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Theorem reperflem 18339
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 18298 . . . . . . 7  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
2 reperflem.3 . . . . . . . 8  |-  S  C_  CC
32sseli 3189 . . . . . . 7  |-  ( u  e.  S  ->  u  e.  CC )
4 recld2.1 . . . . . . . . 9  |-  J  =  ( TopOpen ` fld )
54cnfldtopn 18307 . . . . . . . 8  |-  J  =  ( MetOpen `  ( abs  o. 
-  ) )
65neibl 18063 . . . . . . 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 2639 . . . . . . . . . . . . . . . 16  |-  ( u  e.  S  ->  A. v  e.  RR  ( u  +  v )  e.  S
)
10 rpre 10376 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  RR+  ->  r  e.  RR )
1110rehalfcld 9974 . . . . . . . . . . . . . . . 16  |-  ( r  e.  RR+  ->  ( r  /  2 )  e.  RR )
12 oveq2 5882 . . . . . . . . . . . . . . . . . 18  |-  ( v  =  ( r  / 
2 )  ->  (
u  +  v )  =  ( u  +  ( r  /  2
) ) )
1312eleq1d 2362 . . . . . . . . . . . . . . . . 17  |-  ( v  =  ( r  / 
2 )  ->  (
( u  +  v )  e.  S  <->  ( u  +  ( r  / 
2 ) )  e.  S ) )
1413rspccva 2896 . . . . . . . . . . . . . . . 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 3191 . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . 15  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1918cnmetdval 18296 . . . . . . . . . . . . . 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 10418 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  e.  RR+ )
2322rpcnd 10408 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  e.  CC )
2417, 23pncan2d 9175 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) )  -  u
)  =  ( r  /  2 ) )
2524fveq2d 5545 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( abs `  (
( u  +  ( r  /  2 ) )  -  u ) )  =  ( abs `  ( r  /  2
) ) )
2622rpred 10406 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  e.  RR )
2722rpge0d 10410 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
0  <_  ( r  /  2 ) )
2826, 27absidd 11921 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( abs `  (
r  /  2 ) )  =  ( r  /  2 ) )
2920, 25, 283eqtrd 2332 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) ) ( abs 
o.  -  ) u
)  =  ( r  /  2 ) )
30 rphalflt 10396 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  ( r  /  2 )  < 
r )
3130adantl 452 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  <  r )
3229, 31eqbrtrd 4059 . . . . . . . . . . 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 10377 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  r  e. 
RR* )
3534adantl 452 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
r  e.  RR* )
36 elbl3 17967 . . . . . . . . . . . 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 10411 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  =/=  0 )
4024, 39eqnetrd 2477 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) )  -  u
)  =/=  0 )
41 subeq0 9089 . . . . . . . . . . . . . 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 2494 . . . . . . . . . . . 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 3762 . . . . . . . . . . 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 3522 . . . . . . . . . 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 3407 . . . . . . . . . 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 3500 . . . . . . . . . . 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 2680 . . . . . . 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 2638 . . . 4  |-  ( u  e.  S  ->  A. n  e.  ( ( nei `  J
) `  { u } ) ( n  i^i  ( S  \  { u } ) )  =/=  (/) )
584cnfldtop 18309 . . . . . 6  |-  J  e. 
Top
594cnfldtopon 18308 . . . . . . . 8  |-  J  e.  (TopOn `  CC )
6059toponunii 16686 . . . . . . 7  |-  CC  =  U. J
6160islp2 16893 . . . . . 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 3197 . 2  |-  S  C_  ( ( limPt `  J
) `  S )
66 eqid 2296 . . . 4  |-  ( Jt  S )  =  ( Jt  S )
6760, 66restperf 16930 . . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   class class class wbr 4039    o. ccom 4709   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    + caddc 8756   RR*cxr 8882    < clt 8883    - cmin 9053    / cdiv 9439   2c2 9811   RR+crp 10370   abscabs 11735   ↾t crest 13341   TopOpenctopn 13342   * Metcxmt 16385   ballcbl 16387  ℂfldccnfld 16393   Topctop 16647   neicnei 16850   limPtclp 16882  Perfcperf 16883
This theorem is referenced by:  reperf  18340  cnperf  18341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-xms 17901  df-ms 17902
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