ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nnnn0addcl Unicode version

Theorem nnnn0addcl 9120
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 9092 . 2  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 nnaddcl 8853 . . 3  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  +  N
)  e.  NN )
3 oveq2 5832 . . . . 5  |-  ( N  =  0  ->  ( M  +  N )  =  ( M  + 
0 ) )
4 nncn 8841 . . . . . 6  |-  ( M  e.  NN  ->  M  e.  CC )
54addid1d 8024 . . . . 5  |-  ( M  e.  NN  ->  ( M  +  0 )  =  M )
63, 5sylan9eqr 2212 . . . 4  |-  ( ( M  e.  NN  /\  N  =  0 )  ->  ( M  +  N )  =  M )
7 simpl 108 . . . 4  |-  ( ( M  e.  NN  /\  N  =  0 )  ->  M  e.  NN )
86, 7eqeltrd 2234 . . 3  |-  ( ( M  e.  NN  /\  N  =  0 )  ->  ( M  +  N )  e.  NN )
92, 8jaodan 787 . 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 698    = wceq 1335    e. wcel 2128  (class class class)co 5824   0cc0 7732    + caddc 7735   NNcn 8833   NN0cn0 9090
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-sep 4082  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addass 7834  ax-i2m1 7837  ax-0id 7840
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-iota 5135  df-fv 5178  df-ov 5827  df-inn 8834  df-n0 9091
This theorem is referenced by:  nn0nnaddcl  9121  elz2  9235  bcxmas  11386
  Copyright terms: Public domain W3C validator