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Theorem reperflem 20417
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 20374 . . . . . . 7  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
2 reperflem.3 . . . . . . . 8  |-  S  C_  CC
32sseli 3373 . . . . . . 7  |-  ( u  e.  S  ->  u  e.  CC )
4 recld2.1 . . . . . . . . 9  |-  J  =  ( TopOpen ` fld )
54cnfldtopn 20383 . . . . . . . 8  |-  J  =  ( MetOpen `  ( abs  o. 
-  ) )
65neibl 20098 . . . . . . 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 2820 . . . . . . . . . . . . . . . 16  |-  ( u  e.  S  ->  A. v  e.  RR  ( u  +  v )  e.  S
)
10 rpre 11018 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  RR+  ->  r  e.  RR )
1110rehalfcld 10592 . . . . . . . . . . . . . . . 16  |-  ( r  e.  RR+  ->  ( r  /  2 )  e.  RR )
12 oveq2 6120 . . . . . . . . . . . . . . . . . 18  |-  ( v  =  ( r  / 
2 )  ->  (
u  +  v )  =  ( u  +  ( r  /  2
) ) )
1312eleq1d 2509 . . . . . . . . . . . . . . . . 17  |-  ( v  =  ( r  / 
2 )  ->  (
( u  +  v )  e.  S  <->  ( u  +  ( r  / 
2 ) )  e.  S ) )
1413rspccva 3093 . . . . . . . . . . . . . . . 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 3375 . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . 15  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1918cnmetdval 20372 . . . . . . . . . . . . . 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 11060 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  e.  RR+ )
2322rpcnd 11050 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  e.  CC )
2417, 23pncan2d 9742 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) )  -  u
)  =  ( r  /  2 ) )
2524fveq2d 5716 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( abs `  (
( u  +  ( r  /  2 ) )  -  u ) )  =  ( abs `  ( r  /  2
) ) )
2622rpred 11048 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  e.  RR )
2722rpge0d 11052 . . . . . . . . . . . . . 14  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
0  <_  ( r  /  2 ) )
2826, 27absidd 12930 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( abs `  (
r  /  2 ) )  =  ( r  /  2 ) )
2920, 25, 283eqtrd 2479 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) ) ( abs 
o.  -  ) u
)  =  ( r  /  2 ) )
30 rphalflt 11038 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  ( r  /  2 )  < 
r )
3130adantl 466 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  <  r )
3229, 31eqbrtrd 4333 . . . . . . . . . . 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 11019 . . . . . . . . . . . . 13  |-  ( r  e.  RR+  ->  r  e. 
RR* )
3534adantl 466 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
r  e.  RR* )
36 elbl3 19989 . . . . . . . . . . . 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 1219 . . . . . . . . . . 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 11053 . . . . . . . . . . . . 13  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( r  /  2
)  =/=  0 )
4024, 39eqnetrd 2654 . . . . . . . . . . . 12  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( ( u  +  ( r  /  2
) )  -  u
)  =/=  0 )
4116, 17, 40subne0ad 9751 . . . . . . . . . . 11  |-  ( ( u  e.  S  /\  r  e.  RR+ )  -> 
( u  +  ( r  /  2 ) )  =/=  u )
42 eldifsn 4021 . . . . . . . . . . 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 3754 . . . . . . . . . 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 3596 . . . . . . . . . 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 3691 . . . . . . . . . . 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 2862 . . . . . . 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 2819 . . . 4  |-  ( u  e.  S  ->  A. n  e.  ( ( nei `  J
) `  { u } ) ( n  i^i  ( S  \  { u } ) )  =/=  (/) )
554cnfldtop 20385 . . . . . 6  |-  J  e. 
Top
564cnfldtopon 20384 . . . . . . . 8  |-  J  e.  (TopOn `  CC )
5756toponunii 18559 . . . . . . 7  |-  CC  =  U. J
5857islp2 18771 . . . . . 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 1304 . . . . 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 3381 . 2  |-  S  C_  ( ( limPt `  J
) `  S )
63 eqid 2443 . . . 4  |-  ( Jt  S )  =  ( Jt  S )
6457, 63restperf 18810 . . 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 1369    e. wcel 1756    =/= wne 2620   A.wral 2736   E.wrex 2737    \ cdif 3346    i^i cin 3348    C_ wss 3349   (/)c0 3658   {csn 3898   class class class wbr 4313    o. ccom 4865   ` cfv 5439  (class class class)co 6112   CCcc 9301   RRcr 9302   0cc0 9303    + caddc 9306   RR*cxr 9438    < clt 9439    - cmin 9616    / cdiv 10014   2c2 10392   RR+crp 11012   abscabs 12744   ↾t crest 14380   TopOpenctopn 14381   *Metcxmt 17823   ballcbl 17825  ℂfldccnfld 17840   Topctop 18520   neicnei 18723   limPtclp 18760  Perfcperf 18761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fi 7682  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-fz 11459  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-plusg 14272  df-mulr 14273  df-starv 14274  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-rest 14382  df-topn 14383  df-topgen 14403  df-psmet 17831  df-xmet 17832  df-met 17833  df-bl 17834  df-mopn 17835  df-cnfld 17841  df-top 18525  df-bases 18527  df-topon 18528  df-topsp 18529  df-cld 18645  df-ntr 18646  df-cls 18647  df-nei 18724  df-lp 18762  df-perf 18763  df-xms 19917  df-ms 19918
This theorem is referenced by:  reperf  20418  cnperf  20419
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