MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elznn0 Structured version   Unicode version

Theorem elznn0 10653
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 10640 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
2 elnn0 10573 . . . . . 6  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
32a1i 11 . . . . 5  |-  ( N  e.  RR  ->  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) ) )
4 elnn0 10573 . . . . . 6  |-  ( -u N  e.  NN0  <->  ( -u N  e.  NN  \/  -u N  =  0 ) )
5 recn 9364 . . . . . . . . 9  |-  ( N  e.  RR  ->  N  e.  CC )
6 0cn 9370 . . . . . . . . 9  |-  0  e.  CC
7 negcon1 9653 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  0  e.  CC )  ->  ( -u N  =  0  <->  -u 0  =  N ) )
85, 6, 7sylancl 662 . . . . . . . 8  |-  ( N  e.  RR  ->  ( -u N  =  0  <->  -u 0  =  N ) )
9 neg0 9647 . . . . . . . . . 10  |-  -u 0  =  0
109eqeq1i 2445 . . . . . . . . 9  |-  ( -u
0  =  N  <->  0  =  N )
11 eqcom 2440 . . . . . . . . 9  |-  ( 0  =  N  <->  N  = 
0 )
1210, 11bitri 249 . . . . . . . 8  |-  ( -u
0  =  N  <->  N  = 
0 )
138, 12syl6bb 261 . . . . . . 7  |-  ( N  e.  RR  ->  ( -u N  =  0  <->  N  =  0 ) )
1413orbi2d 701 . . . . . 6  |-  ( N  e.  RR  ->  (
( -u N  e.  NN  \/  -u N  =  0 )  <->  ( -u N  e.  NN  \/  N  =  0 ) ) )
154, 14syl5bb 257 . . . . 5  |-  ( N  e.  RR  ->  ( -u N  e.  NN0  <->  ( -u N  e.  NN  \/  N  =  0 ) ) )
163, 15orbi12d 709 . . . 4  |-  ( N  e.  RR  ->  (
( N  e.  NN0  \/  -u N  e.  NN0 ) 
<->  ( ( N  e.  NN  \/  N  =  0 )  \/  ( -u N  e.  NN  \/  N  =  0 ) ) ) )
17 3orass 968 . . . . 5  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( N  =  0  \/  ( N  e.  NN  \/  -u N  e.  NN ) ) )
18 orcom 387 . . . . 5  |-  ( ( N  =  0  \/  ( N  e.  NN  \/  -u N  e.  NN ) )  <->  ( ( N  e.  NN  \/  -u N  e.  NN )  \/  N  =  0 ) )
19 orordir 531 . . . . 5  |-  ( ( ( N  e.  NN  \/  -u N  e.  NN )  \/  N  = 
0 )  <->  ( ( N  e.  NN  \/  N  =  0 )  \/  ( -u N  e.  NN  \/  N  =  0 ) ) )
2017, 18, 193bitrri 272 . . . 4  |-  ( ( ( N  e.  NN  \/  N  =  0
)  \/  ( -u N  e.  NN  \/  N  =  0 ) )  <->  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) )
2116, 20syl6rbb 262 . . 3  |-  ( N  e.  RR  ->  (
( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) 
<->  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
2221pm5.32i 637 . 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 249 1  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 964    = wceq 1369    e. wcel 1756   CCcc 9272   RRcr 9273   0cc0 9274   -ucneg 9588   NNcn 10314   NN0cn0 10571   ZZcz 10638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-ltxr 9415  df-sub 9589  df-neg 9590  df-n0 10572  df-z 10639
This theorem is referenced by:  elz2  10655  zmulcl  10685  expnegz  11890  expaddzlem  11899  odd2np1  13584  mulgz  15639  mulgdirlem  15642  mulgdir  15643  mulgass  15648  mulgdi  16305  cxpmul2z  22116  gxneg  23721  gxadd  23730  gxmul  23733  rexzrexnn0  29113  pell1234qrdich  29173  pell14qrexpcl  29179  pell14qrdich  29181  rmxnn  29265  jm2.19lem4  29312
  Copyright terms: Public domain W3C validator