Theorem elznn0 9659

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 9646	. 2 |- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
2		elnn0 9565	. . . . . 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 9565	. . . . . 6 |- ( -u N e. NN0 <-> ( -u N e. NN \/ -u N = 0 ) )
5		recn 8312	. . . . . . . . 9 |- ( N e. RR -> N e. CC )
6		0cn 8318	. . . . . . . . 9 |- 0 e. CC
7		negcon1 8578	. . . . . . . . 9 |- ( ( N e. CC /\ 0 e. CC ) -> ( -u N = 0 <-> -u 0 = N ) )
8	5, 6, 7	sylancl 417	. . . . . . . 8 |- ( N e. RR -> ( -u N = 0 <-> -u 0 = N ) )
9		neg0 8572	. . . . . . . . . 10 |- -u 0 = 0
10	9	eqeq1i 2246	. . . . . . . . 9 |- ( -u 0 = N <-> 0 = N )
11		eqcom 2240	. . . . . . . . 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 802	. . . . . 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 805	. . . 4 |- ( N e. RR -> ( ( N e. NN0 \/ -u N e. NN0 ) <-> ( ( N e. NN \/ N = 0 ) \/ ( -u N e. NN \/ N = 0 ) ) ) )
17		3orass 1012	. . . . 5 |- ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N = 0 \/ ( N e. NN \/ -u N e. NN ) ) )
18		orcom 740	. . . . 5 |- ( ( N = 0 \/ ( N e. NN \/ -u N e. NN ) ) <-> ( ( N e. NN \/ -u N e. NN ) \/ N = 0 ) )
19		orordir 786	. . . . 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 458	. 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 ) ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   \/ wo 720   \/ w3o 1008   = wceq 1402   e. wcel 2209   CCcc 8177   RRcr 8178   0cc0 8179   -ucneg 8498   NNcn 9304   NN0cn0 9563   ZZcz 9644

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-setind 4684  ax-resscn 8271  ax-1cn 8272  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-sub 8499  df-neg 8500  df-n0 9564  df-z 9645

This theorem is used by:  peano2z  9680  zmulcl  9698  elz2  9716  expnegzap  11010  expaddzaplem  11019  odd2np1  12640  bezoutlemzz  12779  bezoutlemaz  12780  bezoutlembz  12781  mulgz  13953  mulgdirlem  13956  mulgdir  13957  mulgass  13962