MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bcth Unicode version

Theorem bcth 19270
Description: Baire's Category Theorem. If a nonempty metric space is complete, it is nonmeager in itself. In other words, no open set in the metric space can be the countable union of rare closed subsets (where rare means having an empty interior), so some subset  M `
 k must have a nonempty interior. Theorem 4.7-2 of [Kreyszig] p. 247. (The terminology "meager" and "nonmeager" is used by Kreyszig to replace Baire's "of the first category" and "of the second category." The latter terms are going out of favor to avoid confusion with category theory.) See bcthlem5 19269 for an overview of the proof. (Contributed by NM, 28-Oct-2007.) (Proof shortened by Mario Carneiro, 6-Jan-2014.)
Hypothesis
Ref Expression
bcth.2  |-  J  =  ( MetOpen `  D )
Assertion
Ref Expression
bcth  |-  ( ( D  e.  ( CMet `  X )  /\  M : NN --> ( Clsd `  J
)  /\  ( ( int `  J ) `  U. ran  M )  =/=  (/) )  ->  E. k  e.  NN  ( ( int `  J ) `  ( M `  k )
)  =/=  (/) )
Distinct variable groups:    D, k    k, J    k, M    k, X

Proof of Theorem bcth
Dummy variables  n  r  x  z  g  m  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bcth.2 . . . . . 6  |-  J  =  ( MetOpen `  D )
2 simpll 731 . . . . . 6  |-  ( ( ( D  e.  (
CMet `  X )  /\  M : NN --> ( Clsd `  J ) )  /\  A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) )  ->  D  e.  ( CMet `  X
) )
3 eleq1 2495 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
x  e.  X  <->  y  e.  X ) )
4 eleq1 2495 . . . . . . . . . . 11  |-  ( r  =  m  ->  (
r  e.  RR+  <->  m  e.  RR+ ) )
53, 4bi2anan9 844 . . . . . . . . . 10  |-  ( ( x  =  y  /\  r  =  m )  ->  ( ( x  e.  X  /\  r  e.  RR+ )  <->  ( y  e.  X  /\  m  e.  RR+ ) ) )
6 simpr 448 . . . . . . . . . . . 12  |-  ( ( x  =  y  /\  r  =  m )  ->  r  =  m )
76breq1d 4214 . . . . . . . . . . 11  |-  ( ( x  =  y  /\  r  =  m )  ->  ( r  <  (
1  /  k )  <-> 
m  <  ( 1  /  k ) ) )
8 oveq12 6081 . . . . . . . . . . . . 13  |-  ( ( x  =  y  /\  r  =  m )  ->  ( x ( ball `  D ) r )  =  ( y (
ball `  D )
m ) )
98fveq2d 5723 . . . . . . . . . . . 12  |-  ( ( x  =  y  /\  r  =  m )  ->  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  =  ( ( cls `  J
) `  ( y
( ball `  D )
m ) ) )
109sseq1d 3367 . . . . . . . . . . 11  |-  ( ( x  =  y  /\  r  =  m )  ->  ( ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  k )
)  <->  ( ( cls `  J ) `  (
y ( ball `  D
) m ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  k )
) ) )
117, 10anbi12d 692 . . . . . . . . . 10  |-  ( ( x  =  y  /\  r  =  m )  ->  ( ( r  < 
( 1  /  k
)  /\  ( ( cls `  J ) `  ( x ( ball `  D ) r ) )  C_  ( (
( ball `  D ) `  z )  \  ( M `  k )
) )  <->  ( m  <  ( 1  /  k
)  /\  ( ( cls `  J ) `  ( y ( ball `  D ) m ) )  C_  ( (
( ball `  D ) `  z )  \  ( M `  k )
) ) ) )
125, 11anbi12d 692 . . . . . . . . 9  |-  ( ( x  =  y  /\  r  =  m )  ->  ( ( ( x  e.  X  /\  r  e.  RR+ )  /\  (
r  <  ( 1  /  k )  /\  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  C_  ( ( ( ball `  D ) `  z
)  \  ( M `  k ) ) ) )  <->  ( ( y  e.  X  /\  m  e.  RR+ )  /\  (
m  <  ( 1  /  k )  /\  ( ( cls `  J
) `  ( y
( ball `  D )
m ) )  C_  ( ( ( ball `  D ) `  z
)  \  ( M `  k ) ) ) ) ) )
1312cbvopabv 4269 . . . . . . . 8  |-  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  k )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  k )
) ) ) }  =  { <. y ,  m >.  |  (
( y  e.  X  /\  m  e.  RR+ )  /\  ( m  <  (
1  /  k )  /\  ( ( cls `  J ) `  (
y ( ball `  D
) m ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  k )
) ) ) }
14 oveq2 6080 . . . . . . . . . . . 12  |-  ( k  =  n  ->  (
1  /  k )  =  ( 1  /  n ) )
1514breq2d 4216 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
m  <  ( 1  /  k )  <->  m  <  ( 1  /  n ) ) )
16 fveq2 5719 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  ( M `  k )  =  ( M `  n ) )
1716difeq2d 3457 . . . . . . . . . . . 12  |-  ( k  =  n  ->  (
( ( ball `  D
) `  z )  \  ( M `  k ) )  =  ( ( ( ball `  D ) `  z
)  \  ( M `  n ) ) )
1817sseq2d 3368 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
( ( cls `  J
) `  ( y
( ball `  D )
m ) )  C_  ( ( ( ball `  D ) `  z
)  \  ( M `  k ) )  <->  ( ( cls `  J ) `  ( y ( ball `  D ) m ) )  C_  ( (
( ball `  D ) `  z )  \  ( M `  n )
) ) )
1915, 18anbi12d 692 . . . . . . . . . 10  |-  ( k  =  n  ->  (
( m  <  (
1  /  k )  /\  ( ( cls `  J ) `  (
y ( ball `  D
) m ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  k )
) )  <->  ( m  <  ( 1  /  n
)  /\  ( ( cls `  J ) `  ( y ( ball `  D ) m ) )  C_  ( (
( ball `  D ) `  z )  \  ( M `  n )
) ) ) )
2019anbi2d 685 . . . . . . . . 9  |-  ( k  =  n  ->  (
( ( y  e.  X  /\  m  e.  RR+ )  /\  (
m  <  ( 1  /  k )  /\  ( ( cls `  J
) `  ( y
( ball `  D )
m ) )  C_  ( ( ( ball `  D ) `  z
)  \  ( M `  k ) ) ) )  <->  ( ( y  e.  X  /\  m  e.  RR+ )  /\  (
m  <  ( 1  /  n )  /\  ( ( cls `  J
) `  ( y
( ball `  D )
m ) )  C_  ( ( ( ball `  D ) `  z
)  \  ( M `  n ) ) ) ) ) )
2120opabbidv 4263 . . . . . . . 8  |-  ( k  =  n  ->  { <. y ,  m >.  |  ( ( y  e.  X  /\  m  e.  RR+ )  /\  ( m  <  (
1  /  k )  /\  ( ( cls `  J ) `  (
y ( ball `  D
) m ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  k )
) ) ) }  =  { <. y ,  m >.  |  (
( y  e.  X  /\  m  e.  RR+ )  /\  ( m  <  (
1  /  n )  /\  ( ( cls `  J ) `  (
y ( ball `  D
) m ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  n )
) ) ) } )
2213, 21syl5eq 2479 . . . . . . 7  |-  ( k  =  n  ->  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  k )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  k )
) ) ) }  =  { <. y ,  m >.  |  (
( y  e.  X  /\  m  e.  RR+ )  /\  ( m  <  (
1  /  n )  /\  ( ( cls `  J ) `  (
y ( ball `  D
) m ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  n )
) ) ) } )
23 fveq2 5719 . . . . . . . . . . . 12  |-  ( z  =  g  ->  (
( ball `  D ) `  z )  =  ( ( ball `  D
) `  g )
)
2423difeq1d 3456 . . . . . . . . . . 11  |-  ( z  =  g  ->  (
( ( ball `  D
) `  z )  \  ( M `  n ) )  =  ( ( ( ball `  D ) `  g
)  \  ( M `  n ) ) )
2524sseq2d 3368 . . . . . . . . . 10  |-  ( z  =  g  ->  (
( ( cls `  J
) `  ( y
( ball `  D )
m ) )  C_  ( ( ( ball `  D ) `  z
)  \  ( M `  n ) )  <->  ( ( cls `  J ) `  ( y ( ball `  D ) m ) )  C_  ( (
( ball `  D ) `  g )  \  ( M `  n )
) ) )
2625anbi2d 685 . . . . . . . . 9  |-  ( z  =  g  ->  (
( m  <  (
1  /  n )  /\  ( ( cls `  J ) `  (
y ( ball `  D
) m ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  n )
) )  <->  ( m  <  ( 1  /  n
)  /\  ( ( cls `  J ) `  ( y ( ball `  D ) m ) )  C_  ( (
( ball `  D ) `  g )  \  ( M `  n )
) ) ) )
2726anbi2d 685 . . . . . . . 8  |-  ( z  =  g  ->  (
( ( y  e.  X  /\  m  e.  RR+ )  /\  (
m  <  ( 1  /  n )  /\  ( ( cls `  J
) `  ( y
( ball `  D )
m ) )  C_  ( ( ( ball `  D ) `  z
)  \  ( M `  n ) ) ) )  <->  ( ( y  e.  X  /\  m  e.  RR+ )  /\  (
m  <  ( 1  /  n )  /\  ( ( cls `  J
) `  ( y
( ball `  D )
m ) )  C_  ( ( ( ball `  D ) `  g
)  \  ( M `  n ) ) ) ) ) )
2827opabbidv 4263 . . . . . . 7  |-  ( z  =  g  ->  { <. y ,  m >.  |  ( ( y  e.  X  /\  m  e.  RR+ )  /\  ( m  <  (
1  /  n )  /\  ( ( cls `  J ) `  (
y ( ball `  D
) m ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  n )
) ) ) }  =  { <. y ,  m >.  |  (
( y  e.  X  /\  m  e.  RR+ )  /\  ( m  <  (
1  /  n )  /\  ( ( cls `  J ) `  (
y ( ball `  D
) m ) ) 
C_  ( ( (
ball `  D ) `  g )  \  ( M `  n )
) ) ) } )
2922, 28cbvmpt2v 6143 . . . . . 6  |-  ( k  e.  NN ,  z  e.  ( X  X.  RR+ )  |->  { <. x ,  r >.  |  ( ( x  e.  X  /\  r  e.  RR+ )  /\  ( r  <  (
1  /  k )  /\  ( ( cls `  J ) `  (
x ( ball `  D
) r ) ) 
C_  ( ( (
ball `  D ) `  z )  \  ( M `  k )
) ) ) } )  =  ( n  e.  NN ,  g  e.  ( X  X.  RR+ )  |->  { <. y ,  m >.  |  (
( y  e.  X  /\  m  e.  RR+ )  /\  ( m  <  (
1  /  n )  /\  ( ( cls `  J ) `  (
y ( ball `  D
) m ) ) 
C_  ( ( (
ball `  D ) `  g )  \  ( M `  n )
) ) ) } )
30 simplr 732 . . . . . 6  |-  ( ( ( D  e.  (
CMet `  X )  /\  M : NN --> ( Clsd `  J ) )  /\  A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) )  ->  M : NN
--> ( Clsd `  J
) )
31 simpr 448 . . . . . . 7  |-  ( ( ( D  e.  (
CMet `  X )  /\  M : NN --> ( Clsd `  J ) )  /\  A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) )  ->  A. k  e.  NN  ( ( int `  J ) `  ( M `  k )
)  =  (/) )
3216fveq2d 5723 . . . . . . . . 9  |-  ( k  =  n  ->  (
( int `  J
) `  ( M `  k ) )  =  ( ( int `  J
) `  ( M `  n ) ) )
3332eqeq1d 2443 . . . . . . . 8  |-  ( k  =  n  ->  (
( ( int `  J
) `  ( M `  k ) )  =  (/) 
<->  ( ( int `  J
) `  ( M `  n ) )  =  (/) ) )
3433cbvralv 2924 . . . . . . 7  |-  ( A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) 
<-> 
A. n  e.  NN  ( ( int `  J
) `  ( M `  n ) )  =  (/) )
3531, 34sylib 189 . . . . . 6  |-  ( ( ( D  e.  (
CMet `  X )  /\  M : NN --> ( Clsd `  J ) )  /\  A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) )  ->  A. n  e.  NN  ( ( int `  J ) `  ( M `  n )
)  =  (/) )
361, 2, 29, 30, 35bcthlem5 19269 . . . . 5  |-  ( ( ( D  e.  (
CMet `  X )  /\  M : NN --> ( Clsd `  J ) )  /\  A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) )  ->  ( ( int `  J ) `
 U. ran  M
)  =  (/) )
3736ex 424 . . . 4  |-  ( ( D  e.  ( CMet `  X )  /\  M : NN --> ( Clsd `  J
) )  ->  ( A. k  e.  NN  ( ( int `  J
) `  ( M `  k ) )  =  (/)  ->  ( ( int `  J ) `  U. ran  M )  =  (/) ) )
3837necon3ad 2634 . . 3  |-  ( ( D  e.  ( CMet `  X )  /\  M : NN --> ( Clsd `  J
) )  ->  (
( ( int `  J
) `  U. ran  M
)  =/=  (/)  ->  -.  A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) ) )
39383impia 1150 . 2  |-  ( ( D  e.  ( CMet `  X )  /\  M : NN --> ( Clsd `  J
)  /\  ( ( int `  J ) `  U. ran  M )  =/=  (/) )  ->  -.  A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) )
40 df-ne 2600 . . . 4  |-  ( ( ( int `  J
) `  ( M `  k ) )  =/=  (/) 
<->  -.  ( ( int `  J ) `  ( M `  k )
)  =  (/) )
4140rexbii 2722 . . 3  |-  ( E. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =/=  (/) 
<->  E. k  e.  NN  -.  ( ( int `  J
) `  ( M `  k ) )  =  (/) )
42 rexnal 2708 . . 3  |-  ( E. k  e.  NN  -.  ( ( int `  J
) `  ( M `  k ) )  =  (/) 
<->  -.  A. k  e.  NN  ( ( int `  J ) `  ( M `  k )
)  =  (/) )
4341, 42bitri 241 . 2  |-  ( E. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =/=  (/) 
<->  -.  A. k  e.  NN  ( ( int `  J ) `  ( M `  k )
)  =  (/) )
4439, 43sylibr 204 1  |-  ( ( D  e.  ( CMet `  X )  /\  M : NN --> ( Clsd `  J
)  /\  ( ( int `  J ) `  U. ran  M )  =/=  (/) )  ->  E. k  e.  NN  ( ( int `  J ) `  ( M `  k )
)  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698    \ cdif 3309    C_ wss 3312   (/)c0 3620   U.cuni 4007   class class class wbr 4204   {copab 4257    X. cxp 4867   ran crn 4870   -->wf 5441   ` cfv 5445  (class class class)co 6072    e. cmpt2 6074   1c1 8980    < clt 9109    / cdiv 9666   NNcn 9989   RR+crp 10601   ballcbl 16676   MetOpencmopn 16679   Clsdccld 17068   intcnt 17069   clsccl 17070   CMetcms 19195
This theorem is referenced by:  bcth2  19271  bcth3  19272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-dc 8315  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-er 6896  df-map 7011  df-pm 7012  df-en 7101  df-dom 7102  df-sdom 7103  df-sup 7437  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-n0 10211  df-z 10272  df-uz 10478  df-q 10564  df-rp 10602  df-xneg 10699  df-xadd 10700  df-xmul 10701  df-ico 10911  df-rest 13638  df-topgen 13655  df-psmet 16682  df-xmet 16683  df-met 16684  df-bl 16685  df-mopn 16686  df-fbas 16687  df-fg 16688  df-top 16951  df-bases 16953  df-topon 16954  df-cld 17071  df-ntr 17072  df-cls 17073  df-nei 17150  df-lm 17281  df-fil 17866  df-fm 17958  df-flim 17959  df-flf 17960  df-cfil 19196  df-cau 19197  df-cmet 19198
  Copyright terms: Public domain W3C validator