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Theorem evennn02n 11785
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 2220 . . . . . . . 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 8908 . . . . . . . . . . . 12  |-  2  e.  RR
43a1i 9 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  2  e.  RR )
5 zre 9176 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  n  e.  RR )
65adantl 275 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  n  e.  RR )
7 2pos 8929 . . . . . . . . . . . 12  |-  0  <  2
87a1i 9 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  0  <  2 )
9 nn0ge0 9120 . . . . . . . . . . . 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 8730 . . . . . . . . . . 11  |-  ( ( ( 2  e.  RR  /\  n  e.  RR )  /\  ( 0  <  2  /\  0  <_ 
( 2  x.  n
) ) )  -> 
0  <_  n )
124, 6, 8, 10, 11syl22anc 1221 . . . . . . . . . 10  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  0  <_  n )
13 elnn0z 9185 . . . . . . . . . 10  |-  ( n  e.  NN0  <->  ( n  e.  ZZ  /\  0  <_  n ) )
142, 12, 13sylanbrc 414 . . . . . . . . 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 392 . . . 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 2454 . 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 9190 . . 3  |-  NN0  C_  ZZ
23 rexss 3195 . . 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 11777 . . 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 1335    e. wcel 2128   E.wrex 2436    C_ wss 3102   class class class wbr 3967  (class class class)co 5826   RRcr 7733   0cc0 7734    x. cmul 7739    < clt 7914    <_ cle 7915   2c2 8889   NN0cn0 9095   ZZcz 9172    || cdvds 11694
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-mulrcl 7833  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-ltadd 7850  ax-pre-mulgt0 7851
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-br 3968  df-opab 4028  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-iota 5137  df-fun 5174  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-inn 8839  df-2 8897  df-n0 9096  df-z 9173  df-dvds 11695
This theorem is referenced by: (None)
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