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Theorem elnnz 9467
Description: Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
Assertion
Ref Expression
elnnz |- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
Proof of Theorem elnnz
StepHypRef Expression
1 nnre 9128 . . . 4 |- ( N e. NN -> N e. RR )
2 orc 717 . . . 4 |- ( N e. NN -> ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) )
3 nngt0 9146 . . . 4 |- ( N e. NN -> 0 < N )
41, 2, 3jca31 309 . . 3 |- ( N e. NN -> ( ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) /\ 0 < N ) )
5 idd 21 . . . . . . 7 |- ( ( N e. RR /\ 0 < N ) -> ( N e. NN -> N e. NN ) )
6 lt0neg2 8627 . . . . . . . . . . . 12 |- ( N e. RR -> ( 0 < N <-> -u N < 0 ) )
7 renegcl 8418 . . . . . . . . . . . . 13 |- ( N e. RR -> -u N e. RR )
8 0re 8157 . . . . . . . . . . . . 13 |- 0 e. RR
9 ltnsym 8243 . . . . . . . . . . . . 13 |- ( ( -u N e. RR /\ 0 e. RR ) -> ( -u N < 0 -> -. 0 < -u N ) )
107, 8, 9sylancl 413 . . . . . . . . . . . 12 |- ( N e. RR -> ( -u N < 0 -> -. 0 < -u N ) )
116, 10sylbid 150 . . . . . . . . . . 11 |- ( N e. RR -> ( 0 < N -> -. 0 < -u N ) )
1211imp 124 . . . . . . . . . 10 |- ( ( N e. RR /\ 0 < N ) -> -. 0 < -u N )
13 nngt0 9146 . . . . . . . . . 10 |- ( -u N e. NN -> 0 < -u N )
1412, 13nsyl 631 . . . . . . . . 9 |- ( ( N e. RR /\ 0 < N ) -> -. -u N e. NN )
15 gt0ne0 8585 . . . . . . . . . 10 |- ( ( N e. RR /\ 0 < N ) -> N =/= 0 )
1615neneqd 2421 . . . . . . . . 9 |- ( ( N e. RR /\ 0 < N ) -> -. N = 0 )
17 ioran 757 . . . . . . . . 9 |- ( -. ( -u N e. NN \/ N = 0 ) <-> ( -. -u N e. NN /\ -. N = 0 ) )
1814, 16, 17sylanbrc 417 . . . . . . . 8 |- ( ( N e. RR /\ 0 < N ) -> -. ( -u N e. NN \/ N = 0 ) )
1918pm2.21d 622 . . . . . . 7 |- ( ( N e. RR /\ 0 < N ) -> ( ( -u N e. NN \/ N = 0 ) -> N e. NN ) )
205, 19jaod 722 . . . . . 6 |- ( ( N e. RR /\ 0 < N ) -> ( ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) -> N e. NN ) )
2120ex 115 . . . . 5 |- ( N e. RR -> ( 0 < N -> ( ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) -> N e. NN ) ) )
2221com23 78 . . . 4 |- ( N e. RR -> ( ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) -> ( 0 < N -> N e. NN ) ) )
2322imp31 256 . . 3 |- ( ( ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) /\ 0 < N ) -> N e. NN )
244, 23impbii 126 . 2 |- ( N e. NN <-> ( ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) /\ 0 < N ) )
25 elz 9459 . . . 4 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
26 3orrot 1008 . . . . . 6 |- ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N e. NN \/ -u N e. NN \/ N = 0 ) )
27 3orass 1005 . . . . . 6 |- ( ( N e. NN \/ -u N e. NN \/ N = 0 ) <-> ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) )
2826, 27bitri 184 . . . . 5 |- ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) )
2928anbi2i 457 . . . 4 |- ( ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) <-> ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) )
3025, 29bitri 184 . . 3 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) )
3130anbi1i 458 . 2 |- ( ( N e. ZZ /\ 0 < N ) <-> ( ( N e. RR /\ ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) /\ 0 < N ) )
3224, 31bitr4i 187 1 |- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   \/ w3o 1001   = wceq 1395   e. wcel 2200   class class class wbr 4083   RRcr 8009   0cc0 8010   < clt 8192   -ucneg 8329   NNcn 9121   ZZcz 9457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-ltadd 8126
This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-z 9458
This theorem is referenced by:  nnssz  9474  elnnz1  9480  znnsub  9509  nn0ge0div  9545  msqznn  9558  elpq  9856  elfz1b  10298  lbfzo0  10393  fzo1fzo0n0  10395  elfzo0z  10396  fzofzim  10400  elfzodifsumelfzo  10419  exp3val  10775  nnesq  10893  swrdlsw  11217  pfxccatin12lem3  11280  nnabscl  11627  cvgratnnlemabsle  12054  p1modz1  12321  nndivdvds  12323  zdvdsdc  12339  oddge22np1  12408  evennn2n  12410  nno  12433  nnoddm1d2  12437  divalglemex  12449  divalglemeuneg  12450  divalg  12451  ndvdsadd  12458  bitsfzolem  12481  sqgcd  12566  qredeu  12635  prmind2  12658  sqrt2irrlem  12699  sqrt2irrap  12718  qgt0numnn  12737  oddprm  12798  pythagtriplem6  12809  pythagtriplem11  12813  pythagtriplem13  12815  pythagtriplem19  12821  pc2dvds  12869  pcadd  12879  4sqlem11  12940  4sqlem12  12941  mulgval  13675  mulgfng  13677  subgmulg  13741  znidomb  14638  sgmnncl  15678  mersenne  15687  gausslemma2dlem1a  15753  lgseisenlem1  15765  lgsquadlem1  15772  lgsquadlem2  15773  2sqlem8  15818
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