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Theorem evennn02n 11904
Description: A nonnegative integer is even iff it is twice another nonnegative integer. (Contributed by AV, 12-Aug-2021.)
Assertion
Ref Expression
evennn02n  |-  ( N  e.  NN0  ->  ( 2 
||  N  <->  E. n  e.  NN0  ( 2  x.  n )  =  N ) )
Distinct variable group:    n, N

Proof of Theorem evennn02n
StepHypRef Expression
1 eleq1 2251 . . . . . . . 8  |-  ( ( 2  x.  n )  =  N  ->  (
( 2  x.  n
)  e.  NN0  <->  N  e.  NN0 ) )
2 simpr 110 . . . . . . . . . 10  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  n  e.  ZZ )
3 2re 9006 . . . . . . . . . . . 12  |-  2  e.  RR
43a1i 9 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  2  e.  RR )
5 zre 9274 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  n  e.  RR )
65adantl 277 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  n  e.  RR )
7 2pos 9027 . . . . . . . . . . . 12  |-  0  <  2
87a1i 9 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  0  <  2 )
9 nn0ge0 9218 . . . . . . . . . . . 12  |-  ( ( 2  x.  n )  e.  NN0  ->  0  <_ 
( 2  x.  n
) )
109adantr 276 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  0  <_  ( 2  x.  n ) )
11 prodge0 8828 . . . . . . . . . . 11  |-  ( ( ( 2  e.  RR  /\  n  e.  RR )  /\  ( 0  <  2  /\  0  <_ 
( 2  x.  n
) ) )  -> 
0  <_  n )
124, 6, 8, 10, 11syl22anc 1249 . . . . . . . . . 10  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  0  <_  n )
13 elnn0z 9283 . . . . . . . . . 10  |-  ( n  e.  NN0  <->  ( n  e.  ZZ  /\  0  <_  n ) )
142, 12, 13sylanbrc 417 . . . . . . . . 9  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  n  e.  NN0 )
1514ex 115 . . . . . . . 8  |-  ( ( 2  x.  n )  e.  NN0  ->  ( n  e.  ZZ  ->  n  e.  NN0 ) )
161, 15syl6bir 164 . . . . . . 7  |-  ( ( 2  x.  n )  =  N  ->  ( N  e.  NN0  ->  (
n  e.  ZZ  ->  n  e.  NN0 ) ) )
1716com13 80 . . . . . 6  |-  ( n  e.  ZZ  ->  ( N  e.  NN0  ->  (
( 2  x.  n
)  =  N  ->  n  e.  NN0 ) ) )
1817impcom 125 . . . . 5  |-  ( ( N  e.  NN0  /\  n  e.  ZZ )  ->  ( ( 2  x.  n )  =  N  ->  n  e.  NN0 ) )
1918pm4.71rd 394 . . . 4  |-  ( ( N  e.  NN0  /\  n  e.  ZZ )  ->  ( ( 2  x.  n )  =  N  <-> 
( n  e.  NN0  /\  ( 2  x.  n
)  =  N ) ) )
2019bicomd 141 . . 3  |-  ( ( N  e.  NN0  /\  n  e.  ZZ )  ->  ( ( n  e. 
NN0  /\  ( 2  x.  n )  =  N )  <->  ( 2  x.  n )  =  N ) )
2120rexbidva 2486 . 2  |-  ( N  e.  NN0  ->  ( E. n  e.  ZZ  (
n  e.  NN0  /\  ( 2  x.  n
)  =  N )  <->  E. n  e.  ZZ  ( 2  x.  n
)  =  N ) )
22 nn0ssz 9288 . . 3  |-  NN0  C_  ZZ
23 rexss 3236 . . 3  |-  ( NN0  C_  ZZ  ->  ( E. n  e.  NN0  ( 2  x.  n )  =  N  <->  E. n  e.  ZZ  ( n  e.  NN0  /\  ( 2  x.  n
)  =  N ) ) )
2422, 23mp1i 10 . 2  |-  ( N  e.  NN0  ->  ( E. n  e.  NN0  (
2  x.  n )  =  N  <->  E. n  e.  ZZ  ( n  e. 
NN0  /\  ( 2  x.  n )  =  N ) ) )
25 even2n 11896 . . 3  |-  ( 2 
||  N  <->  E. n  e.  ZZ  ( 2  x.  n )  =  N )
2625a1i 9 . 2  |-  ( N  e.  NN0  ->  ( 2 
||  N  <->  E. n  e.  ZZ  ( 2  x.  n )  =  N ) )
2721, 24, 263bitr4rd 221 1  |-  ( N  e.  NN0  ->  ( 2 
||  N  <->  E. n  e.  NN0  ( 2  x.  n )  =  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1363    e. wcel 2159   E.wrex 2468    C_ wss 3143   class class class wbr 4017  (class class class)co 5890   RRcr 7827   0cc0 7828    x. cmul 7833    < clt 8009    <_ cle 8010   2c2 8987   NN0cn0 9193   ZZcz 9270    || 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-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-ltadd 7944  ax-pre-mulgt0 7945
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  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-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-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-iota 5192  df-fun 5232  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-inn 8937  df-2 8995  df-n0 9194  df-z 9271  df-dvds 11812
This theorem is referenced by: (None)
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