 3  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  +  N
)  e.  NN )
3 oveq2 5748 . . . . 5  |-  ( N  =  0  ->  ( M  +  N )  =  ( M  + 
0 ) )
4 nncn 8685 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  CC )
54addid1d 7875 . . . . 5  |-  ( M  e.  NN  ->  ( M  +  0 )  =  M )
63, 5sylan9eqr 2170 . . . 4  |-  ( ( M  e.  NN  /\  N  =  0 )  ->  ( M  +  N )  =  M )
7 simpl 108 . . . 4  |-  ( ( M  e.  NN  /\  N  =  0 )  ->  M  e.  NN )
86, 7eqeltrd 2192 . . 3  |-  ( ( M  e.  NN  /\  N  =  0 )  ->  ( M  +  N )  e.  NN )
92, 8jaodan 769 . 2  |-  ( ( M  e.  NN  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( M  +  N )  e.  NN )
101, 9sylan2b 283 1  |-  ( ( M  e.  NN  /\  N  e.  NN0 )  -> 
( M  +  N
)  e.  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680    = wceq 1314    e. wcel 1463  (class class class)co 5740   0cc0 7584    + caddc 7587   NNcn 8677   NN0cn0 8928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addass 7686  ax-i2m1 7689  ax-0id 7692
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-iota 5056  df-fv 5099  df-ov 5743  df-inn 8678  df-n0 8929
This theorem is referenced by:  nn0nnaddcl  8959  elz2  9073  bcxmas  11198
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