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Theorem nn0o1gt2 11636
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 9002 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 elnnnn0c 9045 . . . . 5  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  1  <_  N ) )
3 1z 9103 . . . . . . . 8  |-  1  e.  ZZ
4 nn0z 9097 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  ZZ )
5 zleloe 9124 . . . . . . . 8  |-  ( ( 1  e.  ZZ  /\  N  e.  ZZ )  ->  ( 1  <_  N  <->  ( 1  <  N  \/  1  =  N )
) )
63, 4, 5sylancr 411 . . . . . . 7  |-  ( N  e.  NN0  ->  ( 1  <_  N  <->  ( 1  <  N  \/  1  =  N ) ) )
7 1zzd 9104 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  1  e.  ZZ )
8 zltp1le 9131 . . . . . . . . . . . . 13  |-  ( ( 1  e.  ZZ  /\  N  e.  ZZ )  ->  ( 1  <  N  <->  ( 1  +  1 )  <_  N ) )
97, 4, 8syl2anc 409 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( 1  <  N  <->  ( 1  +  1 )  <_  N ) )
10 1p1e2 8860 . . . . . . . . . . . . . 14  |-  ( 1  +  1 )  =  2
1110breq1i 3943 . . . . . . . . . . . . 13  |-  ( ( 1  +  1 )  <_  N  <->  2  <_  N )
1211a1i 9 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( ( 1  +  1 )  <_  N  <->  2  <_  N ) )
13 2z 9105 . . . . . . . . . . . . 13  |-  2  e.  ZZ
14 zleloe 9124 . . . . . . . . . . . . 13  |-  ( ( 2  e.  ZZ  /\  N  e.  ZZ )  ->  ( 2  <_  N  <->  ( 2  <  N  \/  2  =  N )
) )
1513, 4, 14sylancr 411 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( 2  <_  N  <->  ( 2  <  N  \/  2  =  N ) ) )
169, 12, 153bitrd 213 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( 1  <  N  <->  ( 2  <  N  \/  2  =  N ) ) )
17 olc 701 . . . . . . . . . . . . . 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 5788 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  =  2  ->  ( N  +  1 )  =  ( 2  +  1 ) )
2019oveq1d 5796 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  2  ->  (
( N  +  1 )  /  2 )  =  ( ( 2  +  1 )  / 
2 ) )
2120eqcoms 2143 . . . . . . . . . . . . . . . . . 18  |-  ( 2  =  N  ->  (
( N  +  1 )  /  2 )  =  ( ( 2  +  1 )  / 
2 ) )
2221adantl 275 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  2  =  N )  ->  ( ( N  + 
1 )  /  2
)  =  ( ( 2  +  1 )  /  2 ) )
23 2p1e3 8876 . . . . . . . . . . . . . . . . . 18  |-  ( 2  +  1 )  =  3
2423oveq1i 5791 . . . . . . . . . . . . . . . . 17  |-  ( ( 2  +  1 )  /  2 )  =  ( 3  /  2
)
2522, 24eqtrdi 2189 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  2  =  N )  ->  ( ( N  + 
1 )  /  2
)  =  ( 3  /  2 ) )
2625eleq1d 2209 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  2  =  N )  ->  ( ( ( N  +  1 )  / 
2 )  e.  NN0  <->  (
3  /  2 )  e.  NN0 ) )
27 3halfnz 9171 . . . . . . . . . . . . . . . 16  |-  -.  (
3  /  2 )  e.  ZZ
28 nn0z 9097 . . . . . . . . . . . . . . . . 17  |-  ( ( 3  /  2 )  e.  NN0  ->  ( 3  /  2 )  e.  ZZ )
2928pm2.24d 612 . . . . . . . . . . . . . . . 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 162 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  2  =  N )  ->  ( ( ( N  +  1 )  / 
2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N
) ) )
3231expcom 115 . . . . . . . . . . . . 13  |-  ( 2  =  N  ->  ( N  e.  NN0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) ) )
3318, 32jaoi 706 . . . . . . . . . . . 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 149 . . . . . . . . . 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 702 . . . . . . . . . . 11  |-  ( N  =  1  ->  ( N  =  1  \/  2  <  N ) )
3837eqcoms 2143 . . . . . . . . . 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 706 . . . . . . . 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 149 . . . . . 6  |-  ( N  e.  NN0  ->  ( 1  <_  N  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) ) )
4342imp 123 . . . . 5  |-  ( ( N  e.  NN0  /\  1  <_  N )  -> 
( ( ( N  +  1 )  / 
2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N
) ) )
442, 43sylbi 120 . . . 4  |-  ( N  e.  NN  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) )
45 oveq1 5788 . . . . . . . 8  |-  ( N  =  0  ->  ( N  +  1 )  =  ( 0  +  1 ) )
46 0p1e1 8857 . . . . . . . 8  |-  ( 0  +  1 )  =  1
4745, 46eqtrdi 2189 . . . . . . 7  |-  ( N  =  0  ->  ( N  +  1 )  =  1 )
4847oveq1d 5796 . . . . . 6  |-  ( N  =  0  ->  (
( N  +  1 )  /  2 )  =  ( 1  / 
2 ) )
4948eleq1d 2209 . . . . 5  |-  ( N  =  0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  <->  ( 1  /  2 )  e. 
NN0 ) )
50 halfnz 9170 . . . . . 6  |-  -.  (
1  /  2 )  e.  ZZ
51 nn0z 9097 . . . . . . 7  |-  ( ( 1  /  2 )  e.  NN0  ->  ( 1  /  2 )  e.  ZZ )
5251pm2.24d 612 . . . . . 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 162 . . . 4  |-  ( N  =  0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) )
5544, 54jaoi 706 . . 3  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  ( ( ( N  +  1 )  /  2 )  e. 
NN0  ->  ( N  =  1  \/  2  < 
N ) ) )
561, 55sylbi 120 . 2  |-  ( N  e.  NN0  ->  ( ( ( N  +  1 )  /  2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) )
5756imp 123 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 103    <-> wb 104    \/ wo 698    = wceq 1332    e. wcel 1481   class class class wbr 3936  (class class class)co 5781   0cc0 7643   1c1 7644    + caddc 7646    < clt 7823    <_ cle 7824    / cdiv 8455   NNcn 8743   2c2 8794   3c3 8795   NN0cn0 9000   ZZcz 9077
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-br 3937  df-opab 3997  df-id 4222  df-po 4225  df-iso 4226  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-2 8802  df-3 8803  df-4 8804  df-n0 9001  df-z 9078
This theorem is referenced by:  nno  11637  nn0o  11638
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