Theorem peano2z 9515

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 9483	. . 3 |- ( N e. ZZ -> N e. RR )
2		1red 8194	. . 3 |- ( N e. ZZ -> 1 e. RR )
3	1, 2	readdcld 8209	. 2 |- ( N e. ZZ -> ( N + 1 ) e. RR )
4		elznn0nn 9493	. . . . 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 797	. . . 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 9442	. . . . 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 8194	. . . . . . . . 9 |- ( ( N e. ZZ /\ -u N e. NN ) -> 1 e. RR )
13	11, 12	readdcld 8209	. . . . . . . 8 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( N + 1 ) e. RR )
14	13	renegcld 8559	. . . . . . 7 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) e. RR )
15	14	recnd 8208	. . . . . 6 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) e. CC )
16	11	recnd 8208	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ -u N e. NN ) -> N e. CC )
17		1cnd 8195	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ -u N e. NN ) -> 1 e. CC )
18	16, 17	negdid 8503	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) = ( -u N + -u 1 ) )
19	18	oveq1d 6033	. . . . . . . . . 10 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = ( ( -u N + -u 1 ) + 1 ) )
20	16	negcld 8477	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u N e. CC )
21		neg1cn 9248	. . . . . . . . . . . 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 8202	. . . . . . . . . 10 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( ( -u N + -u 1 ) + 1 ) = ( -u N + ( -u 1 + 1 ) ) )
24	19, 23	eqtrd 2264	. . . . . . . . 9 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = ( -u N + ( -u 1 + 1 ) ) )
25		ax-1cn 8125	. . . . . . . . . . 11 |- 1 e. CC
26		1pneg1e0 9254	. . . . . . . . . . 11 |- ( 1 + -u 1 ) = 0
27	25, 21, 26	addcomli 8324	. . . . . . . . . 10 |- ( -u 1 + 1 ) = 0
28	27	oveq2i 6029	. . . . . . . . 9 |- ( -u N + ( -u 1 + 1 ) ) = ( -u N + 0 )
29	24, 28	eqtrdi 2280	. . . . . . . 8 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = ( -u N + 0 ) )
30	20	addridd 8328	. . . . . . . 8 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u N + 0 ) = -u N )
31	29, 30	eqtrd 2264	. . . . . . 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 2308	. . . . . 6 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) e. NN )
34		elnn0nn 9444	. . . . . 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 793	. . 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 9494	. 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 715   e. wcel 2202  (class class class)co 6018   CCcc 8030   RRcr 8031   0cc0 8032   1c1 8033   + caddc 8035   -ucneg 8351   NNcn 9143   NN0cn0 9402   ZZcz 9479

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480

This theorem is referenced by:  zaddcllempos  9516  peano2zm  9517  zleltp1  9535  btwnnz  9574  peano2uz2  9587  uzind  9591  uzind2  9592  peano2zd  9605  eluzp1m1  9780  eluzp1p1  9782  peano2uz  9817  zltaddlt1le  10242  fzp1disj  10315  elfzp1b  10332  fzneuz  10336  fzp1nel  10339  fzval3  10450  fzossfzop1  10458  rebtwn2zlemstep  10513  flhalf  10563  frec2uzsucd  10664  zesq  10921  hashfzp1  11089  odd2np1lem  12451  odd2np1  12452  mulsucdiv2z  12464  oddp1d2  12469  zob  12470  ltoddhalfle  12472  fldivp1  12939  lgsdir2lem2  15777