Theorem peano2z 9353

Description: Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)

Assertion

Ref	Expression
peano2z	|- ( N e. ZZ -> ( N + 1 ) e. ZZ )

Proof of Theorem peano2z

Step	Hyp	Ref	Expression
1		zre 9321	. . 3 |- ( N e. ZZ -> N e. RR )
2		1red 8034	. . 3 |- ( N e. ZZ -> 1 e. RR )
3	1, 2	readdcld 8049	. 2 |- ( N e. ZZ -> ( N + 1 ) e. RR )
4		elznn0nn 9331	. . . . 5 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
5	4	biimpi 120	. . . 4 |- ( N e. ZZ -> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
6	1	biantrurd 305	. . . . 5 |- ( N e. ZZ -> ( -u N e. NN <-> ( N e. RR /\ -u N e. NN ) ) )
7	6	orbi2d 791	. . . 4 |- ( N e. ZZ -> ( ( N e. NN0 \/ -u N e. NN ) <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) ) )
8	5, 7	mpbird 167	. . 3 |- ( N e. ZZ -> ( N e. NN0 \/ -u N e. NN ) )
9		peano2nn0 9280	. . . . 5 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
10	9	a1i 9	. . . 4 |- ( N e. ZZ -> ( N e. NN0 -> ( N + 1 ) e. NN0 ) )
11	1	adantr 276	. . . . . . . . 9 |- ( ( N e. ZZ /\ -u N e. NN ) -> N e. RR )
12		1red 8034	. . . . . . . . 9 |- ( ( N e. ZZ /\ -u N e. NN ) -> 1 e. RR )
13	11, 12	readdcld 8049	. . . . . . . 8 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( N + 1 ) e. RR )
14	13	renegcld 8399	. . . . . . 7 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) e. RR )
15	14	recnd 8048	. . . . . 6 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) e. CC )
16	11	recnd 8048	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ -u N e. NN ) -> N e. CC )
17		1cnd 8035	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ -u N e. NN ) -> 1 e. CC )
18	16, 17	negdid 8343	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) = ( -u N + -u 1 ) )
19	18	oveq1d 5933	. . . . . . . . . 10 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = ( ( -u N + -u 1 ) + 1 ) )
20	16	negcld 8317	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u N e. CC )
21		neg1cn 9087	. . . . . . . . . . . 12 |- -u 1 e. CC
22	21	a1i 9	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u 1 e. CC )
23	20, 22, 17	addassd 8042	. . . . . . . . . 10 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( ( -u N + -u 1 ) + 1 ) = ( -u N + ( -u 1 + 1 ) ) )
24	19, 23	eqtrd 2226	. . . . . . . . 9 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = ( -u N + ( -u 1 + 1 ) ) )
25		ax-1cn 7965	. . . . . . . . . . 11 |- 1 e. CC
26		1pneg1e0 9093	. . . . . . . . . . 11 |- ( 1 + -u 1 ) = 0
27	25, 21, 26	addcomli 8164	. . . . . . . . . 10 |- ( -u 1 + 1 ) = 0
28	27	oveq2i 5929	. . . . . . . . 9 |- ( -u N + ( -u 1 + 1 ) ) = ( -u N + 0 )
29	24, 28	eqtrdi 2242	. . . . . . . 8 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = ( -u N + 0 ) )
30	20	addridd 8168	. . . . . . . 8 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u N + 0 ) = -u N )
31	29, 30	eqtrd 2226	. . . . . . 7 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = -u N )
32		simpr 110	. . . . . . 7 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u N e. NN )
33	31, 32	eqeltrd 2270	. . . . . 6 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) e. NN )
34		elnn0nn 9282	. . . . . 6 |- ( -u ( N + 1 ) e. NN0 <-> ( -u ( N + 1 ) e. CC /\ ( -u ( N + 1 ) + 1 ) e. NN ) )
35	15, 33, 34	sylanbrc 417	. . . . 5 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) e. NN0 )
36	35	ex 115	. . . 4 |- ( N e. ZZ -> ( -u N e. NN -> -u ( N + 1 ) e. NN0 ) )
37	10, 36	orim12d 787	. . 3 |- ( N e. ZZ -> ( ( N e. NN0 \/ -u N e. NN ) -> ( ( N + 1 ) e. NN0 \/ -u ( N + 1 ) e. NN0 ) ) )
38	8, 37	mpd 13	. 2 |- ( N e. ZZ -> ( ( N + 1 ) e. NN0 \/ -u ( N + 1 ) e. NN0 ) )
39		elznn0 9332	. 2 |- ( ( N + 1 ) e. ZZ <-> ( ( N + 1 ) e. RR /\ ( ( N + 1 ) e. NN0 \/ -u ( N + 1 ) e. NN0 ) ) )
40	3, 38, 39	sylanbrc 417	1 |- ( N e. ZZ -> ( N + 1 ) e. ZZ )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   e. wcel 2164  (class class class)co 5918   CCcc 7870   RRcr 7871   0cc0 7872   1c1 7873   + caddc 7875   -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-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  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-int 3871  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-inn 8983  df-n0 9241  df-z 9318

This theorem is referenced by:  zaddcllempos  9354  peano2zm  9355  zleltp1  9372  btwnnz  9411  peano2uz2  9424  uzind  9428  uzind2  9429  peano2zd  9442  eluzp1m1  9616  eluzp1p1  9618  peano2uz  9648  zltaddlt1le  10073  fzp1disj  10146  elfzp1b  10163  fzneuz  10167  fzp1nel  10170  fzval3  10271  fzossfzop1  10279  rebtwn2zlemstep  10321  flhalf  10371  frec2uzsucd  10472  zesq  10729  hashfzp1  10895  odd2np1lem  12013  odd2np1  12014  mulsucdiv2z  12026  oddp1d2  12031  zob  12032  ltoddhalfle  12034  fldivp1  12486  lgsdir2lem2  15145