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Theorem evennn02n 11427
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 2177 . . . . . . . 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 8700 . . . . . . . . . . . 12  |-  2  e.  RR
43a1i 9 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  2  e.  RR )
5 zre 8962 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  n  e.  RR )
65adantl 273 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  n  e.  RR )
7 2pos 8721 . . . . . . . . . . . 12  |-  0  <  2
87a1i 9 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  0  <  2 )
9 nn0ge0 8906 . . . . . . . . . . . 12  |-  ( ( 2  x.  n )  e.  NN0  ->  0  <_ 
( 2  x.  n
) )
109adantr 272 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  0  <_  ( 2  x.  n ) )
11 prodge0 8522 . . . . . . . . . . 11  |-  ( ( ( 2  e.  RR  /\  n  e.  RR )  /\  ( 0  <  2  /\  0  <_ 
( 2  x.  n
) ) )  -> 
0  <_  n )
124, 6, 8, 10, 11syl22anc 1200 . . . . . . . . . 10  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  0  <_  n )
13 elnn0z 8971 . . . . . . . . . 10  |-  ( n  e.  NN0  <->  ( n  e.  ZZ  /\  0  <_  n ) )
142, 12, 13sylanbrc 411 . . . . . . . . 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 389 . . . 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 2408 . 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 8976 . . 3  |-  NN0  C_  ZZ
23 rexss 3130 . . 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 11419 . . 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 1314    e. wcel 1463   E.wrex 2391    C_ wss 3037   class class class wbr 3895  (class class class)co 5728   RRcr 7546   0cc0 7547    x. cmul 7552    < clt 7724    <_ cle 7725   2c2 8681   NN0cn0 8881   ZZcz 8958    || cdvds 11341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-ltadd 7661  ax-pre-mulgt0 7662
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-2 8689  df-n0 8882  df-z 8959  df-dvds 11342
This theorem is referenced by: (None)
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