ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elznn0 Unicode version

Theorem elznn0 8317
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 8304 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
2 elnn0 8241 . . . . . 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 8241 . . . . . 6  |-  ( -u N  e.  NN0  <->  ( -u N  e.  NN  \/  -u N  =  0 ) )
5 recn 7072 . . . . . . . . 9  |-  ( N  e.  RR  ->  N  e.  CC )
6 0cn 7077 . . . . . . . . 9  |-  0  e.  CC
7 negcon1 7326 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  0  e.  CC )  ->  ( -u N  =  0  <->  -u 0  =  N ) )
85, 6, 7sylancl 398 . . . . . . . 8  |-  ( N  e.  RR  ->  ( -u N  =  0  <->  -u 0  =  N ) )
9 neg0 7320 . . . . . . . . . 10  |-  -u 0  =  0
109eqeq1i 2063 . . . . . . . . 9  |-  ( -u
0  =  N  <->  0  =  N )
11 eqcom 2058 . . . . . . . . 9  |-  ( 0  =  N  <->  N  = 
0 )
1210, 11bitri 177 . . . . . . . 8  |-  ( -u
0  =  N  <->  N  = 
0 )
138, 12syl6bb 189 . . . . . . 7  |-  ( N  e.  RR  ->  ( -u N  =  0  <->  N  =  0 ) )
1413orbi2d 714 . . . . . 6  |-  ( N  e.  RR  ->  (
( -u N  e.  NN  \/  -u N  =  0 )  <->  ( -u N  e.  NN  \/  N  =  0 ) ) )
154, 14syl5bb 185 . . . . 5  |-  ( N  e.  RR  ->  ( -u N  e.  NN0  <->  ( -u N  e.  NN  \/  N  =  0 ) ) )
163, 15orbi12d 717 . . . 4  |-  ( N  e.  RR  ->  (
( N  e.  NN0  \/  -u N  e.  NN0 ) 
<->  ( ( N  e.  NN  \/  N  =  0 )  \/  ( -u N  e.  NN  \/  N  =  0 ) ) ) )
17 3orass 899 . . . . 5  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( N  =  0  \/  ( N  e.  NN  \/  -u N  e.  NN ) ) )
18 orcom 657 . . . . 5  |-  ( ( N  =  0  \/  ( N  e.  NN  \/  -u N  e.  NN ) )  <->  ( ( N  e.  NN  \/  -u N  e.  NN )  \/  N  =  0 ) )
19 orordir 701 . . . . 5  |-  ( ( ( N  e.  NN  \/  -u N  e.  NN )  \/  N  = 
0 )  <->  ( ( N  e.  NN  \/  N  =  0 )  \/  ( -u N  e.  NN  \/  N  =  0 ) ) )
2017, 18, 193bitrri 200 . . . 4  |-  ( ( ( N  e.  NN  \/  N  =  0
)  \/  ( -u N  e.  NN  \/  N  =  0 ) )  <->  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) )
2116, 20syl6rbb 190 . . 3  |-  ( N  e.  RR  ->  (
( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) 
<->  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
2221pm5.32i 435 . 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 177 1  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    \/ wo 639    \/ w3o 895    = wceq 1259    e. wcel 1409   CCcc 6945   RRcr 6946   0cc0 6947   -ucneg 7246   NNcn 7990   NN0cn0 8239   ZZcz 8302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290  ax-resscn 7034  ax-1cn 7035  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-sub 7247  df-neg 7248  df-n0 8240  df-z 8303
This theorem is referenced by:  peano2z  8338  zmulcl  8355  elz2  8370  expnegzap  9454  expaddzaplem  9463  odd2np1  10184
  Copyright terms: Public domain W3C validator