Theorem peano2z 9482

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 9450	. . 3 |- ( N e. ZZ -> N e. RR )
2		1red 8161	. . 3 |- ( N e. ZZ -> 1 e. RR )
3	1, 2	readdcld 8176	. 2 |- ( N e. ZZ -> ( N + 1 ) e. RR )
4		elznn0nn 9460	. . . . 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 795	. . . 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 9409	. . . . 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 8161	. . . . . . . . 9 |- ( ( N e. ZZ /\ -u N e. NN ) -> 1 e. RR )
13	11, 12	readdcld 8176	. . . . . . . 8 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( N + 1 ) e. RR )
14	13	renegcld 8526	. . . . . . 7 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) e. RR )
15	14	recnd 8175	. . . . . 6 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) e. CC )
16	11	recnd 8175	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ -u N e. NN ) -> N e. CC )
17		1cnd 8162	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ -u N e. NN ) -> 1 e. CC )
18	16, 17	negdid 8470	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) = ( -u N + -u 1 ) )
19	18	oveq1d 6016	. . . . . . . . . 10 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = ( ( -u N + -u 1 ) + 1 ) )
20	16	negcld 8444	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u N e. CC )
21		neg1cn 9215	. . . . . . . . . . . 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 8169	. . . . . . . . . 10 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( ( -u N + -u 1 ) + 1 ) = ( -u N + ( -u 1 + 1 ) ) )
24	19, 23	eqtrd 2262	. . . . . . . . 9 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = ( -u N + ( -u 1 + 1 ) ) )
25		ax-1cn 8092	. . . . . . . . . . 11 |- 1 e. CC
26		1pneg1e0 9221	. . . . . . . . . . 11 |- ( 1 + -u 1 ) = 0
27	25, 21, 26	addcomli 8291	. . . . . . . . . 10 |- ( -u 1 + 1 ) = 0
28	27	oveq2i 6012	. . . . . . . . 9 |- ( -u N + ( -u 1 + 1 ) ) = ( -u N + 0 )
29	24, 28	eqtrdi 2278	. . . . . . . 8 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = ( -u N + 0 ) )
30	20	addridd 8295	. . . . . . . 8 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u N + 0 ) = -u N )
31	29, 30	eqtrd 2262	. . . . . . 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 2306	. . . . . 6 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) e. NN )
34		elnn0nn 9411	. . . . . 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 791	. . 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 9461	. 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 713   e. wcel 2200  (class class class)co 6001   CCcc 7997   RRcr 7998   0cc0 7999   1c1 8000   + caddc 8002   -ucneg 8318   NNcn 9110   NN0cn0 9369   ZZcz 9446

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447

This theorem is referenced by:  zaddcllempos  9483  peano2zm  9484  zleltp1  9502  btwnnz  9541  peano2uz2  9554  uzind  9558  uzind2  9559  peano2zd  9572  eluzp1m1  9746  eluzp1p1  9748  peano2uz  9778  zltaddlt1le  10203  fzp1disj  10276  elfzp1b  10293  fzneuz  10297  fzp1nel  10300  fzval3  10410  fzossfzop1  10418  rebtwn2zlemstep  10472  flhalf  10522  frec2uzsucd  10623  zesq  10880  hashfzp1  11046  odd2np1lem  12383  odd2np1  12384  mulsucdiv2z  12396  oddp1d2  12401  zob  12402  ltoddhalfle  12404  fldivp1  12871  lgsdir2lem2  15708