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Theorem peano2z 9048
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 9016 . . 3  |-  ( N  e.  ZZ  ->  N  e.  RR )
2 1red 7749 . . 3  |-  ( N  e.  ZZ  ->  1  e.  RR )
31, 2readdcld 7763 . 2  |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  RR )
4 elznn0nn 9026 . . . . 5  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
54biimpi 119 . . . 4  |-  ( N  e.  ZZ  ->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
61biantrurd 303 . . . . 5  |-  ( N  e.  ZZ  ->  ( -u N  e.  NN  <->  ( N  e.  RR  /\  -u N  e.  NN ) ) )
76orbi2d 764 . . . 4  |-  ( N  e.  ZZ  ->  (
( N  e.  NN0  \/  -u N  e.  NN ) 
<->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) ) )
85, 7mpbird 166 . . 3  |-  ( N  e.  ZZ  ->  ( N  e.  NN0  \/  -u N  e.  NN ) )
9 peano2nn0 8975 . . . . 5  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
109a1i 9 . . . 4  |-  ( N  e.  ZZ  ->  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN0 ) )
111adantr 274 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  N  e.  RR )
12 1red 7749 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  1  e.  RR )
1311, 12readdcld 7763 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( N  + 
1 )  e.  RR )
1413renegcld 8110 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u ( N  + 
1 )  e.  RR )
1514recnd 7762 . . . . . 6  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u ( N  + 
1 )  e.  CC )
1611recnd 7762 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  N  e.  CC )
17 1cnd 7750 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  1  e.  CC )
1816, 17negdid 8054 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u ( N  + 
1 )  =  (
-u N  +  -u
1 ) )
1918oveq1d 5757 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u ( N  +  1 )  +  1 )  =  ( ( -u N  +  -u 1 )  +  1 ) )
2016negcld 8028 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u N  e.  CC )
21 neg1cn 8789 . . . . . . . . . . . 12  |-  -u 1  e.  CC
2221a1i 9 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u 1  e.  CC )
2320, 22, 17addassd 7756 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( ( -u N  +  -u 1 )  +  1 )  =  ( -u N  +  ( -u 1  +  1 ) ) )
2419, 23eqtrd 2150 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u ( N  +  1 )  +  1 )  =  ( -u N  +  ( -u 1  +  1 ) ) )
25 ax-1cn 7681 . . . . . . . . . . 11  |-  1  e.  CC
26 1pneg1e0 8795 . . . . . . . . . . 11  |-  ( 1  +  -u 1 )  =  0
2725, 21, 26addcomli 7875 . . . . . . . . . 10  |-  ( -u
1  +  1 )  =  0
2827oveq2i 5753 . . . . . . . . 9  |-  ( -u N  +  ( -u 1  +  1 ) )  =  ( -u N  +  0 )
2924, 28syl6eq 2166 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u ( N  +  1 )  +  1 )  =  ( -u N  + 
0 ) )
3020addid1d 7879 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u N  +  0 )  = 
-u N )
3129, 30eqtrd 2150 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u ( N  +  1 )  +  1 )  = 
-u N )
32 simpr 109 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u N  e.  NN )
3331, 32eqeltrd 2194 . . . . . 6  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u ( N  +  1 )  +  1 )  e.  NN )
34 elnn0nn 8977 . . . . . 6  |-  ( -u ( N  +  1
)  e.  NN0  <->  ( -u ( N  +  1 )  e.  CC  /\  ( -u ( N  +  1 )  +  1 )  e.  NN ) )
3515, 33, 34sylanbrc 413 . . . . 5  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u ( N  + 
1 )  e.  NN0 )
3635ex 114 . . . 4  |-  ( N  e.  ZZ  ->  ( -u N  e.  NN  ->  -u ( N  +  1
)  e.  NN0 )
)
3710, 36orim12d 760 . . 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 9027 . 2  |-  ( ( N  +  1 )  e.  ZZ  <->  ( ( N  +  1 )  e.  RR  /\  (
( N  +  1 )  e.  NN0  \/  -u ( N  +  1 )  e.  NN0 )
) )
403, 38, 39sylanbrc 413 1  |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  ZZ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 682    e. wcel 1465  (class class class)co 5742   CCcc 7586   RRcr 7587   0cc0 7588   1c1 7589    + caddc 7591   -ucneg 7902   NNcn 8684   NN0cn0 8935   ZZcz 9012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-sub 7903  df-neg 7904  df-inn 8685  df-n0 8936  df-z 9013
This theorem is referenced by:  zaddcllempos  9049  peano2zm  9050  zleltp1  9067  btwnnz  9103  peano2uz2  9116  uzind  9120  uzind2  9121  peano2zd  9134  eluzp1m1  9305  eluzp1p1  9307  peano2uz  9334  zltaddlt1le  9744  fzp1disj  9815  elfzp1b  9832  fzneuz  9836  fzp1nel  9839  fzval3  9936  fzossfzop1  9944  rebtwn2zlemstep  9985  flhalf  10030  frec2uzsucd  10129  zesq  10365  hashfzp1  10525  odd2np1lem  11481  odd2np1  11482  mulsucdiv2z  11494  oddp1d2  11499  zob  11500  ltoddhalfle  11502
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