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Theorem evennn02n 11491
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 2180 . . . . . . . 8  |-  ( ( 2  x.  n )  =  N  ->  (
( 2  x.  n
)  e.  NN0  <->  N  e.  NN0 ) )
2 simpr 109 . . . . . . . . . 10  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  n  e.  ZZ )
3 2re 8754 . . . . . . . . . . . 12  |-  2  e.  RR
43a1i 9 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  2  e.  RR )
5 zre 9016 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  n  e.  RR )
65adantl 275 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  n  e.  RR )
7 2pos 8775 . . . . . . . . . . . 12  |-  0  <  2
87a1i 9 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  0  <  2 )
9 nn0ge0 8960 . . . . . . . . . . . 12  |-  ( ( 2  x.  n )  e.  NN0  ->  0  <_ 
( 2  x.  n
) )
109adantr 274 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  0  <_  ( 2  x.  n ) )
11 prodge0 8576 . . . . . . . . . . 11  |-  ( ( ( 2  e.  RR  /\  n  e.  RR )  /\  ( 0  <  2  /\  0  <_ 
( 2  x.  n
) ) )  -> 
0  <_  n )
124, 6, 8, 10, 11syl22anc 1202 . . . . . . . . . 10  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  0  <_  n )
13 elnn0z 9025 . . . . . . . . . 10  |-  ( n  e.  NN0  <->  ( n  e.  ZZ  /\  0  <_  n ) )
142, 12, 13sylanbrc 413 . . . . . . . . 9  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  n  e.  NN0 )
1514ex 114 . . . . . . . 8  |-  ( ( 2  x.  n )  e.  NN0  ->  ( n  e.  ZZ  ->  n  e.  NN0 ) )
161, 15syl6bir 163 . . . . . . 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 124 . . . . 5  |-  ( ( N  e.  NN0  /\  n  e.  ZZ )  ->  ( ( 2  x.  n )  =  N  ->  n  e.  NN0 ) )
1918pm4.71rd 391 . . . 4  |-  ( ( N  e.  NN0  /\  n  e.  ZZ )  ->  ( ( 2  x.  n )  =  N  <-> 
( n  e.  NN0  /\  ( 2  x.  n
)  =  N ) ) )
2019bicomd 140 . . 3  |-  ( ( N  e.  NN0  /\  n  e.  ZZ )  ->  ( ( n  e. 
NN0  /\  ( 2  x.  n )  =  N )  <->  ( 2  x.  n )  =  N ) )
2120rexbidva 2411 . 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 9030 . . 3  |-  NN0  C_  ZZ
23 rexss 3134 . . 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 11483 . . 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 220 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 103    <-> wb 104    = wceq 1316    e. wcel 1465   E.wrex 2394    C_ wss 3041   class class class wbr 3899  (class class class)co 5742   RRcr 7587   0cc0 7588    x. cmul 7593    < clt 7768    <_ cle 7769   2c2 8735   NN0cn0 8935   ZZcz 9012    || 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-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704  ax-pre-mulgt0 7705
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  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-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-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  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-inn 8685  df-2 8743  df-n0 8936  df-z 9013  df-dvds 11406
This theorem is referenced by: (None)
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