Theorem peano2z 9379

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 9347	. . 3 |- ( N e. ZZ -> N e. RR )
2		1red 8058	. . 3 |- ( N e. ZZ -> 1 e. RR )
3	1, 2	readdcld 8073	. 2 |- ( N e. ZZ -> ( N + 1 ) e. RR )
4		elznn0nn 9357	. . . . 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 9306	. . . . 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 8058	. . . . . . . . 9 |- ( ( N e. ZZ /\ -u N e. NN ) -> 1 e. RR )
13	11, 12	readdcld 8073	. . . . . . . 8 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( N + 1 ) e. RR )
14	13	renegcld 8423	. . . . . . 7 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) e. RR )
15	14	recnd 8072	. . . . . 6 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) e. CC )
16	11	recnd 8072	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ -u N e. NN ) -> N e. CC )
17		1cnd 8059	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ -u N e. NN ) -> 1 e. CC )
18	16, 17	negdid 8367	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) = ( -u N + -u 1 ) )
19	18	oveq1d 5940	. . . . . . . . . 10 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = ( ( -u N + -u 1 ) + 1 ) )
20	16	negcld 8341	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u N e. CC )
21		neg1cn 9112	. . . . . . . . . . . 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 8066	. . . . . . . . . 10 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( ( -u N + -u 1 ) + 1 ) = ( -u N + ( -u 1 + 1 ) ) )
24	19, 23	eqtrd 2229	. . . . . . . . 9 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = ( -u N + ( -u 1 + 1 ) ) )
25		ax-1cn 7989	. . . . . . . . . . 11 |- 1 e. CC
26		1pneg1e0 9118	. . . . . . . . . . 11 |- ( 1 + -u 1 ) = 0
27	25, 21, 26	addcomli 8188	. . . . . . . . . 10 |- ( -u 1 + 1 ) = 0
28	27	oveq2i 5936	. . . . . . . . 9 |- ( -u N + ( -u 1 + 1 ) ) = ( -u N + 0 )
29	24, 28	eqtrdi 2245	. . . . . . . 8 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = ( -u N + 0 ) )
30	20	addridd 8192	. . . . . . . 8 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u N + 0 ) = -u N )
31	29, 30	eqtrd 2229	. . . . . . 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 2273	. . . . . 6 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) e. NN )
34		elnn0nn 9308	. . . . . 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 9358	. 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 2167  (class class class)co 5925   CCcc 7894   RRcr 7895   0cc0 7896   1c1 7897   + caddc 7899   -ucneg 8215   NNcn 9007   NN0cn0 9266   ZZcz 9343

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344

This theorem is referenced by:  zaddcllempos  9380  peano2zm  9381  zleltp1  9398  btwnnz  9437  peano2uz2  9450  uzind  9454  uzind2  9455  peano2zd  9468  eluzp1m1  9642  eluzp1p1  9644  peano2uz  9674  zltaddlt1le  10099  fzp1disj  10172  elfzp1b  10189  fzneuz  10193  fzp1nel  10196  fzval3  10297  fzossfzop1  10305  rebtwn2zlemstep  10359  flhalf  10409  frec2uzsucd  10510  zesq  10767  hashfzp1  10933  odd2np1lem  12054  odd2np1  12055  mulsucdiv2z  12067  oddp1d2  12072  zob  12073  ltoddhalfle  12075  fldivp1  12542  lgsdir2lem2  15354