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Theorem nn0o1gt2 10985
Description: An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.)
Assertion
Ref Expression
nn0o1gt2  |-  ( ( N  e.  NN0  /\  ( ( N  + 
1 )  /  2
)  e.  NN0 )  ->  ( N  =  1  \/  2  <  N
) )

Proof of Theorem nn0o1gt2
StepHypRef Expression
1 elnn0 8645 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 elnnnn0c 8688 . . . . 5  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  1  <_  N ) )
3 1z 8746 . . . . . . . 8  |-  1  e.  ZZ
4 nn0z 8740 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ZZ )
5 zleloe 8767 . . . . . . . 8  |-  ( ( 1  e.  ZZ  /\  N  e.  ZZ )  ->  ( 1  <_  N  <->  ( 1  <  N  \/  1  =  N )
) )
63, 4, 5sylancr 405 . . . . . . 7  |-  ( N  e.  NN0  ->  ( 1  <_  N  <->  ( 1  <  N  \/  1  =  N ) ) )
7 1zzd 8747 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  1  e.  ZZ )
8 zltp1le 8774 . . . . . . . . . . . . 13  |-  ( ( 1  e.  ZZ  /\  N  e.  ZZ )  ->  ( 1  <  N  <->  ( 1  +  1 )  <_  N ) )
97, 4, 8syl2anc 403 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( 1  <  N  <->  ( 1  +  1 )  <_  N ) )
10 1p1e2 8509 . . . . . . . . . . . . . 14  |-  ( 1  +  1 )  =  2
1110breq1i 3844 . . . . . . . . . . . . 13  |-  ( ( 1  +  1 )  <_  N  <->  2  <_  N )
1211a1i 9 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( ( 1  +  1 )  <_  N  <->  2  <_  N ) )
13 2z 8748 . . . . . . . . . . . . 13  |-  2  e.  ZZ
14 zleloe 8767 . . . . . . . . . . . . 13  |-  ( ( 2  e.  ZZ  /\  N  e.  ZZ )  ->  ( 2  <_  N  <->  ( 2  <  N  \/  2  =  N )
) )
1513, 4, 14sylancr 405 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( 2  <_  N  <->  ( 2  <  N  \/  2  =  N ) ) )
169, 12, 153bitrd 212 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( 1  <  N  <->  ( 2  <  N  \/  2  =  N ) ) )
17 olc 667 . . . . . . . . . . . . . 14  |-  ( 2  <  N  ->  ( N  =  1  \/  2  <  N ) )
18172a1d 23 . . . . . . . . . . . . 13  |-  ( 2  <  N  ->  ( N  e.  NN0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) ) )
19 oveq1 5641 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  =  2  ->  ( N  +  1 )  =  ( 2  +  1 ) )
2019oveq1d 5649 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  2  ->  (
( N  +  1 )  /  2 )  =  ( ( 2  +  1 )  / 
2 ) )
2120eqcoms 2091 . . . . . . . . . . . . . . . . . 18  |-  ( 2  =  N  ->  (
( N  +  1 )  /  2 )  =  ( ( 2  +  1 )  / 
2 ) )
2221adantl 271 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  2  =  N )  ->  ( ( N  + 
1 )  /  2
)  =  ( ( 2  +  1 )  /  2 ) )
23 2p1e3 8519 . . . . . . . . . . . . . . . . . 18  |-  ( 2  +  1 )  =  3
2423oveq1i 5644 . . . . . . . . . . . . . . . . 17  |-  ( ( 2  +  1 )  /  2 )  =  ( 3  /  2
)
2522, 24syl6eq 2136 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  2  =  N )  ->  ( ( N  + 
1 )  /  2
)  =  ( 3  /  2 ) )
2625eleq1d 2156 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  2  =  N )  ->  ( ( ( N  +  1 )  / 
2 )  e.  NN0  <->  (
3  /  2 )  e.  NN0 ) )
27 3halfnz 8813 . . . . . . . . . . . . . . . 16  |-  -.  (
3  /  2 )  e.  ZZ
28 nn0z 8740 . . . . . . . . . . . . . . . . 17  |-  ( ( 3  /  2 )  e.  NN0  ->  ( 3  /  2 )  e.  ZZ )
2928pm2.24d 587 . . . . . . . . . . . . . . . 16  |-  ( ( 3  /  2 )  e.  NN0  ->  ( -.  ( 3  /  2
)  e.  ZZ  ->  ( N  =  1  \/  2  <  N ) ) )
3027, 29mpi 15 . . . . . . . . . . . . . . 15  |-  ( ( 3  /  2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N ) )
3126, 30syl6bi 161 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  2  =  N )  ->  ( ( ( N  +  1 )  / 
2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N
) ) )
3231expcom 114 . . . . . . . . . . . . 13  |-  ( 2  =  N  ->  ( N  e.  NN0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) ) )
3318, 32jaoi 671 . . . . . . . . . . . 12  |-  ( ( 2  <  N  \/  2  =  N )  ->  ( N  e.  NN0  ->  ( ( ( N  +  1 )  / 
2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N
) ) ) )
3433com12 30 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( ( 2  <  N  \/  2  =  N )  ->  ( ( ( N  +  1 )  / 
2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N
) ) ) )
3516, 34sylbid 148 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( 1  <  N  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) ) )
3635com12 30 . . . . . . . . 9  |-  ( 1  <  N  ->  ( N  e.  NN0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) ) )
37 orc 668 . . . . . . . . . . 11  |-  ( N  =  1  ->  ( N  =  1  \/  2  <  N ) )
3837eqcoms 2091 . . . . . . . . . 10  |-  ( 1  =  N  ->  ( N  =  1  \/  2  <  N ) )
39382a1d 23 . . . . . . . . 9  |-  ( 1  =  N  ->  ( N  e.  NN0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) ) )
4036, 39jaoi 671 . . . . . . . 8  |-  ( ( 1  <  N  \/  1  =  N )  ->  ( N  e.  NN0  ->  ( ( ( N  +  1 )  / 
2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N
) ) ) )
4140com12 30 . . . . . . 7  |-  ( N  e.  NN0  ->  ( ( 1  <  N  \/  1  =  N )  ->  ( ( ( N  +  1 )  / 
2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N
) ) ) )
426, 41sylbid 148 . . . . . 6  |-  ( N  e.  NN0  ->  ( 1  <_  N  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) ) )
4342imp 122 . . . . 5  |-  ( ( N  e.  NN0  /\  1  <_  N )  -> 
( ( ( N  +  1 )  / 
2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N
) ) )
442, 43sylbi 119 . . . 4  |-  ( N  e.  NN  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) )
45 oveq1 5641 . . . . . . . 8  |-  ( N  =  0  ->  ( N  +  1 )  =  ( 0  +  1 ) )
46 0p1e1 8507 . . . . . . . 8  |-  ( 0  +  1 )  =  1
4745, 46syl6eq 2136 . . . . . . 7  |-  ( N  =  0  ->  ( N  +  1 )  =  1 )
4847oveq1d 5649 . . . . . 6  |-  ( N  =  0  ->  (
( N  +  1 )  /  2 )  =  ( 1  / 
2 ) )
4948eleq1d 2156 . . . . 5  |-  ( N  =  0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  <->  ( 1  /  2 )  e. 
NN0 ) )
50 halfnz 8812 . . . . . 6  |-  -.  (
1  /  2 )  e.  ZZ
51 nn0z 8740 . . . . . . 7  |-  ( ( 1  /  2 )  e.  NN0  ->  ( 1  /  2 )  e.  ZZ )
5251pm2.24d 587 . . . . . 6  |-  ( ( 1  /  2 )  e.  NN0  ->  ( -.  ( 1  /  2
)  e.  ZZ  ->  ( N  =  1  \/  2  <  N ) ) )
5350, 52mpi 15 . . . . 5  |-  ( ( 1  /  2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N ) )
5449, 53syl6bi 161 . . . 4  |-  ( N  =  0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) )
5544, 54jaoi 671 . . 3  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  ( ( ( N  +  1 )  /  2 )  e. 
NN0  ->  ( N  =  1  \/  2  < 
N ) ) )
561, 55sylbi 119 . 2  |-  ( N  e.  NN0  ->  ( ( ( N  +  1 )  /  2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) )
5756imp 122 1  |-  ( ( N  e.  NN0  /\  ( ( N  + 
1 )  /  2
)  e.  NN0 )  ->  ( N  =  1  \/  2  <  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   0cc0 7329   1c1 7330    + caddc 7332    < clt 7501    <_ cle 7502    / cdiv 8113   NNcn 8394   2c2 8444   3c3 8445   NN0cn0 8643   ZZcz 8720
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-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
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-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721
This theorem is referenced by:  nno  10986  nn0o  10987
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