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Theorem nneoor 8818
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 5641 . . . . . 6  |-  ( j  =  1  ->  (
j  +  1 )  =  ( 1  +  1 ) )
21oveq1d 5649 . . . . 5  |-  ( j  =  1  ->  (
( j  +  1 )  /  2 )  =  ( ( 1  +  1 )  / 
2 ) )
32eleq1d 2156 . . . 4  |-  ( j  =  1  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( (
1  +  1 )  /  2 )  e.  NN ) )
4 oveq1 5641 . . . . 5  |-  ( j  =  1  ->  (
j  /  2 )  =  ( 1  / 
2 ) )
54eleq1d 2156 . . . 4  |-  ( j  =  1  ->  (
( j  /  2
)  e.  NN  <->  ( 1  /  2 )  e.  NN ) )
63, 5orbi12d 742 . . 3  |-  ( j  =  1  ->  (
( ( ( j  +  1 )  / 
2 )  e.  NN  \/  ( j  /  2
)  e.  NN )  <-> 
( ( ( 1  +  1 )  / 
2 )  e.  NN  \/  ( 1  /  2
)  e.  NN ) ) )
7 oveq1 5641 . . . . . 6  |-  ( j  =  k  ->  (
j  +  1 )  =  ( k  +  1 ) )
87oveq1d 5649 . . . . 5  |-  ( j  =  k  ->  (
( j  +  1 )  /  2 )  =  ( ( k  +  1 )  / 
2 ) )
98eleq1d 2156 . . . 4  |-  ( j  =  k  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( (
k  +  1 )  /  2 )  e.  NN ) )
10 oveq1 5641 . . . . 5  |-  ( j  =  k  ->  (
j  /  2 )  =  ( k  / 
2 ) )
1110eleq1d 2156 . . . 4  |-  ( j  =  k  ->  (
( j  /  2
)  e.  NN  <->  ( k  /  2 )  e.  NN ) )
129, 11orbi12d 742 . . 3  |-  ( j  =  k  ->  (
( ( ( j  +  1 )  / 
2 )  e.  NN  \/  ( j  /  2
)  e.  NN )  <-> 
( ( ( k  +  1 )  / 
2 )  e.  NN  \/  ( k  /  2
)  e.  NN ) ) )
13 oveq1 5641 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
j  +  1 )  =  ( ( k  +  1 )  +  1 ) )
1413oveq1d 5649 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( j  +  1 )  /  2 )  =  ( ( ( k  +  1 )  +  1 )  / 
2 ) )
1514eleq1d 2156 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( (
( k  +  1 )  +  1 )  /  2 )  e.  NN ) )
16 oveq1 5641 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
j  /  2 )  =  ( ( k  +  1 )  / 
2 ) )
1716eleq1d 2156 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( j  /  2
)  e.  NN  <->  ( (
k  +  1 )  /  2 )  e.  NN ) )
1815, 17orbi12d 742 . . 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 5641 . . . . . 6  |-  ( j  =  N  ->  (
j  +  1 )  =  ( N  + 
1 ) )
2019oveq1d 5649 . . . . 5  |-  ( j  =  N  ->  (
( j  +  1 )  /  2 )  =  ( ( N  +  1 )  / 
2 ) )
2120eleq1d 2156 . . . 4  |-  ( j  =  N  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( ( N  +  1 )  /  2 )  e.  NN ) )
22 oveq1 5641 . . . . 5  |-  ( j  =  N  ->  (
j  /  2 )  =  ( N  / 
2 ) )
2322eleq1d 2156 . . . 4  |-  ( j  =  N  ->  (
( j  /  2
)  e.  NN  <->  ( N  /  2 )  e.  NN ) )
2421, 23orbi12d 742 . . 3  |-  ( j  =  N  ->  (
( ( ( j  +  1 )  / 
2 )  e.  NN  \/  ( j  /  2
)  e.  NN )  <-> 
( ( ( N  +  1 )  / 
2 )  e.  NN  \/  ( N  /  2
)  e.  NN ) ) )
25 df-2 8452 . . . . . . 7  |-  2  =  ( 1  +  1 )
2625oveq1i 5644 . . . . . 6  |-  ( 2  /  2 )  =  ( ( 1  +  1 )  /  2
)
27 2div2e1 8518 . . . . . 6  |-  ( 2  /  2 )  =  1
2826, 27eqtr3i 2110 . . . . 5  |-  ( ( 1  +  1 )  /  2 )  =  1
29 1nn 8405 . . . . 5  |-  1  e.  NN
3028, 29eqeltri 2160 . . . 4  |-  ( ( 1  +  1 )  /  2 )  e.  NN
3130orci 685 . . 3  |-  ( ( ( 1  +  1 )  /  2 )  e.  NN  \/  (
1  /  2 )  e.  NN )
32 peano2nn 8406 . . . . . 6  |-  ( ( k  /  2 )  e.  NN  ->  (
( k  /  2
)  +  1 )  e.  NN )
33 nncn 8402 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  CC )
34 add1p1 8635 . . . . . . . . . 10  |-  ( k  e.  CC  ->  (
( k  +  1 )  +  1 )  =  ( k  +  2 ) )
3534oveq1d 5649 . . . . . . . . 9  |-  ( k  e.  CC  ->  (
( ( k  +  1 )  +  1 )  /  2 )  =  ( ( k  +  2 )  / 
2 ) )
36 2cn 8464 . . . . . . . . . . 11  |-  2  e.  CC
37 2ap0 8486 . . . . . . . . . . . 12  |-  2 #  0
3836, 37pm3.2i 266 . . . . . . . . . . 11  |-  ( 2  e.  CC  /\  2 #  0 )
39 divdirap 8138 . . . . . . . . . . 11  |-  ( ( k  e.  CC  /\  2  e.  CC  /\  (
2  e.  CC  /\  2 #  0 ) )  -> 
( ( k  +  2 )  /  2
)  =  ( ( k  /  2 )  +  ( 2  / 
2 ) ) )
4036, 38, 39mp3an23 1265 . . . . . . . . . 10  |-  ( k  e.  CC  ->  (
( k  +  2 )  /  2 )  =  ( ( k  /  2 )  +  ( 2  /  2
) ) )
4127oveq2i 5645 . . . . . . . . . 10  |-  ( ( k  /  2 )  +  ( 2  / 
2 ) )  =  ( ( k  / 
2 )  +  1 )
4240, 41syl6eq 2136 . . . . . . . . 9  |-  ( k  e.  CC  ->  (
( k  +  2 )  /  2 )  =  ( ( k  /  2 )  +  1 ) )
4335, 42eqtrd 2120 . . . . . . . 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 2156 . . . . . 6  |-  ( k  e.  NN  ->  (
( ( ( k  +  1 )  +  1 )  /  2
)  e.  NN  <->  ( (
k  /  2 )  +  1 )  e.  NN ) )
4632, 45syl5ibr 154 . . . . 5  |-  ( k  e.  NN  ->  (
( k  /  2
)  e.  NN  ->  ( ( ( k  +  1 )  +  1 )  /  2 )  e.  NN ) )
4746orim2d 737 . . . 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 682 . . . 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 159 . . 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 8410 . 2  |-  ( N  e.  NN  ->  (
( ( N  + 
1 )  /  2
)  e.  NN  \/  ( N  /  2
)  e.  NN ) )
5150orcomd 683 1  |-  ( N  e.  NN  ->  (
( N  /  2
)  e.  NN  \/  ( ( N  + 
1 )  /  2
)  e.  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664    = wceq 1289    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   CCcc 7327   0cc0 7329   1c1 7330    + caddc 7332   # cap 8034    / cdiv 8113   NNcn 8394   2c2 8444
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-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
This theorem is referenced by:  nneo  8819  zeo  8821
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