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Theorem nn0o1gt2 10449
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 8357 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 elnnnn0c 8400 . . . . 5  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  1  <_  N ) )
3 1z 8458 . . . . . . . 8  |-  1  e.  ZZ
4 nn0z 8452 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ZZ )
5 zleloe 8479 . . . . . . . 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 8459 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  1  e.  ZZ )
8 zltp1le 8486 . . . . . . . . . . . . 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 8222 . . . . . . . . . . . . . 14  |-  ( 1  +  1 )  =  2
1110breq1i 3800 . . . . . . . . . . . . 13  |-  ( ( 1  +  1 )  <_  N  <->  2  <_  N )
1211a1i 9 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( ( 1  +  1 )  <_  N  <->  2  <_  N ) )
13 2z 8460 . . . . . . . . . . . . 13  |-  2  e.  ZZ
14 zleloe 8479 . . . . . . . . . . . . 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 665 . . . . . . . . . . . . . 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 5550 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  =  2  ->  ( N  +  1 )  =  ( 2  +  1 ) )
2019oveq1d 5558 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  2  ->  (
( N  +  1 )  /  2 )  =  ( ( 2  +  1 )  / 
2 ) )
2120eqcoms 2085 . . . . . . . . . . . . . . . . . 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 8232 . . . . . . . . . . . . . . . . . 18  |-  ( 2  +  1 )  =  3
2423oveq1i 5553 . . . . . . . . . . . . . . . . 17  |-  ( ( 2  +  1 )  /  2 )  =  ( 3  /  2
)
2522, 24syl6eq 2130 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  2  =  N )  ->  ( ( N  + 
1 )  /  2
)  =  ( 3  /  2 ) )
2625eleq1d 2148 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  2  =  N )  ->  ( ( ( N  +  1 )  / 
2 )  e.  NN0  <->  (
3  /  2 )  e.  NN0 ) )
27 3halfnz 8525 . . . . . . . . . . . . . . . 16  |-  -.  (
3  /  2 )  e.  ZZ
28 nn0z 8452 . . . . . . . . . . . . . . . . 17  |-  ( ( 3  /  2 )  e.  NN0  ->  ( 3  /  2 )  e.  ZZ )
2928pm2.24d 585 . . . . . . . . . . . . . . . 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 669 . . . . . . . . . . . 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 666 . . . . . . . . . . 11  |-  ( N  =  1  ->  ( N  =  1  \/  2  <  N ) )
3837eqcoms 2085 . . . . . . . . . 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 669 . . . . . . . 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 5550 . . . . . . . 8  |-  ( N  =  0  ->  ( N  +  1 )  =  ( 0  +  1 ) )
46 0p1e1 8220 . . . . . . . 8  |-  ( 0  +  1 )  =  1
4745, 46syl6eq 2130 . . . . . . 7  |-  ( N  =  0  ->  ( N  +  1 )  =  1 )
4847oveq1d 5558 . . . . . 6  |-  ( N  =  0  ->  (
( N  +  1 )  /  2 )  =  ( 1  / 
2 ) )
4948eleq1d 2148 . . . . 5  |-  ( N  =  0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  <->  ( 1  /  2 )  e. 
NN0 ) )
50 halfnz 8524 . . . . . 6  |-  -.  (
1  /  2 )  e.  ZZ
51 nn0z 8452 . . . . . . 7  |-  ( ( 1  /  2 )  e.  NN0  ->  ( 1  /  2 )  e.  ZZ )
5251pm2.24d 585 . . . . . 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 669 . . 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 662    = wceq 1285    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   0cc0 7043   1c1 7044    + caddc 7046    < clt 7215    <_ cle 7216    / cdiv 7827   NNcn 8106   2c2 8156   3c3 8157   NN0cn0 8355   ZZcz 8432
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-id 4056  df-po 4059  df-iso 4060  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-2 8165  df-3 8166  df-4 8167  df-n0 8356  df-z 8433
This theorem is referenced by:  nno  10450  nn0o  10451
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