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Theorem oddge22np1 11903
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 2251 . . . . . . . 8  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  (
( ( 2  x.  n )  +  1 )  e.  ( ZZ>= ` 
2 )  <->  N  e.  ( ZZ>= `  2 )
) )
2 nn0z 9290 . . . . . . . . . . 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 9551 . . . . . . . . . . . 12  |-  ( ( ( 2  x.  n
)  +  1 )  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  e.  ZZ  /\  2  <_ 
( ( 2  x.  n )  +  1 ) ) )
5 2re 9006 . . . . . . . . . . . . . . . . 17  |-  2  e.  RR
65a1i 9 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN0  ->  2  e.  RR )
7 1red 7989 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN0  ->  1  e.  RR )
8 2nn0 9210 . . . . . . . . . . . . . . . . . . 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 9251 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN0  ->  ( 2  x.  n )  e. 
NN0 )
1211nn0red 9247 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN0  ->  ( 2  x.  n )  e.  RR )
136, 7, 12lesubaddd 8516 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN0  ->  ( ( 2  -  1 )  <_  ( 2  x.  n )  <->  2  <_  ( ( 2  x.  n
)  +  1 ) ) )
14 2m1e1 9054 . . . . . . . . . . . . . . . . 17  |-  ( 2  -  1 )  =  1
1514breq1i 4024 . . . . . . . . . . . . . . . 16  |-  ( ( 2  -  1 )  <_  ( 2  x.  n )  <->  1  <_  ( 2  x.  n ) )
16 nn0re 9202 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN0  ->  n  e.  RR )
17 2pos 9027 . . . . . . . . . . . . . . . . . . . 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 8851 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  RR  /\  n  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( 1  /  2 )  <_  n 
<->  1  <_  ( 2  x.  n ) ) )
217, 16, 19, 20syl3anc 1248 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN0  ->  ( ( 1  /  2 )  <_  n  <->  1  <_  ( 2  x.  n ) ) )
22 halfgt0 9151 . . . . . . . . . . . . . . . . . 18  |-  0  <  ( 1  /  2
)
23 0red 7975 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN0  ->  0  e.  RR )
24 halfre 9149 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1  /  2 )  e.  RR
2524a1i 9 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN0  ->  ( 1  /  2 )  e.  RR )
26 ltletr 8064 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 0  e.  RR  /\  ( 1  /  2
)  e.  RR  /\  n  e.  RR )  ->  ( ( 0  < 
( 1  /  2
)  /\  ( 1  /  2 )  <_  n )  ->  0  <  n ) )
2723, 25, 16, 26syl3anc 1248 . . . . . . . . . . . . . . . . . 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 1021 . . . . . . . . . . . 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 9280 . . . . . . . . . 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 2486 . 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 9196 . . 3  |-  NN  C_  NN0
46 rexss 3236 . . 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 9586 . . 3  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN0 )
49 oddnn02np1 11902 . . 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 979    = wceq 1363    e. wcel 2159   E.wrex 2468    C_ wss 3143   class class class wbr 4017   ` cfv 5230  (class class class)co 5890   RRcr 7827   0cc0 7828   1c1 7829    + caddc 7831    x. cmul 7833    < clt 8009    <_ cle 8010    - cmin 8145    / cdiv 8646   NNcn 8936   2c2 8987   NN0cn0 9193   ZZcz 9270   ZZ>=cuz 9545    || cdvds 11811
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-xor 1386  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-po 4310  df-iso 4311  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-2 8995  df-n0 9194  df-z 9271  df-uz 9546  df-dvds 11812
This theorem is referenced by: (None)
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