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Theorem nneoor 9050
Description: A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
Assertion
Ref Expression
nneoor  |-  ( N  e.  NN  ->  (
( N  /  2
)  e.  NN  \/  ( ( N  + 
1 )  /  2
)  e.  NN ) )

Proof of Theorem nneoor
Dummy variables  j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5733 . . . . . 6  |-  ( j  =  1  ->  (
j  +  1 )  =  ( 1  +  1 ) )
21oveq1d 5741 . . . . 5  |-  ( j  =  1  ->  (
( j  +  1 )  /  2 )  =  ( ( 1  +  1 )  / 
2 ) )
32eleq1d 2181 . . . 4  |-  ( j  =  1  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( (
1  +  1 )  /  2 )  e.  NN ) )
4 oveq1 5733 . . . . 5  |-  ( j  =  1  ->  (
j  /  2 )  =  ( 1  / 
2 ) )
54eleq1d 2181 . . . 4  |-  ( j  =  1  ->  (
( j  /  2
)  e.  NN  <->  ( 1  /  2 )  e.  NN ) )
63, 5orbi12d 765 . . 3  |-  ( j  =  1  ->  (
( ( ( j  +  1 )  / 
2 )  e.  NN  \/  ( j  /  2
)  e.  NN )  <-> 
( ( ( 1  +  1 )  / 
2 )  e.  NN  \/  ( 1  /  2
)  e.  NN ) ) )
7 oveq1 5733 . . . . . 6  |-  ( j  =  k  ->  (
j  +  1 )  =  ( k  +  1 ) )
87oveq1d 5741 . . . . 5  |-  ( j  =  k  ->  (
( j  +  1 )  /  2 )  =  ( ( k  +  1 )  / 
2 ) )
98eleq1d 2181 . . . 4  |-  ( j  =  k  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( (
k  +  1 )  /  2 )  e.  NN ) )
10 oveq1 5733 . . . . 5  |-  ( j  =  k  ->  (
j  /  2 )  =  ( k  / 
2 ) )
1110eleq1d 2181 . . . 4  |-  ( j  =  k  ->  (
( j  /  2
)  e.  NN  <->  ( k  /  2 )  e.  NN ) )
129, 11orbi12d 765 . . 3  |-  ( j  =  k  ->  (
( ( ( j  +  1 )  / 
2 )  e.  NN  \/  ( j  /  2
)  e.  NN )  <-> 
( ( ( k  +  1 )  / 
2 )  e.  NN  \/  ( k  /  2
)  e.  NN ) ) )
13 oveq1 5733 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
j  +  1 )  =  ( ( k  +  1 )  +  1 ) )
1413oveq1d 5741 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( j  +  1 )  /  2 )  =  ( ( ( k  +  1 )  +  1 )  / 
2 ) )
1514eleq1d 2181 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( (
( k  +  1 )  +  1 )  /  2 )  e.  NN ) )
16 oveq1 5733 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
j  /  2 )  =  ( ( k  +  1 )  / 
2 ) )
1716eleq1d 2181 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( j  /  2
)  e.  NN  <->  ( (
k  +  1 )  /  2 )  e.  NN ) )
1815, 17orbi12d 765 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( ( ( j  +  1 )  / 
2 )  e.  NN  \/  ( j  /  2
)  e.  NN )  <-> 
( ( ( ( k  +  1 )  +  1 )  / 
2 )  e.  NN  \/  ( ( k  +  1 )  /  2
)  e.  NN ) ) )
19 oveq1 5733 . . . . . 6  |-  ( j  =  N  ->  (
j  +  1 )  =  ( N  + 
1 ) )
2019oveq1d 5741 . . . . 5  |-  ( j  =  N  ->  (
( j  +  1 )  /  2 )  =  ( ( N  +  1 )  / 
2 ) )
2120eleq1d 2181 . . . 4  |-  ( j  =  N  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( ( N  +  1 )  /  2 )  e.  NN ) )
22 oveq1 5733 . . . . 5  |-  ( j  =  N  ->  (
j  /  2 )  =  ( N  / 
2 ) )
2322eleq1d 2181 . . . 4  |-  ( j  =  N  ->  (
( j  /  2
)  e.  NN  <->  ( N  /  2 )  e.  NN ) )
2421, 23orbi12d 765 . . 3  |-  ( j  =  N  ->  (
( ( ( j  +  1 )  / 
2 )  e.  NN  \/  ( j  /  2
)  e.  NN )  <-> 
( ( ( N  +  1 )  / 
2 )  e.  NN  \/  ( N  /  2
)  e.  NN ) ) )
25 df-2 8682 . . . . . . 7  |-  2  =  ( 1  +  1 )
2625oveq1i 5736 . . . . . 6  |-  ( 2  /  2 )  =  ( ( 1  +  1 )  /  2
)
27 2div2e1 8749 . . . . . 6  |-  ( 2  /  2 )  =  1
2826, 27eqtr3i 2135 . . . . 5  |-  ( ( 1  +  1 )  /  2 )  =  1
29 1nn 8634 . . . . 5  |-  1  e.  NN
3028, 29eqeltri 2185 . . . 4  |-  ( ( 1  +  1 )  /  2 )  e.  NN
3130orci 703 . . 3  |-  ( ( ( 1  +  1 )  /  2 )  e.  NN  \/  (
1  /  2 )  e.  NN )
32 peano2nn 8635 . . . . . 6  |-  ( ( k  /  2 )  e.  NN  ->  (
( k  /  2
)  +  1 )  e.  NN )
33 nncn 8631 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  CC )
34 add1p1 8866 . . . . . . . . . 10  |-  ( k  e.  CC  ->  (
( k  +  1 )  +  1 )  =  ( k  +  2 ) )
3534oveq1d 5741 . . . . . . . . 9  |-  ( k  e.  CC  ->  (
( ( k  +  1 )  +  1 )  /  2 )  =  ( ( k  +  2 )  / 
2 ) )
36 2cn 8694 . . . . . . . . . . 11  |-  2  e.  CC
37 2ap0 8716 . . . . . . . . . . . 12  |-  2 #  0
3836, 37pm3.2i 268 . . . . . . . . . . 11  |-  ( 2  e.  CC  /\  2 #  0 )
39 divdirap 8363 . . . . . . . . . . 11  |-  ( ( k  e.  CC  /\  2  e.  CC  /\  (
2  e.  CC  /\  2 #  0 ) )  -> 
( ( k  +  2 )  /  2
)  =  ( ( k  /  2 )  +  ( 2  / 
2 ) ) )
4036, 38, 39mp3an23 1288 . . . . . . . . . 10  |-  ( k  e.  CC  ->  (
( k  +  2 )  /  2 )  =  ( ( k  /  2 )  +  ( 2  /  2
) ) )
4127oveq2i 5737 . . . . . . . . . 10  |-  ( ( k  /  2 )  +  ( 2  / 
2 ) )  =  ( ( k  / 
2 )  +  1 )
4240, 41syl6eq 2161 . . . . . . . . 9  |-  ( k  e.  CC  ->  (
( k  +  2 )  /  2 )  =  ( ( k  /  2 )  +  1 ) )
4335, 42eqtrd 2145 . . . . . . . 8  |-  ( k  e.  CC  ->  (
( ( k  +  1 )  +  1 )  /  2 )  =  ( ( k  /  2 )  +  1 ) )
4433, 43syl 14 . . . . . . 7  |-  ( k  e.  NN  ->  (
( ( k  +  1 )  +  1 )  /  2 )  =  ( ( k  /  2 )  +  1 ) )
4544eleq1d 2181 . . . . . 6  |-  ( k  e.  NN  ->  (
( ( ( k  +  1 )  +  1 )  /  2
)  e.  NN  <->  ( (
k  /  2 )  +  1 )  e.  NN ) )
4632, 45syl5ibr 155 . . . . 5  |-  ( k  e.  NN  ->  (
( k  /  2
)  e.  NN  ->  ( ( ( k  +  1 )  +  1 )  /  2 )  e.  NN ) )
4746orim2d 760 . . . 4  |-  ( k  e.  NN  ->  (
( ( ( k  +  1 )  / 
2 )  e.  NN  \/  ( k  /  2
)  e.  NN )  ->  ( ( ( k  +  1 )  /  2 )  e.  NN  \/  ( ( ( k  +  1 )  +  1 )  /  2 )  e.  NN ) ) )
48 orcom 700 . . . 4  |-  ( ( ( ( k  +  1 )  /  2
)  e.  NN  \/  ( ( ( k  +  1 )  +  1 )  /  2
)  e.  NN )  <-> 
( ( ( ( k  +  1 )  +  1 )  / 
2 )  e.  NN  \/  ( ( k  +  1 )  /  2
)  e.  NN ) )
4947, 48syl6ib 160 . . 3  |-  ( k  e.  NN  ->  (
( ( ( k  +  1 )  / 
2 )  e.  NN  \/  ( k  /  2
)  e.  NN )  ->  ( ( ( ( k  +  1 )  +  1 )  /  2 )  e.  NN  \/  ( ( k  +  1 )  /  2 )  e.  NN ) ) )
506, 12, 18, 24, 31, 49nnind 8639 . 2  |-  ( N  e.  NN  ->  (
( ( N  + 
1 )  /  2
)  e.  NN  \/  ( N  /  2
)  e.  NN ) )
5150orcomd 701 1  |-  ( N  e.  NN  ->  (
( N  /  2
)  e.  NN  \/  ( ( N  + 
1 )  /  2
)  e.  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680    = wceq 1312    e. wcel 1461   class class class wbr 3893  (class class class)co 5726   CCcc 7538   0cc0 7540   1c1 7541    + caddc 7543   # cap 8254    / cdiv 8338   NNcn 8623   2c2 8674
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7629  ax-resscn 7630  ax-1cn 7631  ax-1re 7632  ax-icn 7633  ax-addcl 7634  ax-addrcl 7635  ax-mulcl 7636  ax-mulrcl 7637  ax-addcom 7638  ax-mulcom 7639  ax-addass 7640  ax-mulass 7641  ax-distr 7642  ax-i2m1 7643  ax-0lt1 7644  ax-1rid 7645  ax-0id 7646  ax-rnegex 7647  ax-precex 7648  ax-cnre 7649  ax-pre-ltirr 7650  ax-pre-ltwlin 7651  ax-pre-lttrn 7652  ax-pre-apti 7653  ax-pre-ltadd 7654  ax-pre-mulgt0 7655  ax-pre-mulext 7656
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-br 3894  df-opab 3948  df-id 4173  df-po 4176  df-iso 4177  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-pnf 7719  df-mnf 7720  df-xr 7721  df-ltxr 7722  df-le 7723  df-sub 7851  df-neg 7852  df-reap 8248  df-ap 8255  df-div 8339  df-inn 8624  df-2 8682
This theorem is referenced by:  nneo  9051  zeo  9053
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