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Theorem peano2z 8338
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
StepHypRef Expression
1 zre 8306 . . 3  |-  ( N  e.  ZZ  ->  N  e.  RR )
2 1red 7100 . . 3  |-  ( N  e.  ZZ  ->  1  e.  RR )
31, 2readdcld 7114 . 2  |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  RR )
4 elznn0nn 8316 . . . . 5  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
54biimpi 117 . . . 4  |-  ( N  e.  ZZ  ->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
61biantrurd 293 . . . . 5  |-  ( N  e.  ZZ  ->  ( -u N  e.  NN  <->  ( N  e.  RR  /\  -u N  e.  NN ) ) )
76orbi2d 714 . . . 4  |-  ( N  e.  ZZ  ->  (
( N  e.  NN0  \/  -u N  e.  NN ) 
<->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) ) )
85, 7mpbird 160 . . 3  |-  ( N  e.  ZZ  ->  ( N  e.  NN0  \/  -u N  e.  NN ) )
9 peano2nn0 8279 . . . . 5  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
109a1i 9 . . . 4  |-  ( N  e.  ZZ  ->  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN0 ) )
111adantr 265 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  N  e.  RR )
12 1red 7100 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  1  e.  RR )
1311, 12readdcld 7114 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( N  + 
1 )  e.  RR )
1413renegcld 7450 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u ( N  + 
1 )  e.  RR )
1514recnd 7113 . . . . . 6  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u ( N  + 
1 )  e.  CC )
1611recnd 7113 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  N  e.  CC )
17 1cnd 7101 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  1  e.  CC )
1816, 17negdid 7398 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u ( N  + 
1 )  =  (
-u N  +  -u
1 ) )
1918oveq1d 5555 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u ( N  +  1 )  +  1 )  =  ( ( -u N  +  -u 1 )  +  1 ) )
2016negcld 7372 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u N  e.  CC )
21 neg1cn 8095 . . . . . . . . . . . 12  |-  -u 1  e.  CC
2221a1i 9 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u 1  e.  CC )
2320, 22, 17addassd 7107 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( ( -u N  +  -u 1 )  +  1 )  =  ( -u N  +  ( -u 1  +  1 ) ) )
2419, 23eqtrd 2088 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u ( N  +  1 )  +  1 )  =  ( -u N  +  ( -u 1  +  1 ) ) )
25 ax-1cn 7035 . . . . . . . . . . 11  |-  1  e.  CC
26 1pneg1e0 8101 . . . . . . . . . . 11  |-  ( 1  +  -u 1 )  =  0
2725, 21, 26addcomli 7219 . . . . . . . . . 10  |-  ( -u
1  +  1 )  =  0
2827oveq2i 5551 . . . . . . . . 9  |-  ( -u N  +  ( -u 1  +  1 ) )  =  ( -u N  +  0 )
2924, 28syl6eq 2104 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u ( N  +  1 )  +  1 )  =  ( -u N  + 
0 ) )
3020addid1d 7223 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u N  +  0 )  = 
-u N )
3129, 30eqtrd 2088 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u ( N  +  1 )  +  1 )  = 
-u N )
32 simpr 107 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u N  e.  NN )
3331, 32eqeltrd 2130 . . . . . 6  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u ( N  +  1 )  +  1 )  e.  NN )
34 elnn0nn 8281 . . . . . 6  |-  ( -u ( N  +  1
)  e.  NN0  <->  ( -u ( N  +  1 )  e.  CC  /\  ( -u ( N  +  1 )  +  1 )  e.  NN ) )
3515, 33, 34sylanbrc 402 . . . . 5  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u ( N  + 
1 )  e.  NN0 )
3635ex 112 . . . 4  |-  ( N  e.  ZZ  ->  ( -u N  e.  NN  ->  -u ( N  +  1
)  e.  NN0 )
)
3710, 36orim12d 710 . . 3  |-  ( N  e.  ZZ  ->  (
( N  e.  NN0  \/  -u N  e.  NN )  ->  ( ( N  +  1 )  e. 
NN0  \/  -u ( N  +  1 )  e. 
NN0 ) ) )
388, 37mpd 13 . 2  |-  ( N  e.  ZZ  ->  (
( N  +  1 )  e.  NN0  \/  -u ( N  +  1 )  e.  NN0 )
)
39 elznn0 8317 . 2  |-  ( ( N  +  1 )  e.  ZZ  <->  ( ( N  +  1 )  e.  RR  /\  (
( N  +  1 )  e.  NN0  \/  -u ( N  +  1 )  e.  NN0 )
) )
403, 38, 39sylanbrc 402 1  |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  ZZ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    \/ wo 639    e. wcel 1409  (class class class)co 5540   CCcc 6945   RRcr 6946   0cc0 6947   1c1 6948    + caddc 6950   -ucneg 7246   NNcn 7990   NN0cn0 8239   ZZcz 8302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-addass 7044  ax-distr 7046  ax-i2m1 7047  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-sub 7247  df-neg 7248  df-inn 7991  df-n0 8240  df-z 8303
This theorem is referenced by:  zaddcllempos  8339  peano2zm  8340  zleltp1  8357  btwnnz  8392  peano2uz2  8404  uzind  8408  uzind2  8409  peano2zd  8422  eluzp1m1  8592  eluzp1p1  8594  peano2uz  8622  zltaddlt1le  8975  fzp1disj  9044  elfzp1b  9061  fzneuz  9065  fzp1nel  9068  fzval3  9162  fzossfzop1  9170  rebtwn2zlemstep  9209  flhalf  9252  frec2uzzd  9350  frec2uzsucd  9351  zesq  9535  odd2np1lem  10183  odd2np1  10184  mulsucdiv2z  10197  oddp1d2  10202  zob  10203  ltoddhalfle  10205
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