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Theorem elznn0 9076
Description: Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
Assertion
Ref Expression
elznn0  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )

Proof of Theorem elznn0
StepHypRef Expression
1 elz 9063 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
2 elnn0 8986 . . . . . 6  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
32a1i 9 . . . . 5  |-  ( N  e.  RR  ->  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) ) )
4 elnn0 8986 . . . . . 6  |-  ( -u N  e.  NN0  <->  ( -u N  e.  NN  \/  -u N  =  0 ) )
5 recn 7760 . . . . . . . . 9  |-  ( N  e.  RR  ->  N  e.  CC )
6 0cn 7765 . . . . . . . . 9  |-  0  e.  CC
7 negcon1 8021 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  0  e.  CC )  ->  ( -u N  =  0  <->  -u 0  =  N ) )
85, 6, 7sylancl 409 . . . . . . . 8  |-  ( N  e.  RR  ->  ( -u N  =  0  <->  -u 0  =  N ) )
9 neg0 8015 . . . . . . . . . 10  |-  -u 0  =  0
109eqeq1i 2147 . . . . . . . . 9  |-  ( -u
0  =  N  <->  0  =  N )
11 eqcom 2141 . . . . . . . . 9  |-  ( 0  =  N  <->  N  = 
0 )
1210, 11bitri 183 . . . . . . . 8  |-  ( -u
0  =  N  <->  N  = 
0 )
138, 12syl6bb 195 . . . . . . 7  |-  ( N  e.  RR  ->  ( -u N  =  0  <->  N  =  0 ) )
1413orbi2d 779 . . . . . 6  |-  ( N  e.  RR  ->  (
( -u N  e.  NN  \/  -u N  =  0 )  <->  ( -u N  e.  NN  \/  N  =  0 ) ) )
154, 14syl5bb 191 . . . . 5  |-  ( N  e.  RR  ->  ( -u N  e.  NN0  <->  ( -u N  e.  NN  \/  N  =  0 ) ) )
163, 15orbi12d 782 . . . 4  |-  ( N  e.  RR  ->  (
( N  e.  NN0  \/  -u N  e.  NN0 ) 
<->  ( ( N  e.  NN  \/  N  =  0 )  \/  ( -u N  e.  NN  \/  N  =  0 ) ) ) )
17 3orass 965 . . . . 5  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( N  =  0  \/  ( N  e.  NN  \/  -u N  e.  NN ) ) )
18 orcom 717 . . . . 5  |-  ( ( N  =  0  \/  ( N  e.  NN  \/  -u N  e.  NN ) )  <->  ( ( N  e.  NN  \/  -u N  e.  NN )  \/  N  =  0 ) )
19 orordir 763 . . . . 5  |-  ( ( ( N  e.  NN  \/  -u N  e.  NN )  \/  N  = 
0 )  <->  ( ( N  e.  NN  \/  N  =  0 )  \/  ( -u N  e.  NN  \/  N  =  0 ) ) )
2017, 18, 193bitrri 206 . . . 4  |-  ( ( ( N  e.  NN  \/  N  =  0
)  \/  ( -u N  e.  NN  \/  N  =  0 ) )  <->  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) )
2116, 20syl6rbb 196 . . 3  |-  ( N  e.  RR  ->  (
( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) 
<->  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
2221pm5.32i 449 . 2  |-  ( ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) )  <->  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
231, 22bitri 183 1  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    \/ wo 697    \/ w3o 961    = wceq 1331    e. wcel 1480   CCcc 7625   RRcr 7626   0cc0 7627   -ucneg 7941   NNcn 8727   NN0cn0 8984   ZZcz 9061
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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452  ax-resscn 7719  ax-1cn 7720  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7942  df-neg 7943  df-n0 8985  df-z 9062
This theorem is referenced by:  peano2z  9097  zmulcl  9114  elz2  9129  expnegzap  10334  expaddzaplem  10343  odd2np1  11577  bezoutlemzz  11697  bezoutlemaz  11698  bezoutlembz  11699
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