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Theorem nnnn0addcl 9165
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 9137 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnaddcl 8898 . . 3  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  +  N
)  e.  NN )
3 oveq2 5861 . . . . 5  |-  ( N  =  0  ->  ( M  +  N )  =  ( M  + 
0 ) )
4 nncn 8886 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  CC )
54addid1d 8068 . . . . 5  |-  ( M  e.  NN  ->  ( M  +  0 )  =  M )
63, 5sylan9eqr 2225 . . . 4  |-  ( ( M  e.  NN  /\  N  =  0 )  ->  ( M  +  N )  =  M )
7 simpl 108 . . . 4  |-  ( ( M  e.  NN  /\  N  =  0 )  ->  M  e.  NN )
86, 7eqeltrd 2247 . . 3  |-  ( ( M  e.  NN  /\  N  =  0 )  ->  ( M  +  N )  e.  NN )
92, 8jaodan 792 . 2  |-  ( ( M  e.  NN  /\  ( N  e.  NN  \/  N  =  0
) )  ->  ( M  +  N )  e.  NN )
101, 9sylan2b 285 1  |-  ( ( M  e.  NN  /\  N  e.  NN0 )  -> 
( M  +  N
)  e.  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 703    = wceq 1348    e. wcel 2141  (class class class)co 5853   0cc0 7774    + caddc 7777   NNcn 8878   NN0cn0 9135
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addass 7876  ax-i2m1 7879  ax-0id 7882
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856  df-inn 8879  df-n0 9136
This theorem is referenced by:  nn0nnaddcl  9166  elz2  9283  bcxmas  11452
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