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Theorem elnnz 9252
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 8915 . . . 4 |- ( N e. NN -> N e. RR )
2 orc 712 . . . 4 |- ( N e. NN -> ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) )
3 nngt0 8933 . . . 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 8416 . . . . . . . . . . . 12 |- ( N e. RR -> ( 0 < N <-> -u N < 0 ) )
7 renegcl 8208 . . . . . . . . . . . . 13 |- ( N e. RR -> -u N e. RR )
8 0re 7948 . . . . . . . . . . . . 13 |- 0 e. RR
9 ltnsym 8033 . . . . . . . . . . . . 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 8933 . . . . . . . . . 10 |- ( -u N e. NN -> 0 < -u N )
1412, 13nsyl 628 . . . . . . . . 9 |- ( ( N e. RR /\ 0 < N ) -> -. -u N e. NN )
15 gt0ne0 8374 . . . . . . . . . 10 |- ( ( N e. RR /\ 0 < N ) -> N =/= 0 )
1615neneqd 2368 . . . . . . . . 9 |- ( ( N e. RR /\ 0 < N ) -> -. N = 0 )
17 ioran 752 . . . . . . . . 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 619 . . . . . . 7 |- ( ( N e. RR /\ 0 < N ) -> ( ( -u N e. NN \/ N = 0 ) -> N e. NN ) )
205, 19jaod 717 . . . . . 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 9244 . . . 4 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
26 3orrot 984 . . . . . 6 |- ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N e. NN \/ -u N e. NN \/ N = 0 ) )
27 3orass 981 . . . . . 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 708   \/ w3o 977   = wceq 1353   e. wcel 2148   class class class wbr 4000   RRcr 7801   0cc0 7802   < clt 7982   -ucneg 8119   NNcn 8908   ZZcz 9242
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-z 9243
This theorem is referenced by:  nnssz  9259  elnnz1  9265  znnsub  9293  nn0ge0div  9329  msqznn  9342  elpq  9637  elfz1b  10076  lbfzo0  10167  fzo1fzo0n0  10169  elfzo0z  10170  fzofzim  10174  elfzodifsumelfzo  10187  exp3val  10508  nnesq  10625  nnabscl  11093  cvgratnnlemabsle  11519  p1modz1  11785  nndivdvds  11787  zdvdsdc  11803  oddge22np1  11869  evennn2n  11871  nno  11894  nnoddm1d2  11898  divalglemex  11910  divalglemeuneg  11911  divalg  11912  ndvdsadd  11919  sqgcd  12013  qredeu  12080  prmind2  12103  sqrt2irrlem  12144  sqrt2irrap  12163  qgt0numnn  12182  oddprm  12242  pythagtriplem6  12253  pythagtriplem11  12257  pythagtriplem13  12259  pythagtriplem19  12265  pc2dvds  12312  pcadd  12322  mulgval  12875  mulgfng  12876  2sqlem8  14126
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