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Theorem nneoor 9326
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 5872 . . . . . 6  |-  ( j  =  1  ->  (
j  +  1 )  =  ( 1  +  1 ) )
21oveq1d 5880 . . . . 5  |-  ( j  =  1  ->  (
( j  +  1 )  /  2 )  =  ( ( 1  +  1 )  / 
2 ) )
32eleq1d 2244 . . . 4  |-  ( j  =  1  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( (
1  +  1 )  /  2 )  e.  NN ) )
4 oveq1 5872 . . . . 5  |-  ( j  =  1  ->  (
j  /  2 )  =  ( 1  / 
2 ) )
54eleq1d 2244 . . . 4  |-  ( j  =  1  ->  (
( j  /  2
)  e.  NN  <->  ( 1  /  2 )  e.  NN ) )
63, 5orbi12d 793 . . 3  |-  ( j  =  1  ->  (
( ( ( j  +  1 )  / 
2 )  e.  NN  \/  ( j  /  2
)  e.  NN )  <-> 
( ( ( 1  +  1 )  / 
2 )  e.  NN  \/  ( 1  /  2
)  e.  NN ) ) )
7 oveq1 5872 . . . . . 6  |-  ( j  =  k  ->  (
j  +  1 )  =  ( k  +  1 ) )
87oveq1d 5880 . . . . 5  |-  ( j  =  k  ->  (
( j  +  1 )  /  2 )  =  ( ( k  +  1 )  / 
2 ) )
98eleq1d 2244 . . . 4  |-  ( j  =  k  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( (
k  +  1 )  /  2 )  e.  NN ) )
10 oveq1 5872 . . . . 5  |-  ( j  =  k  ->  (
j  /  2 )  =  ( k  / 
2 ) )
1110eleq1d 2244 . . . 4  |-  ( j  =  k  ->  (
( j  /  2
)  e.  NN  <->  ( k  /  2 )  e.  NN ) )
129, 11orbi12d 793 . . 3  |-  ( j  =  k  ->  (
( ( ( j  +  1 )  / 
2 )  e.  NN  \/  ( j  /  2
)  e.  NN )  <-> 
( ( ( k  +  1 )  / 
2 )  e.  NN  \/  ( k  /  2
)  e.  NN ) ) )
13 oveq1 5872 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
j  +  1 )  =  ( ( k  +  1 )  +  1 ) )
1413oveq1d 5880 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( j  +  1 )  /  2 )  =  ( ( ( k  +  1 )  +  1 )  / 
2 ) )
1514eleq1d 2244 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( (
( k  +  1 )  +  1 )  /  2 )  e.  NN ) )
16 oveq1 5872 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
j  /  2 )  =  ( ( k  +  1 )  / 
2 ) )
1716eleq1d 2244 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( j  /  2
)  e.  NN  <->  ( (
k  +  1 )  /  2 )  e.  NN ) )
1815, 17orbi12d 793 . . 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 5872 . . . . . 6  |-  ( j  =  N  ->  (
j  +  1 )  =  ( N  + 
1 ) )
2019oveq1d 5880 . . . . 5  |-  ( j  =  N  ->  (
( j  +  1 )  /  2 )  =  ( ( N  +  1 )  / 
2 ) )
2120eleq1d 2244 . . . 4  |-  ( j  =  N  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( ( N  +  1 )  /  2 )  e.  NN ) )
22 oveq1 5872 . . . . 5  |-  ( j  =  N  ->  (
j  /  2 )  =  ( N  / 
2 ) )
2322eleq1d 2244 . . . 4  |-  ( j  =  N  ->  (
( j  /  2
)  e.  NN  <->  ( N  /  2 )  e.  NN ) )
2421, 23orbi12d 793 . . 3  |-  ( j  =  N  ->  (
( ( ( j  +  1 )  / 
2 )  e.  NN  \/  ( j  /  2
)  e.  NN )  <-> 
( ( ( N  +  1 )  / 
2 )  e.  NN  \/  ( N  /  2
)  e.  NN ) ) )
25 df-2 8949 . . . . . . 7  |-  2  =  ( 1  +  1 )
2625oveq1i 5875 . . . . . 6  |-  ( 2  /  2 )  =  ( ( 1  +  1 )  /  2
)
27 2div2e1 9022 . . . . . 6  |-  ( 2  /  2 )  =  1
2826, 27eqtr3i 2198 . . . . 5  |-  ( ( 1  +  1 )  /  2 )  =  1
29 1nn 8901 . . . . 5  |-  1  e.  NN
3028, 29eqeltri 2248 . . . 4  |-  ( ( 1  +  1 )  /  2 )  e.  NN
3130orci 731 . . 3  |-  ( ( ( 1  +  1 )  /  2 )  e.  NN  \/  (
1  /  2 )  e.  NN )
32 peano2nn 8902 . . . . . 6  |-  ( ( k  /  2 )  e.  NN  ->  (
( k  /  2
)  +  1 )  e.  NN )
33 nncn 8898 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  CC )
34 add1p1 9139 . . . . . . . . . 10  |-  ( k  e.  CC  ->  (
( k  +  1 )  +  1 )  =  ( k  +  2 ) )
3534oveq1d 5880 . . . . . . . . 9  |-  ( k  e.  CC  ->  (
( ( k  +  1 )  +  1 )  /  2 )  =  ( ( k  +  2 )  / 
2 ) )
36 2cn 8961 . . . . . . . . . . 11  |-  2  e.  CC
37 2ap0 8983 . . . . . . . . . . . 12  |-  2 #  0
3836, 37pm3.2i 272 . . . . . . . . . . 11  |-  ( 2  e.  CC  /\  2 #  0 )
39 divdirap 8626 . . . . . . . . . . 11  |-  ( ( k  e.  CC  /\  2  e.  CC  /\  (
2  e.  CC  /\  2 #  0 ) )  -> 
( ( k  +  2 )  /  2
)  =  ( ( k  /  2 )  +  ( 2  / 
2 ) ) )
4036, 38, 39mp3an23 1329 . . . . . . . . . 10  |-  ( k  e.  CC  ->  (
( k  +  2 )  /  2 )  =  ( ( k  /  2 )  +  ( 2  /  2
) ) )
4127oveq2i 5876 . . . . . . . . . 10  |-  ( ( k  /  2 )  +  ( 2  / 
2 ) )  =  ( ( k  / 
2 )  +  1 )
4240, 41eqtrdi 2224 . . . . . . . . 9  |-  ( k  e.  CC  ->  (
( k  +  2 )  /  2 )  =  ( ( k  /  2 )  +  1 ) )
4335, 42eqtrd 2208 . . . . . . . 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 2244 . . . . . 6  |-  ( k  e.  NN  ->  (
( ( ( k  +  1 )  +  1 )  /  2
)  e.  NN  <->  ( (
k  /  2 )  +  1 )  e.  NN ) )
4632, 45syl5ibr 156 . . . . 5  |-  ( k  e.  NN  ->  (
( k  /  2
)  e.  NN  ->  ( ( ( k  +  1 )  +  1 )  /  2 )  e.  NN ) )
4746orim2d 788 . . . 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 728 . . . 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 161 . . 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 8906 . 2  |-  ( N  e.  NN  ->  (
( ( N  + 
1 )  /  2
)  e.  NN  \/  ( N  /  2
)  e.  NN ) )
5150orcomd 729 1  |-  ( N  e.  NN  ->  (
( N  /  2
)  e.  NN  \/  ( ( N  + 
1 )  /  2
)  e.  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 708    = wceq 1353    e. wcel 2146   class class class wbr 3998  (class class class)co 5865   CCcc 7784   0cc0 7786   1c1 7787    + caddc 7789   # cap 8512    / cdiv 8601   NNcn 8890   2c2 8941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-id 4287  df-po 4290  df-iso 4291  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949
This theorem is referenced by:  nneo  9327  zeo  9329
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