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Theorem nneoor 9344
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 5876 . . . . . 6  |-  ( j  =  1  ->  (
j  +  1 )  =  ( 1  +  1 ) )
21oveq1d 5884 . . . . 5  |-  ( j  =  1  ->  (
( j  +  1 )  /  2 )  =  ( ( 1  +  1 )  / 
2 ) )
32eleq1d 2246 . . . 4  |-  ( j  =  1  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( (
1  +  1 )  /  2 )  e.  NN ) )
4 oveq1 5876 . . . . 5  |-  ( j  =  1  ->  (
j  /  2 )  =  ( 1  / 
2 ) )
54eleq1d 2246 . . . 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 5876 . . . . . 6  |-  ( j  =  k  ->  (
j  +  1 )  =  ( k  +  1 ) )
87oveq1d 5884 . . . . 5  |-  ( j  =  k  ->  (
( j  +  1 )  /  2 )  =  ( ( k  +  1 )  / 
2 ) )
98eleq1d 2246 . . . 4  |-  ( j  =  k  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( (
k  +  1 )  /  2 )  e.  NN ) )
10 oveq1 5876 . . . . 5  |-  ( j  =  k  ->  (
j  /  2 )  =  ( k  / 
2 ) )
1110eleq1d 2246 . . . 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 5876 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
j  +  1 )  =  ( ( k  +  1 )  +  1 ) )
1413oveq1d 5884 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( j  +  1 )  /  2 )  =  ( ( ( k  +  1 )  +  1 )  / 
2 ) )
1514eleq1d 2246 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( (
( k  +  1 )  +  1 )  /  2 )  e.  NN ) )
16 oveq1 5876 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
j  /  2 )  =  ( ( k  +  1 )  / 
2 ) )
1716eleq1d 2246 . . . 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 5876 . . . . . 6  |-  ( j  =  N  ->  (
j  +  1 )  =  ( N  + 
1 ) )
2019oveq1d 5884 . . . . 5  |-  ( j  =  N  ->  (
( j  +  1 )  /  2 )  =  ( ( N  +  1 )  / 
2 ) )
2120eleq1d 2246 . . . 4  |-  ( j  =  N  ->  (
( ( j  +  1 )  /  2
)  e.  NN  <->  ( ( N  +  1 )  /  2 )  e.  NN ) )
22 oveq1 5876 . . . . 5  |-  ( j  =  N  ->  (
j  /  2 )  =  ( N  / 
2 ) )
2322eleq1d 2246 . . . 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 8967 . . . . . . 7  |-  2  =  ( 1  +  1 )
2625oveq1i 5879 . . . . . 6  |-  ( 2  /  2 )  =  ( ( 1  +  1 )  /  2
)
27 2div2e1 9040 . . . . . 6  |-  ( 2  /  2 )  =  1
2826, 27eqtr3i 2200 . . . . 5  |-  ( ( 1  +  1 )  /  2 )  =  1
29 1nn 8919 . . . . 5  |-  1  e.  NN
3028, 29eqeltri 2250 . . . 4  |-  ( ( 1  +  1 )  /  2 )  e.  NN
3130orci 731 . . 3  |-  ( ( ( 1  +  1 )  /  2 )  e.  NN  \/  (
1  /  2 )  e.  NN )
32 peano2nn 8920 . . . . . 6  |-  ( ( k  /  2 )  e.  NN  ->  (
( k  /  2
)  +  1 )  e.  NN )
33 nncn 8916 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  CC )
34 add1p1 9157 . . . . . . . . . 10  |-  ( k  e.  CC  ->  (
( k  +  1 )  +  1 )  =  ( k  +  2 ) )
3534oveq1d 5884 . . . . . . . . 9  |-  ( k  e.  CC  ->  (
( ( k  +  1 )  +  1 )  /  2 )  =  ( ( k  +  2 )  / 
2 ) )
36 2cn 8979 . . . . . . . . . . 11  |-  2  e.  CC
37 2ap0 9001 . . . . . . . . . . . 12  |-  2 #  0
3836, 37pm3.2i 272 . . . . . . . . . . 11  |-  ( 2  e.  CC  /\  2 #  0 )
39 divdirap 8643 . . . . . . . . . . 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 5880 . . . . . . . . . 10  |-  ( ( k  /  2 )  +  ( 2  / 
2 ) )  =  ( ( k  / 
2 )  +  1 )
4240, 41eqtrdi 2226 . . . . . . . . 9  |-  ( k  e.  CC  ->  (
( k  +  2 )  /  2 )  =  ( ( k  /  2 )  +  1 ) )
4335, 42eqtrd 2210 . . . . . . . 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 2246 . . . . . 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 8924 . 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 2148   class class class wbr 4000  (class class class)co 5869   CCcc 7800   0cc0 7802   1c1 7803    + caddc 7805   # cap 8528    / cdiv 8618   NNcn 8908   2c2 8959
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967
This theorem is referenced by:  nneo  9345  zeo  9347
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