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Theorem evennn02n 11841
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 2233 . . . . . . . 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 8948 . . . . . . . . . . . 12  |-  2  e.  RR
43a1i 9 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  2  e.  RR )
5 zre 9216 . . . . . . . . . . . 12  |-  ( n  e.  ZZ  ->  n  e.  RR )
65adantl 275 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  n  e.  RR )
7 2pos 8969 . . . . . . . . . . . 12  |-  0  <  2
87a1i 9 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  0  <  2 )
9 nn0ge0 9160 . . . . . . . . . . . 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 8770 . . . . . . . . . . 11  |-  ( ( ( 2  e.  RR  /\  n  e.  RR )  /\  ( 0  <  2  /\  0  <_ 
( 2  x.  n
) ) )  -> 
0  <_  n )
124, 6, 8, 10, 11syl22anc 1234 . . . . . . . . . 10  |-  ( ( ( 2  x.  n
)  e.  NN0  /\  n  e.  ZZ )  ->  0  <_  n )
13 elnn0z 9225 . . . . . . . . . 10  |-  ( n  e.  NN0  <->  ( n  e.  ZZ  /\  0  <_  n ) )
142, 12, 13sylanbrc 415 . . . . . . . . 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 2467 . 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 9230 . . 3  |-  NN0  C_  ZZ
23 rexss 3214 . . 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 11833 . . 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 1348    e. wcel 2141   E.wrex 2449    C_ wss 3121   class class class wbr 3989  (class class class)co 5853   RRcr 7773   0cc0 7774    x. cmul 7779    < clt 7954    <_ cle 7955   2c2 8929   NN0cn0 9135   ZZcz 9212    || cdvds 11749
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890  ax-pre-mulgt0 7891
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213  df-dvds 11750
This theorem is referenced by: (None)
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