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Theorem reperflem 21615
Description: A subset of the real numbers that is closed under addition with real numbers is perfect. (Contributed by Mario Carneiro, 26-Dec-2016.)
Hypotheses
Ref Expression
recld2.1  |-  J  =  ( TopOpen ` fld )
reperflem.2  |-  ( ( u  e.  S  /\  v  e.  RR )  ->  ( u  +  v )  e.  S )
reperflem.3  |-  S  C_  CC
Assertion
Ref Expression
reperflem  |-  ( Jt  S )  e. Perf
Distinct variable groups:    u, J    v, u, S
Allowed substitution hint:    J( v)

Proof of Theorem reperflem
Dummy variables  n  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnxmet 21572 . . . . . . 7  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
2 reperflem.3 . . . . . . . 8  |-  S  C_  CC
32sseli 3438 . . . . . . 7  |-  ( u  e.  S  ->  u  e.  CC )
4 recld2.1 . . . . . . . . 9  |-  J  =  ( TopOpen ` fld )
54cnfldtopn 21581 . . . . . . . 8  |-  J  =  ( MetOpen `  ( abs  o. 
-  ) )
65neibl 21296 . . . . . . 7  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  u  e.  CC )  ->  ( n  e.  ( ( nei `  J
) `  { u } )  <->  ( n  C_  CC  /\  E. r  e.  RR+  ( u (
ball `  ( abs  o. 
-  ) ) r )  C_  n )
) )
71, 3, 6sylancr 661 . . . . . 6  |-  ( u  e.  S  ->  (
n  e.  ( ( nei `  J ) `
 { u }
)  <->  ( n  C_  CC  /\  E. r  e.  RR+  ( u ( ball `  ( abs  o.  -  ) ) r ) 
C_  n ) ) )
8 reperflem.2 . . . . . . . . . . . . . . . . 17  |-  ( ( u  e.  S  /\  v  e.  RR )  ->  ( u  +  v )  e.  S )
98ralrimiva 2818 . . . . . . . . . . . . . . . 16  |-  ( u  e.  S  ->  A. v  e.  RR  ( u  +  v )  e.  S
)
10 rpre 11271 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  RR+  ->  r  e.  RR )
1110rehalfcld 10826 . . . . . . . . . . . . . . . 16  |-  ( r  e.  RR+  ->  ( r  /  2 )  e.  RR )
12 oveq2 6286 . . . . . . . . . . . . . . . . . 18  |-  ( v  =  ( r  / 
2 )  ->  (
u  +  v )  =  ( u  +  ( r  /  2
) ) )
1312eleq1d 2471 . . . . . . . . . . . . . . . . 17  |-  ( v  =  ( r  / 
2 )  ->  (
( u  +  v )  e.  S  <->  ( u  +  ( r  / 
2 ) )  e.  S ) )
1413rspccva 3159 . . . . . . . . . . . . . . . 16  |-  ( ( A. v  e.  RR  ( u  +  v
)  e.  S  /\  ( r  /  2
)  e.  RR )  ->  ( u  +  ( r  /  2
) )  e.  S
)
159, 11, 14syl2an 475 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( u  +  ( r  /  2 ) )  e.  S )
162, 15sseldi 3440 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( u  +  ( r  /  2 ) )  e.  CC )
173adantr 463 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  ->  u  e.  CC )
18 eqid 2402 . . . . . . . . . . . . . . 15  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1918cnmetdval 21570 . . . . . . . . . . . . . 14  |-  ( ( ( u  +  ( r  /  2 ) )  e.  CC  /\  u  e.  CC )  ->  ( ( u  +  ( r  /  2
) ) ( abs 
o.  -  ) u
)  =  ( abs `  ( ( u  +  ( r  /  2
) )  -  u
) ) )
2016, 17, 19syl2anc 659 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) ) ( abs 
o.  -  ) u
)  =  ( abs `  ( ( u  +  ( r  /  2
) )  -  u
) ) )
21 simpr 459 . . . . . . . . . . . . . . . . 17  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
r  e.  RR+ )
2221rphalfcld 11316 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  e.  RR+ )
2322rpcnd 11306 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  e.  CC )
2417, 23pncan2d 9969 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) )  -  u
)  =  ( r  /  2 ) )
2524fveq2d 5853 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( abs `  (
( u  +  ( r  /  2 ) )  -  u ) )  =  ( abs `  ( r  /  2
) ) )
2622rpred 11304 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  e.  RR )
2722rpge0d 11308 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
0  <_  ( r  /  2 ) )
2826, 27absidd 13403 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( abs `  (
r  /  2 ) )  =  ( r  /  2 ) )
2920, 25, 283eqtrd 2447 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) ) ( abs 
o.  -  ) u
)  =  ( r  /  2 ) )
30 rphalflt 11292 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  ( r  /  2 )  < 
r )
3130adantl 464 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  <  r )
3229, 31eqbrtrd 4415 . . . . . . . . . . 11  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) ) ( abs 
o.  -  ) u
)  <  r )
331a1i 11 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( abs  o.  -  )  e.  ( *Met `  CC ) )
34 rpxr 11272 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  r  e. 
RR* )
3534adantl 464 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
r  e.  RR* )
36 elbl3 21187 . . . . . . . . . . . 12  |-  ( ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  r  e.  RR* )  /\  ( u  e.  CC  /\  ( u  +  ( r  / 
2 ) )  e.  CC ) )  -> 
( ( u  +  ( r  /  2
) )  e.  ( u ( ball `  ( abs  o.  -  ) ) r )  <->  ( (
u  +  ( r  /  2 ) ) ( abs  o.  -  ) u )  < 
r ) )
3733, 35, 17, 16, 36syl22anc 1231 . . . . . . . . . . 11  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) )  e.  ( u ( ball `  ( abs  o.  -  ) ) r )  <->  ( (
u  +  ( r  /  2 ) ) ( abs  o.  -  ) u )  < 
r ) )
3832, 37mpbird 232 . . . . . . . . . 10  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( u  +  ( r  /  2 ) )  e.  ( u ( ball `  ( abs  o.  -  ) ) r ) )
3922rpne0d 11309 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  =/=  0 )
4024, 39eqnetrd 2696 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) )  -  u
)  =/=  0 )
4116, 17, 40subne0ad 9978 . . . . . . . . . . 11  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( u  +  ( r  /  2 ) )  =/=  u )
42 eldifsn 4097 . . . . . . . . . . 11  |-  ( ( u  +  ( r  /  2 ) )  e.  ( S  \  { u } )  <-> 
( ( u  +  ( r  /  2
) )  e.  S  /\  ( u  +  ( r  /  2 ) )  =/=  u ) )
4315, 41, 42sylanbrc 662 . . . . . . . . . 10  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( u  +  ( r  /  2 ) )  e.  ( S 
\  { u }
) )
44 inelcm 3824 . . . . . . . . . 10  |-  ( ( ( u  +  ( r  /  2 ) )  e.  ( u ( ball `  ( abs  o.  -  ) ) r )  /\  (
u  +  ( r  /  2 ) )  e.  ( S  \  { u } ) )  ->  ( (
u ( ball `  ( abs  o.  -  ) ) r )  i^i  ( S  \  { u }
) )  =/=  (/) )
4538, 43, 44syl2anc 659 . . . . . . . . 9  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u (
ball `  ( abs  o. 
-  ) ) r )  i^i  ( S 
\  { u }
) )  =/=  (/) )
46 ssrin 3664 . . . . . . . . . 10  |-  ( ( u ( ball `  ( abs  o.  -  ) ) r )  C_  n  ->  ( ( u (
ball `  ( abs  o. 
-  ) ) r )  i^i  ( S 
\  { u }
) )  C_  (
n  i^i  ( S  \  { u } ) ) )
47 ssn0 3772 . . . . . . . . . . 11  |-  ( ( ( ( u (
ball `  ( abs  o. 
-  ) ) r )  i^i  ( S 
\  { u }
) )  C_  (
n  i^i  ( S  \  { u } ) )  /\  ( ( u ( ball `  ( abs  o.  -  ) ) r )  i^i  ( S  \  { u }
) )  =/=  (/) )  -> 
( n  i^i  ( S  \  { u }
) )  =/=  (/) )
4847ex 432 . . . . . . . . . 10  |-  ( ( ( u ( ball `  ( abs  o.  -  ) ) r )  i^i  ( S  \  { u } ) )  C_  ( n  i^i  ( S  \  {
u } ) )  ->  ( ( ( u ( ball `  ( abs  o.  -  ) ) r )  i^i  ( S  \  { u }
) )  =/=  (/)  ->  (
n  i^i  ( S  \  { u } ) )  =/=  (/) ) )
4946, 48syl 17 . . . . . . . . 9  |-  ( ( u ( ball `  ( abs  o.  -  ) ) r )  C_  n  ->  ( ( ( u ( ball `  ( abs  o.  -  ) ) r )  i^i  ( S  \  { u }
) )  =/=  (/)  ->  (
n  i^i  ( S  \  { u } ) )  =/=  (/) ) )
5045, 49syl5com 28 . . . . . . . 8  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u (
ball `  ( abs  o. 
-  ) ) r )  C_  n  ->  ( n  i^i  ( S 
\  { u }
) )  =/=  (/) ) )
5150rexlimdva 2896 . . . . . . 7  |-  ( u  e.  S  ->  ( E. r  e.  RR+  (
u ( ball `  ( abs  o.  -  ) ) r )  C_  n  ->  ( n  i^i  ( S  \  { u }
) )  =/=  (/) ) )
5251adantld 465 . . . . . 6  |-  ( u  e.  S  ->  (
( n  C_  CC  /\ 
E. r  e.  RR+  ( u ( ball `  ( abs  o.  -  ) ) r ) 
C_  n )  -> 
( n  i^i  ( S  \  { u }
) )  =/=  (/) ) )
537, 52sylbid 215 . . . . 5  |-  ( u  e.  S  ->  (
n  e.  ( ( nei `  J ) `
 { u }
)  ->  ( n  i^i  ( S  \  {
u } ) )  =/=  (/) ) )
5453ralrimiv 2816 . . . 4  |-  ( u  e.  S  ->  A. n  e.  ( ( nei `  J
) `  { u } ) ( n  i^i  ( S  \  { u } ) )  =/=  (/) )
554cnfldtop 21583 . . . . . 6  |-  J  e. 
Top
564cnfldtopon 21582 . . . . . . . 8  |-  J  e.  (TopOn `  CC )
5756toponunii 19725 . . . . . . 7  |-  CC  =  U. J
5857islp2 19939 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  CC  /\  u  e.  CC )  ->  (
u  e.  ( (
limPt `  J ) `  S )  <->  A. n  e.  ( ( nei `  J
) `  { u } ) ( n  i^i  ( S  \  { u } ) )  =/=  (/) ) )
5955, 2, 58mp3an12 1316 . . . . 5  |-  ( u  e.  CC  ->  (
u  e.  ( (
limPt `  J ) `  S )  <->  A. n  e.  ( ( nei `  J
) `  { u } ) ( n  i^i  ( S  \  { u } ) )  =/=  (/) ) )
603, 59syl 17 . . . 4  |-  ( u  e.  S  ->  (
u  e.  ( (
limPt `  J ) `  S )  <->  A. n  e.  ( ( nei `  J
) `  { u } ) ( n  i^i  ( S  \  { u } ) )  =/=  (/) ) )
6154, 60mpbird 232 . . 3  |-  ( u  e.  S  ->  u  e.  ( ( limPt `  J
) `  S )
)
6261ssriv 3446 . 2  |-  S  C_  ( ( limPt `  J
) `  S )
63 eqid 2402 . . . 4  |-  ( Jt  S )  =  ( Jt  S )
6457, 63restperf 19978 . . 3  |-  ( ( J  e.  Top  /\  S  C_  CC )  -> 
( ( Jt  S )  e. Perf 
<->  S  C_  ( ( limPt `  J ) `  S ) ) )
6555, 2, 64mp2an 670 . 2  |-  ( ( Jt  S )  e. Perf  <->  S  C_  (
( limPt `  J ) `  S ) )
6662, 65mpbir 209 1  |-  ( Jt  S )  e. Perf
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755    \ cdif 3411    i^i cin 3413    C_ wss 3414   (/)c0 3738   {csn 3972   class class class wbr 4395    o. ccom 4827   ` cfv 5569  (class class class)co 6278   CCcc 9520   RRcr 9521   0cc0 9522    + caddc 9525   RR*cxr 9657    < clt 9658    - cmin 9841    / cdiv 10247   2c2 10626   RR+crp 11265   abscabs 13216   ↾t crest 15035   TopOpenctopn 15036   *Metcxmt 18723   ballcbl 18725  ℂfldccnfld 18740   Topctop 19686   neicnei 19891   limPtclp 19928  Perfcperf 19929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fi 7905  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-fz 11727  df-seq 12152  df-exp 12211  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-plusg 14922  df-mulr 14923  df-starv 14924  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-rest 15037  df-topn 15038  df-topgen 15058  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-lp 19930  df-perf 19931  df-xms 21115  df-ms 21116
This theorem is referenced by:  reperf  21616  cnperf  21617
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