Theorem elznn0 8366

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 8353	. 2 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
2		elnn0 8290	. . . . . 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 8290	. . . . . 6 |- ( -u N e. NN0 <-> ( -u N e. NN \/ -u N = 0 ) )
5		recn 7106	. . . . . . . . 9 |- ( N e. RR -> N e. CC )
6		0cn 7111	. . . . . . . . 9 |- 0 e. CC
7		negcon1 7360	. . . . . . . . 9 |- ( ( N e. CC /\ 0 e. CC ) -> ( -u N = 0 <-> -u 0 = N ) )
8	5, 6, 7	sylancl 404	. . . . . . . 8 |- ( N e. RR -> ( -u N = 0 <-> -u 0 = N ) )
9		neg0 7354	. . . . . . . . . 10 |- -u 0 = 0
10	9	eqeq1i 2088	. . . . . . . . 9 |- ( -u 0 = N <-> 0 = N )
11		eqcom 2083	. . . . . . . . 9 |- ( 0 = N <-> N = 0 )
12	10, 11	bitri 182	. . . . . . . 8 |- ( -u 0 = N <-> N = 0 )
13	8, 12	syl6bb 194	. . . . . . 7 |- ( N e. RR -> ( -u N = 0 <-> N = 0 ) )
14	13	orbi2d 736	. . . . . 6 |- ( N e. RR -> ( ( -u N e. NN \/ -u N = 0 ) <-> ( -u N e. NN \/ N = 0 ) ) )
15	4, 14	syl5bb 190	. . . . 5 |- ( N e. RR -> ( -u N e. NN0 <-> ( -u N e. NN \/ N = 0 ) ) )
16	3, 15	orbi12d 739	. . . 4 |- ( N e. RR -> ( ( N e. NN0 \/ -u N e. NN0 ) <-> ( ( N e. NN \/ N = 0 ) \/ ( -u N e. NN \/ N = 0 ) ) ) )
17		3orass 922	. . . . 5 |- ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N = 0 \/ ( N e. NN \/ -u N e. NN ) ) )
18		orcom 679	. . . . 5 |- ( ( N = 0 \/ ( N e. NN \/ -u N e. NN ) ) <-> ( ( N e. NN \/ -u N e. NN ) \/ N = 0 ) )
19		orordir 723	. . . . 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 205	. . . 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 195	. . 3 |- ( N e. RR -> ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N e. NN0 \/ -u N e. NN0 ) ) )
22	21	pm5.32i 441	. 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 182	1 |- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )

Syntax hints:   /\ wa 102   <-> wb 103   \/ wo 661   \/ w3o 918   = wceq 1284   e. wcel 1433   CCcc 6979   RRcr 6980   0cc0 6981   -ucneg 7280   NNcn 8039   NN0cn0 8288   ZZcz 8351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280  ax-resscn 7068  ax-1cn 7069  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-sub 7281  df-neg 7282  df-n0 8289  df-z 8352

This theorem is referenced by:  peano2z  8387  zmulcl  8404  elz2  8419  expnegzap  9510  expaddzaplem  9519  odd2np1  10272  bezoutlemzz  10391  bezoutlemaz  10392  bezoutlembz  10393