Theorem peano2z 9443

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 9411	. . 3 |- ( N e. ZZ -> N e. RR )
2		1red 8122	. . 3 |- ( N e. ZZ -> 1 e. RR )
3	1, 2	readdcld 8137	. 2 |- ( N e. ZZ -> ( N + 1 ) e. RR )
4		elznn0nn 9421	. . . . 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 792	. . . 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 9370	. . . . 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 8122	. . . . . . . . 9 |- ( ( N e. ZZ /\ -u N e. NN ) -> 1 e. RR )
13	11, 12	readdcld 8137	. . . . . . . 8 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( N + 1 ) e. RR )
14	13	renegcld 8487	. . . . . . 7 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) e. RR )
15	14	recnd 8136	. . . . . 6 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) e. CC )
16	11	recnd 8136	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ -u N e. NN ) -> N e. CC )
17		1cnd 8123	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ -u N e. NN ) -> 1 e. CC )
18	16, 17	negdid 8431	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u ( N + 1 ) = ( -u N + -u 1 ) )
19	18	oveq1d 5982	. . . . . . . . . 10 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = ( ( -u N + -u 1 ) + 1 ) )
20	16	negcld 8405	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ -u N e. NN ) -> -u N e. CC )
21		neg1cn 9176	. . . . . . . . . . . 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 8130	. . . . . . . . . 10 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( ( -u N + -u 1 ) + 1 ) = ( -u N + ( -u 1 + 1 ) ) )
24	19, 23	eqtrd 2240	. . . . . . . . 9 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = ( -u N + ( -u 1 + 1 ) ) )
25		ax-1cn 8053	. . . . . . . . . . 11 |- 1 e. CC
26		1pneg1e0 9182	. . . . . . . . . . 11 |- ( 1 + -u 1 ) = 0
27	25, 21, 26	addcomli 8252	. . . . . . . . . 10 |- ( -u 1 + 1 ) = 0
28	27	oveq2i 5978	. . . . . . . . 9 |- ( -u N + ( -u 1 + 1 ) ) = ( -u N + 0 )
29	24, 28	eqtrdi 2256	. . . . . . . 8 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) = ( -u N + 0 ) )
30	20	addridd 8256	. . . . . . . 8 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u N + 0 ) = -u N )
31	29, 30	eqtrd 2240	. . . . . . 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 2284	. . . . . 6 |- ( ( N e. ZZ /\ -u N e. NN ) -> ( -u ( N + 1 ) + 1 ) e. NN )
34		elnn0nn 9372	. . . . . 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 788	. . 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 9422	. 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 710   e. wcel 2178  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   1c1 7961   + caddc 7963   -ucneg 8279   NNcn 9071   NN0cn0 9330   ZZcz 9407

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408

This theorem is referenced by:  zaddcllempos  9444  peano2zm  9445  zleltp1  9463  btwnnz  9502  peano2uz2  9515  uzind  9519  uzind2  9520  peano2zd  9533  eluzp1m1  9707  eluzp1p1  9709  peano2uz  9739  zltaddlt1le  10164  fzp1disj  10237  elfzp1b  10254  fzneuz  10258  fzp1nel  10261  fzval3  10370  fzossfzop1  10378  rebtwn2zlemstep  10432  flhalf  10482  frec2uzsucd  10583  zesq  10840  hashfzp1  11006  odd2np1lem  12298  odd2np1  12299  mulsucdiv2z  12311  oddp1d2  12316  zob  12317  ltoddhalfle  12319  fldivp1  12786  lgsdir2lem2  15621