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Theorem nn0o1gt2 40599
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 10899 . . 3  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
2 elnnnn0c 10943 . . . . 5  |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  1  <_  N ) )
3 1red 9683 . . . . . . . 8  |-  ( N  e.  NN0  ->  1  e.  RR )
4 nn0re 10906 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  e.  RR )
53, 4leloed 9803 . . . . . . 7  |-  ( N  e.  NN0  ->  ( 1  <_  N  <->  ( 1  <  N  \/  1  =  N ) ) )
6 1zzd 10996 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  1  e.  ZZ )
7 nn0z 10988 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  ZZ )
8 zltp1le 11014 . . . . . . . . . . . . 13  |-  ( ( 1  e.  ZZ  /\  N  e.  ZZ )  ->  ( 1  <  N  <->  ( 1  +  1 )  <_  N ) )
96, 7, 8syl2anc 671 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( 1  <  N  <->  ( 1  +  1 )  <_  N ) )
10 1p1e2 10750 . . . . . . . . . . . . . 14  |-  ( 1  +  1 )  =  2
1110breq1i 4422 . . . . . . . . . . . . 13  |-  ( ( 1  +  1 )  <_  N  <->  2  <_  N )
1211a1i 11 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( ( 1  +  1 )  <_  N  <->  2  <_  N ) )
13 2re 10706 . . . . . . . . . . . . . 14  |-  2  e.  RR
1413a1i 11 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  2  e.  RR )
1514, 4leloed 9803 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( 2  <_  N  <->  ( 2  <  N  \/  2  =  N ) ) )
169, 12, 153bitrd 287 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( 1  <  N  <->  ( 2  <  N  \/  2  =  N ) ) )
17 olc 390 . . . . . . . . . . . . . 14  |-  ( 2  <  N  ->  ( N  =  1  \/  2  <  N ) )
18172a1d 27 . . . . . . . . . . . . 13  |-  ( 2  <  N  ->  ( N  e.  NN0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) ) )
19 oveq1 6321 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  =  2  ->  ( N  +  1 )  =  ( 2  +  1 ) )
2019oveq1d 6329 . . . . . . . . . . . . . . . . . . 19  |-  ( N  =  2  ->  (
( N  +  1 )  /  2 )  =  ( ( 2  +  1 )  / 
2 ) )
2120eqcoms 2469 . . . . . . . . . . . . . . . . . 18  |-  ( 2  =  N  ->  (
( N  +  1 )  /  2 )  =  ( ( 2  +  1 )  / 
2 ) )
2221adantl 472 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  2  =  N )  ->  ( ( N  + 
1 )  /  2
)  =  ( ( 2  +  1 )  /  2 ) )
23 2p1e3 10761 . . . . . . . . . . . . . . . . . 18  |-  ( 2  +  1 )  =  3
2423oveq1i 6324 . . . . . . . . . . . . . . . . 17  |-  ( ( 2  +  1 )  /  2 )  =  ( 3  /  2
)
2522, 24syl6eq 2511 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  2  =  N )  ->  ( ( N  + 
1 )  /  2
)  =  ( 3  /  2 ) )
2625eleq1d 2523 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  2  =  N )  ->  ( ( ( N  +  1 )  / 
2 )  e.  NN0  <->  (
3  /  2 )  e.  NN0 ) )
27 3halfnz 40589 . . . . . . . . . . . . . . . 16  |-  -.  (
3  /  2 )  e.  ZZ
28 nn0z 10988 . . . . . . . . . . . . . . . . 17  |-  ( ( 3  /  2 )  e.  NN0  ->  ( 3  /  2 )  e.  ZZ )
2928pm2.24d 139 . . . . . . . . . . . . . . . 16  |-  ( ( 3  /  2 )  e.  NN0  ->  ( -.  ( 3  /  2
)  e.  ZZ  ->  ( N  =  1  \/  2  <  N ) ) )
3027, 29mpi 20 . . . . . . . . . . . . . . 15  |-  ( ( 3  /  2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N ) )
3126, 30syl6bi 236 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  2  =  N )  ->  ( ( ( N  +  1 )  / 
2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N
) ) )
3231expcom 441 . . . . . . . . . . . . 13  |-  ( 2  =  N  ->  ( N  e.  NN0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) ) )
3318, 32jaoi 385 . . . . . . . . . . . 12  |-  ( ( 2  <  N  \/  2  =  N )  ->  ( N  e.  NN0  ->  ( ( ( N  +  1 )  / 
2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N
) ) ) )
3433com12 32 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( ( 2  <  N  \/  2  =  N )  ->  ( ( ( N  +  1 )  / 
2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N
) ) ) )
3516, 34sylbid 223 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( 1  <  N  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) ) )
3635com12 32 . . . . . . . . 9  |-  ( 1  <  N  ->  ( N  e.  NN0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) ) )
37 orc 391 . . . . . . . . . . 11  |-  ( N  =  1  ->  ( N  =  1  \/  2  <  N ) )
3837eqcoms 2469 . . . . . . . . . 10  |-  ( 1  =  N  ->  ( N  =  1  \/  2  <  N ) )
39382a1d 27 . . . . . . . . 9  |-  ( 1  =  N  ->  ( N  e.  NN0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) ) )
4036, 39jaoi 385 . . . . . . . 8  |-  ( ( 1  <  N  \/  1  =  N )  ->  ( N  e.  NN0  ->  ( ( ( N  +  1 )  / 
2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N
) ) ) )
4140com12 32 . . . . . . 7  |-  ( N  e.  NN0  ->  ( ( 1  <  N  \/  1  =  N )  ->  ( ( ( N  +  1 )  / 
2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N
) ) ) )
425, 41sylbid 223 . . . . . 6  |-  ( N  e.  NN0  ->  ( 1  <_  N  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) ) )
4342imp 435 . . . . 5  |-  ( ( N  e.  NN0  /\  1  <_  N )  -> 
( ( ( N  +  1 )  / 
2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N
) ) )
442, 43sylbi 200 . . . 4  |-  ( N  e.  NN  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) )
45 oveq1 6321 . . . . . . . 8  |-  ( N  =  0  ->  ( N  +  1 )  =  ( 0  +  1 ) )
46 0p1e1 10748 . . . . . . . 8  |-  ( 0  +  1 )  =  1
4745, 46syl6eq 2511 . . . . . . 7  |-  ( N  =  0  ->  ( N  +  1 )  =  1 )
4847oveq1d 6329 . . . . . 6  |-  ( N  =  0  ->  (
( N  +  1 )  /  2 )  =  ( 1  / 
2 ) )
4948eleq1d 2523 . . . . 5  |-  ( N  =  0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  <->  ( 1  /  2 )  e. 
NN0 ) )
50 halfnz 11042 . . . . . 6  |-  -.  (
1  /  2 )  e.  ZZ
51 nn0z 10988 . . . . . . 7  |-  ( ( 1  /  2 )  e.  NN0  ->  ( 1  /  2 )  e.  ZZ )
5251pm2.24d 139 . . . . . 6  |-  ( ( 1  /  2 )  e.  NN0  ->  ( -.  ( 1  /  2
)  e.  ZZ  ->  ( N  =  1  \/  2  <  N ) ) )
5350, 52mpi 20 . . . . 5  |-  ( ( 1  /  2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N ) )
5449, 53syl6bi 236 . . . 4  |-  ( N  =  0  ->  (
( ( N  + 
1 )  /  2
)  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) )
5544, 54jaoi 385 . . 3  |-  ( ( N  e.  NN  \/  N  =  0 )  ->  ( ( ( N  +  1 )  /  2 )  e. 
NN0  ->  ( N  =  1  \/  2  < 
N ) ) )
561, 55sylbi 200 . 2  |-  ( N  e.  NN0  ->  ( ( ( N  +  1 )  /  2 )  e.  NN0  ->  ( N  =  1  \/  2  <  N ) ) )
5756imp 435 1  |-  ( ( N  e.  NN0  /\  ( ( N  + 
1 )  /  2
)  e.  NN0 )  ->  ( N  =  1  \/  2  <  N
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    = wceq 1454    e. wcel 1897   class class class wbr 4415  (class class class)co 6314   RRcr 9563   0cc0 9564   1c1 9565    + caddc 9567    < clt 9700    <_ cle 9701    / cdiv 10296   NNcn 10636   2c2 10686   3c3 10687   NN0cn0 10897   ZZcz 10965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-4 10697  df-n0 10898  df-z 10966
This theorem is referenced by:  nno  40600  nn0o  40601
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