Theorem elznn0 9332

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 9319	. 2 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
2		elnn0 9242	. . . . . 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 9242	. . . . . 6 |- ( -u N e. NN0 <-> ( -u N e. NN \/ -u N = 0 ) )
5		recn 8005	. . . . . . . . 9 |- ( N e. RR -> N e. CC )
6		0cn 8011	. . . . . . . . 9 |- 0 e. CC
7		negcon1 8271	. . . . . . . . 9 |- ( ( N e. CC /\ 0 e. CC ) -> ( -u N = 0 <-> -u 0 = N ) )
8	5, 6, 7	sylancl 413	. . . . . . . 8 |- ( N e. RR -> ( -u N = 0 <-> -u 0 = N ) )
9		neg0 8265	. . . . . . . . . 10 |- -u 0 = 0
10	9	eqeq1i 2201	. . . . . . . . 9 |- ( -u 0 = N <-> 0 = N )
11		eqcom 2195	. . . . . . . . 9 |- ( 0 = N <-> N = 0 )
12	10, 11	bitri 184	. . . . . . . 8 |- ( -u 0 = N <-> N = 0 )
13	8, 12	bitrdi 196	. . . . . . 7 |- ( N e. RR -> ( -u N = 0 <-> N = 0 ) )
14	13	orbi2d 791	. . . . . 6 |- ( N e. RR -> ( ( -u N e. NN \/ -u N = 0 ) <-> ( -u N e. NN \/ N = 0 ) ) )
15	4, 14	bitrid 192	. . . . 5 |- ( N e. RR -> ( -u N e. NN0 <-> ( -u N e. NN \/ N = 0 ) ) )
16	3, 15	orbi12d 794	. . . 4 |- ( N e. RR -> ( ( N e. NN0 \/ -u N e. NN0 ) <-> ( ( N e. NN \/ N = 0 ) \/ ( -u N e. NN \/ N = 0 ) ) ) )
17		3orass 983	. . . . 5 |- ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N = 0 \/ ( N e. NN \/ -u N e. NN ) ) )
18		orcom 729	. . . . 5 |- ( ( N = 0 \/ ( N e. NN \/ -u N e. NN ) ) <-> ( ( N e. NN \/ -u N e. NN ) \/ N = 0 ) )
19		orordir 775	. . . . 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 207	. . . 4 |- ( ( ( N e. NN \/ N = 0 ) \/ ( -u N e. NN \/ N = 0 ) ) <-> ( N = 0 \/ N e. NN \/ -u N e. NN ) )
21	16, 20	bitr2di 197	. . 3 |- ( N e. RR -> ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N e. NN0 \/ -u N e. NN0 ) ) )
22	21	pm5.32i 454	. 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 184	1 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   \/ wo 709   \/ w3o 979   = wceq 1364   e. wcel 2164   CCcc 7870   RRcr 7871   0cc0 7872   -ucneg 8191   NNcn 8982   NN0cn0 9240   ZZcz 9317

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569  ax-resscn 7964  ax-1cn 7965  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-sub 8192  df-neg 8193  df-n0 9241  df-z 9318

This theorem is referenced by:  peano2z  9353  zmulcl  9370  elz2  9388  expnegzap  10644  expaddzaplem  10653  odd2np1  12014  bezoutlemzz  12139  bezoutlemaz  12140  bezoutlembz  12141  mulgz  13220  mulgdirlem  13223  mulgdir  13224  mulgass  13229