Theorem elznn0 8819

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

Step	Hyp	Ref	Expression
1		elz 8806	. 2 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
2		elnn0 8729	. . . . . 6 |- ( N e. NN0 <-> ( N e. NN \/ N = 0 ) )
3	2	a1i 9	. . . . 5 |- ( N e. RR -> ( N e. NN0 <-> ( N e. NN \/ N = 0 ) ) )
4		elnn0 8729	. . . . . 6 |- ( -u N e. NN0 <-> ( -u N e. NN \/ -u N = 0 ) )
5		recn 7529	. . . . . . . . 9 |- ( N e. RR -> N e. CC )
6		0cn 7534	. . . . . . . . 9 |- 0 e. CC
7		negcon1 7788	. . . . . . . . 9 |- ( ( N e. CC /\ 0 e. CC ) -> ( -u N = 0 <-> -u 0 = N ) )
8	5, 6, 7	sylancl 405	. . . . . . . 8 |- ( N e. RR -> ( -u N = 0 <-> -u 0 = N ) )
9		neg0 7782	. . . . . . . . . 10 |- -u 0 = 0
10	9	eqeq1i 2096	. . . . . . . . 9 |- ( -u 0 = N <-> 0 = N )
11		eqcom 2091	. . . . . . . . 9 |- ( 0 = N <-> N = 0 )
12	10, 11	bitri 183	. . . . . . . 8 |- ( -u 0 = N <-> N = 0 )
13	8, 12	syl6bb 195	. . . . . . 7 |- ( N e. RR -> ( -u N = 0 <-> N = 0 ) )
14	13	orbi2d 740	. . . . . 6 |- ( N e. RR -> ( ( -u N e. NN \/ -u N = 0 ) <-> ( -u N e. NN \/ N = 0 ) ) )
15	4, 14	syl5bb 191	. . . . 5 |- ( N e. RR -> ( -u N e. NN0 <-> ( -u N e. NN \/ N = 0 ) ) )
16	3, 15	orbi12d 743	. . . 4 |- ( N e. RR -> ( ( N e. NN0 \/ -u N e. NN0 ) <-> ( ( N e. NN \/ N = 0 ) \/ ( -u N e. NN \/ N = 0 ) ) ) )
17		3orass 928	. . . . 5 |- ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N = 0 \/ ( N e. NN \/ -u N e. NN ) ) )
18		orcom 683	. . . . 5 |- ( ( N = 0 \/ ( N e. NN \/ -u N e. NN ) ) <-> ( ( N e. NN \/ -u N e. NN ) \/ N = 0 ) )
19		orordir 727	. . . . 5 |- ( ( ( N e. NN \/ -u N e. NN ) \/ N = 0 ) <-> ( ( N e. NN \/ N = 0 ) \/ ( -u N e. NN \/ N = 0 ) ) )
20	17, 18, 19	3bitrri 206	. . . 4 |- ( ( ( N e. NN \/ N = 0 ) \/ ( -u N e. NN \/ N = 0 ) ) <-> ( N = 0 \/ N e. NN \/ -u N e. NN ) )
21	16, 20	syl6rbb 196	. . 3 |- ( N e. RR -> ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N e. NN0 \/ -u N e. NN0 ) ) )
22	21	pm5.32i 443	. 2 |- ( ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )
23	1, 22	bitri 183	1 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   \/ wo 665   \/ w3o 924   = wceq 1290   e. wcel 1439   CCcc 7402   RRcr 7403   0cc0 7404   -ucneg 7708   NNcn 8476   NN0cn0 8727   ZZcz 8804

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-setind 4366  ax-resscn 7491  ax-1cn 7492  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-distr 7503  ax-i2m1 7504  ax-0id 7507  ax-rnegex 7508  ax-cnre 7510

This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-iota 4993  df-fun 5030  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-sub 7709  df-neg 7710  df-n0 8728  df-z 8805

This theorem is referenced by:  peano2z  8840  zmulcl  8857  elz2  8872  expnegzap  10043  expaddzaplem  10052  odd2np1  11205  bezoutlemzz  11323  bezoutlemaz  11324  bezoutlembz  11325