ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  oddge22np1 Unicode version

Theorem oddge22np1 11905
Description: An integer greater than one is odd iff it is one plus twice a positive integer. (Contributed by AV, 16-Aug-2021.)
Assertion
Ref Expression
oddge22np1  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( -.  2  ||  N  <->  E. n  e.  NN  ( ( 2  x.  n )  +  1 )  =  N ) )
Distinct variable group:    n, N

Proof of Theorem oddge22np1
StepHypRef Expression
1 eleq1 2252 . . . . . . . 8  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  (
( ( 2  x.  n )  +  1 )  e.  ( ZZ>= ` 
2 )  <->  N  e.  ( ZZ>= `  2 )
) )
2 nn0z 9292 . . . . . . . . . . 11  |-  ( n  e.  NN0  ->  n  e.  ZZ )
32adantl 277 . . . . . . . . . 10  |-  ( ( ( ( 2  x.  n )  +  1 )  e.  ( ZZ>= ` 
2 )  /\  n  e.  NN0 )  ->  n  e.  ZZ )
4 eluz2 9553 . . . . . . . . . . . 12  |-  ( ( ( 2  x.  n
)  +  1 )  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  e.  ZZ  /\  2  <_ 
( ( 2  x.  n )  +  1 ) ) )
5 2re 9008 . . . . . . . . . . . . . . . . 17  |-  2  e.  RR
65a1i 9 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN0  ->  2  e.  RR )
7 1red 7991 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN0  ->  1  e.  RR )
8 2nn0 9212 . . . . . . . . . . . . . . . . . . 19  |-  2  e.  NN0
98a1i 9 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN0  ->  2  e. 
NN0 )
10 id 19 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN0  ->  n  e. 
NN0 )
119, 10nn0mulcld 9253 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN0  ->  ( 2  x.  n )  e. 
NN0 )
1211nn0red 9249 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN0  ->  ( 2  x.  n )  e.  RR )
136, 7, 12lesubaddd 8518 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN0  ->  ( ( 2  -  1 )  <_  ( 2  x.  n )  <->  2  <_  ( ( 2  x.  n
)  +  1 ) ) )
14 2m1e1 9056 . . . . . . . . . . . . . . . . 17  |-  ( 2  -  1 )  =  1
1514breq1i 4025 . . . . . . . . . . . . . . . 16  |-  ( ( 2  -  1 )  <_  ( 2  x.  n )  <->  1  <_  ( 2  x.  n ) )
16 nn0re 9204 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN0  ->  n  e.  RR )
17 2pos 9029 . . . . . . . . . . . . . . . . . . . 20  |-  0  <  2
185, 17pm3.2i 272 . . . . . . . . . . . . . . . . . . 19  |-  ( 2  e.  RR  /\  0  <  2 )
1918a1i 9 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN0  ->  ( 2  e.  RR  /\  0  <  2 ) )
20 ledivmul 8853 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  RR  /\  n  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( 1  /  2 )  <_  n 
<->  1  <_  ( 2  x.  n ) ) )
217, 16, 19, 20syl3anc 1249 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN0  ->  ( ( 1  /  2 )  <_  n  <->  1  <_  ( 2  x.  n ) ) )
22 halfgt0 9153 . . . . . . . . . . . . . . . . . 18  |-  0  <  ( 1  /  2
)
23 0red 7977 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN0  ->  0  e.  RR )
24 halfre 9151 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1  /  2 )  e.  RR
2524a1i 9 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN0  ->  ( 1  /  2 )  e.  RR )
26 ltletr 8066 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 0  e.  RR  /\  ( 1  /  2
)  e.  RR  /\  n  e.  RR )  ->  ( ( 0  < 
( 1  /  2
)  /\  ( 1  /  2 )  <_  n )  ->  0  <  n ) )
2723, 25, 16, 26syl3anc 1249 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN0  ->  ( ( 0  <  ( 1  /  2 )  /\  ( 1  /  2
)  <_  n )  ->  0  <  n ) )
2822, 27mpani 430 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN0  ->  ( ( 1  /  2 )  <_  n  ->  0  <  n ) )
2921, 28sylbird 170 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN0  ->  ( 1  <_  ( 2  x.  n )  ->  0  <  n ) )
3015, 29biimtrid 152 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN0  ->  ( ( 2  -  1 )  <_  ( 2  x.  n )  ->  0  <  n ) )
3113, 30sylbird 170 . . . . . . . . . . . . . 14  |-  ( n  e.  NN0  ->  ( 2  <_  ( ( 2  x.  n )  +  1 )  ->  0  <  n ) )
3231com12 30 . . . . . . . . . . . . 13  |-  ( 2  <_  ( ( 2  x.  n )  +  1 )  ->  (
n  e.  NN0  ->  0  <  n ) )
33323ad2ant3 1022 . . . . . . . . . . . 12  |-  ( ( 2  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  e.  ZZ  /\  2  <_  ( ( 2  x.  n )  +  1 ) )  -> 
( n  e.  NN0  ->  0  <  n ) )
344, 33sylbi 121 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  +  1 )  e.  ( ZZ>= `  2
)  ->  ( n  e.  NN0  ->  0  <  n ) )
3534imp 124 . . . . . . . . . 10  |-  ( ( ( ( 2  x.  n )  +  1 )  e.  ( ZZ>= ` 
2 )  /\  n  e.  NN0 )  ->  0  <  n )
36 elnnz 9282 . . . . . . . . . 10  |-  ( n  e.  NN  <->  ( n  e.  ZZ  /\  0  < 
n ) )
373, 35, 36sylanbrc 417 . . . . . . . . 9  |-  ( ( ( ( 2  x.  n )  +  1 )  e.  ( ZZ>= ` 
2 )  /\  n  e.  NN0 )  ->  n  e.  NN )
3837ex 115 . . . . . . . 8  |-  ( ( ( 2  x.  n
)  +  1 )  e.  ( ZZ>= `  2
)  ->  ( n  e.  NN0  ->  n  e.  NN ) )
391, 38syl6bir 164 . . . . . . 7  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  ( N  e.  ( ZZ>= ` 
2 )  ->  (
n  e.  NN0  ->  n  e.  NN ) ) )
4039com13 80 . . . . . 6  |-  ( n  e.  NN0  ->  ( N  e.  ( ZZ>= `  2
)  ->  ( (
( 2  x.  n
)  +  1 )  =  N  ->  n  e.  NN ) ) )
4140impcom 125 . . . . 5  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  n  e.  NN0 )  ->  (
( ( 2  x.  n )  +  1 )  =  N  ->  n  e.  NN )
)
4241pm4.71rd 394 . . . 4  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  n  e.  NN0 )  ->  (
( ( 2  x.  n )  +  1 )  =  N  <->  ( n  e.  NN  /\  ( ( 2  x.  n )  +  1 )  =  N ) ) )
4342bicomd 141 . . 3  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  n  e.  NN0 )  ->  (
( n  e.  NN  /\  ( ( 2  x.  n )  +  1 )  =  N )  <-> 
( ( 2  x.  n )  +  1 )  =  N ) )
4443rexbidva 2487 . 2  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( E. n  e.  NN0  ( n  e.  NN  /\  (
( 2  x.  n
)  +  1 )  =  N )  <->  E. n  e.  NN0  ( ( 2  x.  n )  +  1 )  =  N ) )
45 nnssnn0 9198 . . 3  |-  NN  C_  NN0
46 rexss 3237 . . 3  |-  ( NN  C_  NN0  ->  ( E. n  e.  NN  (
( 2  x.  n
)  +  1 )  =  N  <->  E. n  e.  NN0  ( n  e.  NN  /\  ( ( 2  x.  n )  +  1 )  =  N ) ) )
4745, 46mp1i 10 . 2  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( E. n  e.  NN  (
( 2  x.  n
)  +  1 )  =  N  <->  E. n  e.  NN0  ( n  e.  NN  /\  ( ( 2  x.  n )  +  1 )  =  N ) ) )
48 eluzge2nn0 9588 . . 3  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN0 )
49 oddnn02np1 11904 . . 3  |-  ( N  e.  NN0  ->  ( -.  2  ||  N  <->  E. n  e.  NN0  ( ( 2  x.  n )  +  1 )  =  N ) )
5048, 49syl 14 . 2  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( -.  2  ||  N  <->  E. n  e.  NN0  ( ( 2  x.  n )  +  1 )  =  N ) )
5144, 47, 503bitr4rd 221 1  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( -.  2  ||  N  <->  E. n  e.  NN  ( ( 2  x.  n )  +  1 )  =  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   E.wrex 2469    C_ wss 3144   class class class wbr 4018   ` cfv 5231  (class class class)co 5891   RRcr 7829   0cc0 7830   1c1 7831    + caddc 7833    x. cmul 7835    < clt 8011    <_ cle 8012    - cmin 8147    / cdiv 8648   NNcn 8938   2c2 8989   NN0cn0 9195   ZZcz 9272   ZZ>=cuz 9547    || cdvds 11813
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649  df-inn 8939  df-2 8997  df-n0 9196  df-z 9273  df-uz 9548  df-dvds 11814
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator