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Theorem nnnn0addcl 9206
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 9178 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnaddcl 8939 . . 3  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  +  N
)  e.  NN )
3 oveq2 5883 . . . . 5  |-  ( N  =  0  ->  ( M  +  N )  =  ( M  + 
0 ) )
4 nncn 8927 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  CC )
54addid1d 8106 . . . . 5  |-  ( M  e.  NN  ->  ( M  +  0 )  =  M )
63, 5sylan9eqr 2232 . . . 4  |-  ( ( M  e.  NN  /\  N  =  0 )  ->  ( M  +  N )  =  M )
7 simpl 109 . . . 4  |-  ( ( M  e.  NN  /\  N  =  0 )  ->  M  e.  NN )
86, 7eqeltrd 2254 . . 3  |-  ( ( M  e.  NN  /\  N  =  0 )  ->  ( M  +  N )  e.  NN )
92, 8jaodan 797 . 2  |-  ( ( M  e.  NN  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( M  +  N )  e.  NN )
101, 9sylan2b 287 1  |-  ( ( M  e.  NN  /\  N  e.  NN0 )  -> 
( M  +  N
)  e.  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 708    = wceq 1353    e. wcel 2148  (class class class)co 5875   0cc0 7811    + caddc 7814   NNcn 8919   NN0cn0 9176
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4122  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addass 7913  ax-i2m1 7916  ax-0id 7919
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-iota 5179  df-fv 5225  df-ov 5878  df-inn 8920  df-n0 9177
This theorem is referenced by:  nn0nnaddcl  9207  elz2  9324  bcxmas  11497
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