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Theorem peano2z 8786
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 8754 . . 3  |-  ( N  e.  ZZ  ->  N  e.  RR )
2 1red 7503 . . 3  |-  ( N  e.  ZZ  ->  1  e.  RR )
31, 2readdcld 7517 . 2  |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  RR )
4 elznn0nn 8764 . . . . 5  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
54biimpi 118 . . . 4  |-  ( N  e.  ZZ  ->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
61biantrurd 299 . . . . 5  |-  ( N  e.  ZZ  ->  ( -u N  e.  NN  <->  ( N  e.  RR  /\  -u N  e.  NN ) ) )
76orbi2d 739 . . . 4  |-  ( N  e.  ZZ  ->  (
( N  e.  NN0  \/  -u N  e.  NN ) 
<->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) ) )
85, 7mpbird 165 . . 3  |-  ( N  e.  ZZ  ->  ( N  e.  NN0  \/  -u N  e.  NN ) )
9 peano2nn0 8713 . . . . 5  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
109a1i 9 . . . 4  |-  ( N  e.  ZZ  ->  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN0 ) )
111adantr 270 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  N  e.  RR )
12 1red 7503 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  1  e.  RR )
1311, 12readdcld 7517 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( N  + 
1 )  e.  RR )
1413renegcld 7858 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u ( N  + 
1 )  e.  RR )
1514recnd 7516 . . . . . 6  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u ( N  + 
1 )  e.  CC )
1611recnd 7516 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  N  e.  CC )
17 1cnd 7504 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  1  e.  CC )
1816, 17negdid 7806 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u ( N  + 
1 )  =  (
-u N  +  -u
1 ) )
1918oveq1d 5667 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u ( N  +  1 )  +  1 )  =  ( ( -u N  +  -u 1 )  +  1 ) )
2016negcld 7780 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u N  e.  CC )
21 neg1cn 8527 . . . . . . . . . . . 12  |-  -u 1  e.  CC
2221a1i 9 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u 1  e.  CC )
2320, 22, 17addassd 7510 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( ( -u N  +  -u 1 )  +  1 )  =  ( -u N  +  ( -u 1  +  1 ) ) )
2419, 23eqtrd 2120 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u ( N  +  1 )  +  1 )  =  ( -u N  +  ( -u 1  +  1 ) ) )
25 ax-1cn 7438 . . . . . . . . . . 11  |-  1  e.  CC
26 1pneg1e0 8533 . . . . . . . . . . 11  |-  ( 1  +  -u 1 )  =  0
2725, 21, 26addcomli 7627 . . . . . . . . . 10  |-  ( -u
1  +  1 )  =  0
2827oveq2i 5663 . . . . . . . . 9  |-  ( -u N  +  ( -u 1  +  1 ) )  =  ( -u N  +  0 )
2924, 28syl6eq 2136 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u ( N  +  1 )  +  1 )  =  ( -u N  + 
0 ) )
3020addid1d 7631 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u N  +  0 )  = 
-u N )
3129, 30eqtrd 2120 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u ( N  +  1 )  +  1 )  = 
-u N )
32 simpr 108 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u N  e.  NN )
3331, 32eqeltrd 2164 . . . . . 6  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  ( -u ( N  +  1 )  +  1 )  e.  NN )
34 elnn0nn 8715 . . . . . 6  |-  ( -u ( N  +  1
)  e.  NN0  <->  ( -u ( N  +  1 )  e.  CC  /\  ( -u ( N  +  1 )  +  1 )  e.  NN ) )
3515, 33, 34sylanbrc 408 . . . . 5  |-  ( ( N  e.  ZZ  /\  -u N  e.  NN )  ->  -u ( N  + 
1 )  e.  NN0 )
3635ex 113 . . . 4  |-  ( N  e.  ZZ  ->  ( -u N  e.  NN  ->  -u ( N  +  1
)  e.  NN0 )
)
3710, 36orim12d 735 . . 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 8765 . 2  |-  ( ( N  +  1 )  e.  ZZ  <->  ( ( N  +  1 )  e.  RR  /\  (
( N  +  1 )  e.  NN0  \/  -u ( N  +  1 )  e.  NN0 )
) )
403, 38, 39sylanbrc 408 1  |-  ( N  e.  ZZ  ->  ( N  +  1 )  e.  ZZ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664    e. wcel 1438  (class class class)co 5652   CCcc 7348   RRcr 7349   0cc0 7350   1c1 7351    + caddc 7353   -ucneg 7654   NNcn 8422   NN0cn0 8673   ZZcz 8750
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-sub 7655  df-neg 7656  df-inn 8423  df-n0 8674  df-z 8751
This theorem is referenced by:  zaddcllempos  8787  peano2zm  8788  zleltp1  8805  btwnnz  8840  peano2uz2  8853  uzind  8857  uzind2  8858  peano2zd  8871  eluzp1m1  9042  eluzp1p1  9044  peano2uz  9071  zltaddlt1le  9423  fzp1disj  9494  elfzp1b  9511  fzneuz  9515  fzp1nel  9518  fzval3  9615  fzossfzop1  9623  rebtwn2zlemstep  9664  flhalf  9709  frec2uzsucd  9808  zesq  10072  hashfzp1  10232  odd2np1lem  11150  odd2np1  11151  mulsucdiv2z  11163  oddp1d2  11168  zob  11169  ltoddhalfle  11171
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