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Theorem oddge22np1 11490
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 2180 . . . . . . . 8  |-  ( ( ( 2  x.  n
)  +  1 )  =  N  ->  (
( ( 2  x.  n )  +  1 )  e.  ( ZZ>= ` 
2 )  <->  N  e.  ( ZZ>= `  2 )
) )
2 nn0z 9032 . . . . . . . . . . 11  |-  ( n  e.  NN0  ->  n  e.  ZZ )
32adantl 275 . . . . . . . . . 10  |-  ( ( ( ( 2  x.  n )  +  1 )  e.  ( ZZ>= ` 
2 )  /\  n  e.  NN0 )  ->  n  e.  ZZ )
4 eluz2 9288 . . . . . . . . . . . 12  |-  ( ( ( 2  x.  n
)  +  1 )  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  e.  ZZ  /\  2  <_ 
( ( 2  x.  n )  +  1 ) ) )
5 2re 8754 . . . . . . . . . . . . . . . . 17  |-  2  e.  RR
65a1i 9 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN0  ->  2  e.  RR )
7 1red 7749 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN0  ->  1  e.  RR )
8 2nn0 8952 . . . . . . . . . . . . . . . . . . 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 8993 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN0  ->  ( 2  x.  n )  e. 
NN0 )
1211nn0red 8989 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN0  ->  ( 2  x.  n )  e.  RR )
136, 7, 12lesubaddd 8271 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN0  ->  ( ( 2  -  1 )  <_  ( 2  x.  n )  <->  2  <_  ( ( 2  x.  n
)  +  1 ) ) )
14 2m1e1 8802 . . . . . . . . . . . . . . . . 17  |-  ( 2  -  1 )  =  1
1514breq1i 3906 . . . . . . . . . . . . . . . 16  |-  ( ( 2  -  1 )  <_  ( 2  x.  n )  <->  1  <_  ( 2  x.  n ) )
16 nn0re 8944 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN0  ->  n  e.  RR )
17 2pos 8775 . . . . . . . . . . . . . . . . . . . 20  |-  0  <  2
185, 17pm3.2i 270 . . . . . . . . . . . . . . . . . . 19  |-  ( 2  e.  RR  /\  0  <  2 )
1918a1i 9 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN0  ->  ( 2  e.  RR  /\  0  <  2 ) )
20 ledivmul 8599 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  RR  /\  n  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( 1  /  2 )  <_  n 
<->  1  <_  ( 2  x.  n ) ) )
217, 16, 19, 20syl3anc 1201 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN0  ->  ( ( 1  /  2 )  <_  n  <->  1  <_  ( 2  x.  n ) ) )
22 halfgt0 8893 . . . . . . . . . . . . . . . . . 18  |-  0  <  ( 1  /  2
)
23 0red 7735 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN0  ->  0  e.  RR )
24 halfre 8891 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1  /  2 )  e.  RR
2524a1i 9 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN0  ->  ( 1  /  2 )  e.  RR )
26 ltletr 7821 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 0  e.  RR  /\  ( 1  /  2
)  e.  RR  /\  n  e.  RR )  ->  ( ( 0  < 
( 1  /  2
)  /\  ( 1  /  2 )  <_  n )  ->  0  <  n ) )
2723, 25, 16, 26syl3anc 1201 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN0  ->  ( ( 0  <  ( 1  /  2 )  /\  ( 1  /  2
)  <_  n )  ->  0  <  n ) )
2822, 27mpani 426 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN0  ->  ( ( 1  /  2 )  <_  n  ->  0  <  n ) )
2921, 28sylbird 169 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN0  ->  ( 1  <_  ( 2  x.  n )  ->  0  <  n ) )
3015, 29syl5bi 151 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN0  ->  ( ( 2  -  1 )  <_  ( 2  x.  n )  ->  0  <  n ) )
3113, 30sylbird 169 . . . . . . . . . . . . . 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 989 . . . . . . . . . . . 12  |-  ( ( 2  e.  ZZ  /\  ( ( 2  x.  n )  +  1 )  e.  ZZ  /\  2  <_  ( ( 2  x.  n )  +  1 ) )  -> 
( n  e.  NN0  ->  0  <  n ) )
344, 33sylbi 120 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  +  1 )  e.  ( ZZ>= `  2
)  ->  ( n  e.  NN0  ->  0  <  n ) )
3534imp 123 . . . . . . . . . 10  |-  ( ( ( ( 2  x.  n )  +  1 )  e.  ( ZZ>= ` 
2 )  /\  n  e.  NN0 )  ->  0  <  n )
36 elnnz 9022 . . . . . . . . . 10  |-  ( n  e.  NN  <->  ( n  e.  ZZ  /\  0  < 
n ) )
373, 35, 36sylanbrc 413 . . . . . . . . 9  |-  ( ( ( ( 2  x.  n )  +  1 )  e.  ( ZZ>= ` 
2 )  /\  n  e.  NN0 )  ->  n  e.  NN )
3837ex 114 . . . . . . . 8  |-  ( ( ( 2  x.  n
)  +  1 )  e.  ( ZZ>= `  2
)  ->  ( n  e.  NN0  ->  n  e.  NN ) )
391, 38syl6bir 163 . . . . . . 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 124 . . . . 5  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  n  e.  NN0 )  ->  (
( ( 2  x.  n )  +  1 )  =  N  ->  n  e.  NN )
)
4241pm4.71rd 391 . . . 4  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  n  e.  NN0 )  ->  (
( ( 2  x.  n )  +  1 )  =  N  <->  ( n  e.  NN  /\  ( ( 2  x.  n )  +  1 )  =  N ) ) )
4342bicomd 140 . . 3  |-  ( ( N  e.  ( ZZ>= ` 
2 )  /\  n  e.  NN0 )  ->  (
( n  e.  NN  /\  ( ( 2  x.  n )  +  1 )  =  N )  <-> 
( ( 2  x.  n )  +  1 )  =  N ) )
4443rexbidva 2411 . 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 8938 . . 3  |-  NN  C_  NN0
46 rexss 3134 . . 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 9321 . . 3  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN0 )
49 oddnn02np1 11489 . . 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 220 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 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   E.wrex 2394    C_ wss 3041   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   RRcr 7587   0cc0 7588   1c1 7589    + caddc 7591    x. cmul 7593    < clt 7768    <_ cle 7769    - cmin 7901    / cdiv 8399   NNcn 8684   2c2 8735   NN0cn0 8935   ZZcz 9012   ZZ>=cuz 9282    || cdvds 11405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-xor 1339  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311  df-div 8400  df-inn 8685  df-2 8743  df-n0 8936  df-z 9013  df-uz 9283  df-dvds 11406
This theorem is referenced by: (None)
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