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Theorem reperflem 18323
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 18282 . . . . . . 7  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
2 reperflem.3 . . . . . . . 8  |-  S  C_  CC
32sseli 3176 . . . . . . 7  |-  ( u  e.  S  ->  u  e.  CC )
4 recld2.1 . . . . . . . . 9  |-  J  =  ( TopOpen ` fld )
54cnfldtopn 18291 . . . . . . . 8  |-  J  =  ( MetOpen `  ( abs  o. 
-  ) )
65neibl 18047 . . . . . . 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 644 . . . . . 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 2626 . . . . . . . . . . . . . . . 16  |-  ( u  e.  S  ->  A. v  e.  RR  ( u  +  v )  e.  S
)
10 rpre 10360 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  RR+  ->  r  e.  RR )
1110rehalfcld 9958 . . . . . . . . . . . . . . . 16  |-  ( r  e.  RR+  ->  ( r  /  2 )  e.  RR )
12 oveq2 5866 . . . . . . . . . . . . . . . . . 18  |-  ( v  =  ( r  / 
2 )  ->  (
u  +  v )  =  ( u  +  ( r  /  2
) ) )
1312eleq1d 2349 . . . . . . . . . . . . . . . . 17  |-  ( v  =  ( r  / 
2 )  ->  (
( u  +  v )  e.  S  <->  ( u  +  ( r  / 
2 ) )  e.  S ) )
1413rspccva 2883 . . . . . . . . . . . . . . . 16  |-  ( ( A. v  e.  RR  ( u  +  v
)  e.  S  /\  ( r  /  2
)  e.  RR )  ->  ( u  +  ( r  /  2
) )  e.  S
)
159, 11, 14syl2an 463 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( u  +  ( r  /  2 ) )  e.  S )
162, 15sseldi 3178 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( u  +  ( r  /  2 ) )  e.  CC )
173adantr 451 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  ->  u  e.  CC )
18 eqid 2283 . . . . . . . . . . . . . . 15  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1918cnmetdval 18280 . . . . . . . . . . . . . 14  |-  ( ( ( u  +  ( r  /  2 ) )  e.  CC  /\  u  e.  CC )  ->  ( ( u  +  ( r  /  2
) ) ( abs 
o.  -  ) u
)  =  ( abs `  ( ( u  +  ( r  /  2
) )  -  u
) ) )
2016, 17, 19syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) ) ( abs 
o.  -  ) u
)  =  ( abs `  ( ( u  +  ( r  /  2
) )  -  u
) ) )
21 simpr 447 . . . . . . . . . . . . . . . . 17  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
r  e.  RR+ )
2221rphalfcld 10402 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  e.  RR+ )
2322rpcnd 10392 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  e.  CC )
2417, 23pncan2d 9159 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) )  -  u
)  =  ( r  /  2 ) )
2524fveq2d 5529 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( abs `  (
( u  +  ( r  /  2 ) )  -  u ) )  =  ( abs `  ( r  /  2
) ) )
2622rpred 10390 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  e.  RR )
2722rpge0d 10394 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
0  <_  ( r  /  2 ) )
2826, 27absidd 11905 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( abs `  (
r  /  2 ) )  =  ( r  /  2 ) )
2920, 25, 283eqtrd 2319 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) ) ( abs 
o.  -  ) u
)  =  ( r  /  2 ) )
30 rphalflt 10380 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  ( r  /  2 )  < 
r )
3130adantl 452 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  <  r )
3229, 31eqbrtrd 4043 . . . . . . . . . . 11  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) ) ( abs 
o.  -  ) u
)  <  r )
331a1i 10 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( abs  o.  -  )  e.  ( * Met `  CC ) )
34 rpxr 10361 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  r  e. 
RR* )
3534adantl 452 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
r  e.  RR* )
36 elbl3 17951 . . . . . . . . . . . 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 1183 . . . . . . . . . . 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 223 . . . . . . . . . 10  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( u  +  ( r  /  2 ) )  e.  ( u ( ball `  ( abs  o.  -  ) ) r ) )
3922rpne0d 10395 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  =/=  0 )
4024, 39eqnetrd 2464 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) )  -  u
)  =/=  0 )
41 subeq0 9073 . . . . . . . . . . . . . 14  |-  ( ( ( u  +  ( r  /  2 ) )  e.  CC  /\  u  e.  CC )  ->  ( ( ( u  +  ( r  / 
2 ) )  -  u )  =  0  <-> 
( u  +  ( r  /  2 ) )  =  u ) )
4216, 17, 41syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( ( u  +  ( r  / 
2 ) )  -  u )  =  0  <-> 
( u  +  ( r  /  2 ) )  =  u ) )
4342necon3bid 2481 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( ( u  +  ( r  / 
2 ) )  -  u )  =/=  0  <->  ( u  +  ( r  /  2 ) )  =/=  u ) )
4440, 43mpbid 201 . . . . . . . . . . 11  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( u  +  ( r  /  2 ) )  =/=  u )
45 eldifsn 3749 . . . . . . . . . . 11  |-  ( ( u  +  ( r  /  2 ) )  e.  ( S  \  { u } )  <-> 
( ( u  +  ( r  /  2
) )  e.  S  /\  ( u  +  ( r  /  2 ) )  =/=  u ) )
4615, 44, 45sylanbrc 645 . . . . . . . . . 10  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( u  +  ( r  /  2 ) )  e.  ( S 
\  { u }
) )
47 inelcm 3509 . . . . . . . . . 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 }
) )  =/=  (/) )
4838, 46, 47syl2anc 642 . . . . . . . . 9  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u (
ball `  ( abs  o. 
-  ) ) r )  i^i  ( S 
\  { u }
) )  =/=  (/) )
49 ssrin 3394 . . . . . . . . . 10  |-  ( ( u ( ball `  ( abs  o.  -  ) ) r )  C_  n  ->  ( ( u (
ball `  ( abs  o. 
-  ) ) r )  i^i  ( S 
\  { u }
) )  C_  (
n  i^i  ( S  \  { u } ) ) )
50 ssn0 3487 . . . . . . . . . . 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 }
) )  =/=  (/) )
5150ex 423 . . . . . . . . . 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 } ) )  =/=  (/) ) )
5249, 51syl 15 . . . . . . . . 9  |-  ( ( u ( ball `  ( abs  o.  -  ) ) r )  C_  n  ->  ( ( ( u ( ball `  ( abs  o.  -  ) ) r )  i^i  ( S  \  { u }
) )  =/=  (/)  ->  (
n  i^i  ( S  \  { u } ) )  =/=  (/) ) )
5348, 52syl5com 26 . . . . . . . 8  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u (
ball `  ( abs  o. 
-  ) ) r )  C_  n  ->  ( n  i^i  ( S 
\  { u }
) )  =/=  (/) ) )
5453rexlimdva 2667 . . . . . . 7  |-  ( u  e.  S  ->  ( E. r  e.  RR+  (
u ( ball `  ( abs  o.  -  ) ) r )  C_  n  ->  ( n  i^i  ( S  \  { u }
) )  =/=  (/) ) )
5554adantld 453 . . . . . 6  |-  ( u  e.  S  ->  (
( n  C_  CC  /\ 
E. r  e.  RR+  ( u ( ball `  ( abs  o.  -  ) ) r ) 
C_  n )  -> 
( n  i^i  ( S  \  { u }
) )  =/=  (/) ) )
567, 55sylbid 206 . . . . 5  |-  ( u  e.  S  ->  (
n  e.  ( ( nei `  J ) `
 { u }
)  ->  ( n  i^i  ( S  \  {
u } ) )  =/=  (/) ) )
5756ralrimiv 2625 . . . 4  |-  ( u  e.  S  ->  A. n  e.  ( ( nei `  J
) `  { u } ) ( n  i^i  ( S  \  { u } ) )  =/=  (/) )
584cnfldtop 18293 . . . . . 6  |-  J  e. 
Top
594cnfldtopon 18292 . . . . . . . 8  |-  J  e.  (TopOn `  CC )
6059toponunii 16670 . . . . . . 7  |-  CC  =  U. J
6160islp2 16877 . . . . . 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 } ) )  =/=  (/) ) )
6258, 2, 61mp3an12 1267 . . . . 5  |-  ( u  e.  CC  ->  (
u  e.  ( (
limPt `  J ) `  S )  <->  A. n  e.  ( ( nei `  J
) `  { u } ) ( n  i^i  ( S  \  { u } ) )  =/=  (/) ) )
633, 62syl 15 . . . 4  |-  ( u  e.  S  ->  (
u  e.  ( (
limPt `  J ) `  S )  <->  A. n  e.  ( ( nei `  J
) `  { u } ) ( n  i^i  ( S  \  { u } ) )  =/=  (/) ) )
6457, 63mpbird 223 . . 3  |-  ( u  e.  S  ->  u  e.  ( ( limPt `  J
) `  S )
)
6564ssriv 3184 . 2  |-  S  C_  ( ( limPt `  J
) `  S )
66 eqid 2283 . . . 4  |-  ( Jt  S )  =  ( Jt  S )
6760, 66restperf 16914 . . 3  |-  ( ( J  e.  Top  /\  S  C_  CC )  -> 
( ( Jt  S )  e. Perf 
<->  S  C_  ( ( limPt `  J ) `  S ) ) )
6858, 2, 67mp2an 653 . 2  |-  ( ( Jt  S )  e. Perf  <->  S  C_  (
( limPt `  J ) `  S ) )
6965, 68mpbir 200 1  |-  ( Jt  S )  e. Perf
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023    o. ccom 4693   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740   RR*cxr 8866    < clt 8867    - cmin 9037    / cdiv 9423   2c2 9795   RR+crp 10354   abscabs 11719   ↾t crest 13325   TopOpenctopn 13326   * Metcxmt 16369   ballcbl 16371  ℂfldccnfld 16377   Topctop 16631   neicnei 16834   limPtclp 16866  Perfcperf 16867
This theorem is referenced by:  reperf  18324  cnperf  18325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-xms 17885  df-ms 17886
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