Theorem elznn0 9592

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 9579	. 2 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
2		elnn0 9498	. . . . . 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 9498	. . . . . 6 |- ( -u N e. NN0 <-> ( -u N e. NN \/ -u N = 0 ) )
5		recn 8260	. . . . . . . . 9 |- ( N e. RR -> N e. CC )
6		0cn 8266	. . . . . . . . 9 |- 0 e. CC
7		negcon1 8525	. . . . . . . . 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 8519	. . . . . . . . . 10 |- -u 0 = 0
10	9	eqeq1i 2240	. . . . . . . . 9 |- ( -u 0 = N <-> 0 = N )
11		eqcom 2234	. . . . . . . . 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 798	. . . . . 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 801	. . . 4 |- ( N e. RR -> ( ( N e. NN0 \/ -u N e. NN0 ) <-> ( ( N e. NN \/ N = 0 ) \/ ( -u N e. NN \/ N = 0 ) ) ) )
17		3orass 1008	. . . . 5 |- ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N = 0 \/ ( N e. NN \/ -u N e. NN ) ) )
18		orcom 736	. . . . 5 |- ( ( N = 0 \/ ( N e. NN \/ -u N e. NN ) ) <-> ( ( N e. NN \/ -u N e. NN ) \/ N = 0 ) )
19		orordir 782	. . . . 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 716   \/ w3o 1004   = wceq 1398   e. wcel 2203   CCcc 8125   RRcr 8126   0cc0 8127   -ucneg 8445   NNcn 9237   NN0cn0 9496   ZZcz 9577

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-setind 4659  ax-resscn 8219  ax-1cn 8220  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-sub 8446  df-neg 8447  df-n0 9497  df-z 9578

This theorem is referenced by:  peano2z  9613  zmulcl  9631  elz2  9649  expnegzap  10935  expaddzaplem  10944  odd2np1  12559  bezoutlemzz  12698  bezoutlemaz  12699  bezoutlembz  12700  mulgz  13867  mulgdirlem  13870  mulgdir  13871  mulgass  13876