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Theorem elznn0 9214
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 9201 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) ) )
2 elnn0 9124 . . . . . 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 9124 . . . . . 6  |-  ( -u N  e.  NN0  <->  ( -u N  e.  NN  \/  -u N  =  0 ) )
5 recn 7894 . . . . . . . . 9  |-  ( N  e.  RR  ->  N  e.  CC )
6 0cn 7899 . . . . . . . . 9  |-  0  e.  CC
7 negcon1 8158 . . . . . . . . 9  |-  ( ( N  e.  CC  /\  0  e.  CC )  ->  ( -u N  =  0  <->  -u 0  =  N ) )
85, 6, 7sylancl 411 . . . . . . . 8  |-  ( N  e.  RR  ->  ( -u N  =  0  <->  -u 0  =  N ) )
9 neg0 8152 . . . . . . . . . 10  |-  -u 0  =  0
109eqeq1i 2178 . . . . . . . . 9  |-  ( -u
0  =  N  <->  0  =  N )
11 eqcom 2172 . . . . . . . . 9  |-  ( 0  =  N  <->  N  = 
0 )
1210, 11bitri 183 . . . . . . . 8  |-  ( -u
0  =  N  <->  N  = 
0 )
138, 12bitrdi 195 . . . . . . 7  |-  ( N  e.  RR  ->  ( -u N  =  0  <->  N  =  0 ) )
1413orbi2d 785 . . . . . 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 788 . . . 4  |-  ( N  e.  RR  ->  (
( N  e.  NN0  \/  -u N  e.  NN0 ) 
<->  ( ( N  e.  NN  \/  N  =  0 )  \/  ( -u N  e.  NN  \/  N  =  0 ) ) ) )
17 3orass 976 . . . . 5  |-  ( ( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN )  <-> 
( N  =  0  \/  ( N  e.  NN  \/  -u N  e.  NN ) ) )
18 orcom 723 . . . . 5  |-  ( ( N  =  0  \/  ( N  e.  NN  \/  -u N  e.  NN ) )  <->  ( ( N  e.  NN  \/  -u N  e.  NN )  \/  N  =  0 ) )
19 orordir 769 . . . . 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, 20bitr2di 196 . . 3  |-  ( N  e.  RR  ->  (
( N  =  0  \/  N  e.  NN  \/  -u N  e.  NN ) 
<->  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
2221pm5.32i 451 . 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 703    \/ w3o 972    = wceq 1348    e. wcel 2141   CCcc 7759   RRcr 7760   0cc0 7761   -ucneg 8078   NNcn 8865   NN0cn0 9122   ZZcz 9199
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 609  ax-in2 610  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-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-setind 4519  ax-resscn 7853  ax-1cn 7854  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-addass 7863  ax-distr 7865  ax-i2m1 7866  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-sub 8079  df-neg 8080  df-n0 9123  df-z 9200
This theorem is referenced by:  peano2z  9235  zmulcl  9252  elz2  9270  expnegzap  10497  expaddzaplem  10506  odd2np1  11819  bezoutlemzz  11944  bezoutlemaz  11945  bezoutlembz  11946
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