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Theorem reperflem 21051
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 21008 . . . . . . 7  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
2 reperflem.3 . . . . . . . 8  |-  S  C_  CC
32sseli 3493 . . . . . . 7  |-  ( u  e.  S  ->  u  e.  CC )
4 recld2.1 . . . . . . . . 9  |-  J  =  ( TopOpen ` fld )
54cnfldtopn 21017 . . . . . . . 8  |-  J  =  ( MetOpen `  ( abs  o. 
-  ) )
65neibl 20732 . . . . . . 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 663 . . . . . 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 2871 . . . . . . . . . . . . . . . 16  |-  ( u  e.  S  ->  A. v  e.  RR  ( u  +  v )  e.  S
)
10 rpre 11215 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  RR+  ->  r  e.  RR )
1110rehalfcld 10774 . . . . . . . . . . . . . . . 16  |-  ( r  e.  RR+  ->  ( r  /  2 )  e.  RR )
12 oveq2 6283 . . . . . . . . . . . . . . . . . 18  |-  ( v  =  ( r  / 
2 )  ->  (
u  +  v )  =  ( u  +  ( r  /  2
) ) )
1312eleq1d 2529 . . . . . . . . . . . . . . . . 17  |-  ( v  =  ( r  / 
2 )  ->  (
( u  +  v )  e.  S  <->  ( u  +  ( r  / 
2 ) )  e.  S ) )
1413rspccva 3206 . . . . . . . . . . . . . . . 16  |-  ( ( A. v  e.  RR  ( u  +  v
)  e.  S  /\  ( r  /  2
)  e.  RR )  ->  ( u  +  ( r  /  2
) )  e.  S
)
159, 11, 14syl2an 477 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( u  +  ( r  /  2 ) )  e.  S )
162, 15sseldi 3495 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( u  +  ( r  /  2 ) )  e.  CC )
173adantr 465 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  ->  u  e.  CC )
18 eqid 2460 . . . . . . . . . . . . . . 15  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1918cnmetdval 21006 . . . . . . . . . . . . . 14  |-  ( ( ( u  +  ( r  /  2 ) )  e.  CC  /\  u  e.  CC )  ->  ( ( u  +  ( r  /  2
) ) ( abs 
o.  -  ) u
)  =  ( abs `  ( ( u  +  ( r  /  2
) )  -  u
) ) )
2016, 17, 19syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) ) ( abs 
o.  -  ) u
)  =  ( abs `  ( ( u  +  ( r  /  2
) )  -  u
) ) )
21 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
r  e.  RR+ )
2221rphalfcld 11257 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  e.  RR+ )
2322rpcnd 11247 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  e.  CC )
2417, 23pncan2d 9921 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) )  -  u
)  =  ( r  /  2 ) )
2524fveq2d 5861 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( abs `  (
( u  +  ( r  /  2 ) )  -  u ) )  =  ( abs `  ( r  /  2
) ) )
2622rpred 11245 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  e.  RR )
2722rpge0d 11249 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
0  <_  ( r  /  2 ) )
2826, 27absidd 13203 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( abs `  (
r  /  2 ) )  =  ( r  /  2 ) )
2920, 25, 283eqtrd 2505 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) ) ( abs 
o.  -  ) u
)  =  ( r  /  2 ) )
30 rphalflt 11235 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  ( r  /  2 )  < 
r )
3130adantl 466 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  <  r )
3229, 31eqbrtrd 4460 . . . . . . . . . . 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 11216 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  r  e. 
RR* )
3534adantl 466 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
r  e.  RR* )
36 elbl3 20623 . . . . . . . . . . . 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 1224 . . . . . . . . . . 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 11250 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  =/=  0 )
4024, 39eqnetrd 2753 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) )  -  u
)  =/=  0 )
4116, 17, 40subne0ad 9930 . . . . . . . . . . 11  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( u  +  ( r  /  2 ) )  =/=  u )
42 eldifsn 4145 . . . . . . . . . . 11  |-  ( ( u  +  ( r  /  2 ) )  e.  ( S  \  { u } )  <-> 
( ( u  +  ( r  /  2
) )  e.  S  /\  ( u  +  ( r  /  2 ) )  =/=  u ) )
4315, 41, 42sylanbrc 664 . . . . . . . . . 10  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( u  +  ( r  /  2 ) )  e.  ( S 
\  { u }
) )
44 inelcm 3874 . . . . . . . . . 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 661 . . . . . . . . 9  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u (
ball `  ( abs  o. 
-  ) ) r )  i^i  ( S 
\  { u }
) )  =/=  (/) )
46 ssrin 3716 . . . . . . . . . 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 3811 . . . . . . . . . . 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 434 . . . . . . . . . 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 16 . . . . . . . . 9  |-  ( ( u ( ball `  ( abs  o.  -  ) ) r )  C_  n  ->  ( ( ( u ( ball `  ( abs  o.  -  ) ) r )  i^i  ( S  \  { u }
) )  =/=  (/)  ->  (
n  i^i  ( S  \  { u } ) )  =/=  (/) ) )
5045, 49syl5com 30 . . . . . . . 8  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u (
ball `  ( abs  o. 
-  ) ) r )  C_  n  ->  ( n  i^i  ( S 
\  { u }
) )  =/=  (/) ) )
5150rexlimdva 2948 . . . . . . 7  |-  ( u  e.  S  ->  ( E. r  e.  RR+  (
u ( ball `  ( abs  o.  -  ) ) r )  C_  n  ->  ( n  i^i  ( S  \  { u }
) )  =/=  (/) ) )
5251adantld 467 . . . . . 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 2869 . . . 4  |-  ( u  e.  S  ->  A. n  e.  ( ( nei `  J
) `  { u } ) ( n  i^i  ( S  \  { u } ) )  =/=  (/) )
554cnfldtop 21019 . . . . . 6  |-  J  e. 
Top
564cnfldtopon 21018 . . . . . . . 8  |-  J  e.  (TopOn `  CC )
5756toponunii 19193 . . . . . . 7  |-  CC  =  U. J
5857islp2 19405 . . . . . 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 1309 . . . . 5  |-  ( u  e.  CC  ->  (
u  e.  ( (
limPt `  J ) `  S )  <->  A. n  e.  ( ( nei `  J
) `  { u } ) ( n  i^i  ( S  \  { u } ) )  =/=  (/) ) )
603, 59syl 16 . . . 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 3501 . 2  |-  S  C_  ( ( limPt `  J
) `  S )
63 eqid 2460 . . . 4  |-  ( Jt  S )  =  ( Jt  S )
6457, 63restperf 19444 . . 3  |-  ( ( J  e.  Top  /\  S  C_  CC )  -> 
( ( Jt  S )  e. Perf 
<->  S  C_  ( ( limPt `  J ) `  S ) ) )
6555, 2, 64mp2an 672 . 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 369    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   E.wrex 2808    \ cdif 3466    i^i cin 3468    C_ wss 3469   (/)c0 3778   {csn 4020   class class class wbr 4440    o. ccom 4996   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481    + caddc 9484   RR*cxr 9616    < clt 9617    - cmin 9794    / cdiv 10195   2c2 10574   RR+crp 11209   abscabs 13017   ↾t crest 14665   TopOpenctopn 14666   *Metcxmt 18167   ballcbl 18169  ℂfldccnfld 18184   Topctop 19154   neicnei 19357   limPtclp 19394  Perfcperf 19395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fi 7860  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-fz 11662  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-plusg 14557  df-mulr 14558  df-starv 14559  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-rest 14667  df-topn 14668  df-topgen 14688  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-lp 19396  df-perf 19397  df-xms 20551  df-ms 20552
This theorem is referenced by:  reperf  21052  cnperf  21053
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