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Theorem bcth 18849
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 18848 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 730 . . . . . 6  |-  ( ( ( D  e.  (
CMet `  X )  /\  M : NN --> ( Clsd `  J ) )  /\  A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) )  ->  D  e.  ( CMet `  X
) )
3 eleq1 2418 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
x  e.  X  <->  y  e.  X ) )
4 eleq1 2418 . . . . . . . . . . 11  |-  ( r  =  m  ->  (
r  e.  RR+  <->  m  e.  RR+ ) )
53, 4bi2anan9 843 . . . . . . . . . 10  |-  ( ( x  =  y  /\  r  =  m )  ->  ( ( x  e.  X  /\  r  e.  RR+ )  <->  ( y  e.  X  /\  m  e.  RR+ ) ) )
6 simpr 447 . . . . . . . . . . . 12  |-  ( ( x  =  y  /\  r  =  m )  ->  r  =  m )
76breq1d 4112 . . . . . . . . . . 11  |-  ( ( x  =  y  /\  r  =  m )  ->  ( r  <  (
1  /  k )  <-> 
m  <  ( 1  /  k ) ) )
8 oveq12 5951 . . . . . . . . . . . . 13  |-  ( ( x  =  y  /\  r  =  m )  ->  ( x ( ball `  D ) r )  =  ( y (
ball `  D )
m ) )
98fveq2d 5609 . . . . . . . . . . . 12  |-  ( ( x  =  y  /\  r  =  m )  ->  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  =  ( ( cls `  J
) `  ( y
( ball `  D )
m ) ) )
109sseq1d 3281 . . . . . . . . . . 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 691 . . . . . . . . . 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 691 . . . . . . . . 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 4167 . . . . . . . 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 5950 . . . . . . . . . . . 12  |-  ( k  =  n  ->  (
1  /  k )  =  ( 1  /  n ) )
1514breq2d 4114 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
m  <  ( 1  /  k )  <->  m  <  ( 1  /  n ) ) )
16 fveq2 5605 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  ( M `  k )  =  ( M `  n ) )
1716difeq2d 3370 . . . . . . . . . . . 12  |-  ( k  =  n  ->  (
( ( ball `  D
) `  z )  \  ( M `  k ) )  =  ( ( ( ball `  D ) `  z
)  \  ( M `  n ) ) )
1817sseq2d 3282 . . . . . . . . . . 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 691 . . . . . . . . . 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 684 . . . . . . . . 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 4161 . . . . . . . 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 2402 . . . . . . 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 5605 . . . . . . . . . . . 12  |-  ( z  =  g  ->  (
( ball `  D ) `  z )  =  ( ( ball `  D
) `  g )
)
2423difeq1d 3369 . . . . . . . . . . 11  |-  ( z  =  g  ->  (
( ( ball `  D
) `  z )  \  ( M `  n ) )  =  ( ( ( ball `  D ) `  g
)  \  ( M `  n ) ) )
2524sseq2d 3282 . . . . . . . . . 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 684 . . . . . . . . 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 684 . . . . . . . 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 4161 . . . . . . 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 6010 . . . . . 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 731 . . . . . 6  |-  ( ( ( D  e.  (
CMet `  X )  /\  M : NN --> ( Clsd `  J ) )  /\  A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) )  ->  M : NN
--> ( Clsd `  J
) )
31 simpr 447 . . . . . . 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 5609 . . . . . . . . 9  |-  ( k  =  n  ->  (
( int `  J
) `  ( M `  k ) )  =  ( ( int `  J
) `  ( M `  n ) ) )
3332eqeq1d 2366 . . . . . . . 8  |-  ( k  =  n  ->  (
( ( int `  J
) `  ( M `  k ) )  =  (/) 
<->  ( ( int `  J
) `  ( M `  n ) )  =  (/) ) )
3433cbvralv 2840 . . . . . . 7  |-  ( A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) 
<-> 
A. n  e.  NN  ( ( int `  J
) `  ( M `  n ) )  =  (/) )
3531, 34sylib 188 . . . . . 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 18848 . . . . 5  |-  ( ( ( D  e.  (
CMet `  X )  /\  M : NN --> ( Clsd `  J ) )  /\  A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) )  ->  ( ( int `  J ) `
 U. ran  M
)  =  (/) )
3736ex 423 . . . 4  |-  ( ( D  e.  ( CMet `  X )  /\  M : NN --> ( Clsd `  J
) )  ->  ( A. k  e.  NN  ( ( int `  J
) `  ( M `  k ) )  =  (/)  ->  ( ( int `  J ) `  U. ran  M )  =  (/) ) )
3837necon3ad 2557 . . 3  |-  ( ( D  e.  ( CMet `  X )  /\  M : NN --> ( Clsd `  J
) )  ->  (
( ( int `  J
) `  U. ran  M
)  =/=  (/)  ->  -.  A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) ) )
39383impia 1148 . 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 2523 . . . 4  |-  ( ( ( int `  J
) `  ( M `  k ) )  =/=  (/) 
<->  -.  ( ( int `  J ) `  ( M `  k )
)  =  (/) )
4140rexbii 2644 . . 3  |-  ( E. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =/=  (/) 
<->  E. k  e.  NN  -.  ( ( int `  J
) `  ( M `  k ) )  =  (/) )
42 rexnal 2630 . . 3  |-  ( E. k  e.  NN  -.  ( ( int `  J
) `  ( M `  k ) )  =  (/) 
<->  -.  A. k  e.  NN  ( ( int `  J ) `  ( M `  k )
)  =  (/) )
4341, 42bitri 240 . 2  |-  ( E. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =/=  (/) 
<->  -.  A. k  e.  NN  ( ( int `  J ) `  ( M `  k )
)  =  (/) )
4439, 43sylibr 203 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   E.wrex 2620    \ cdif 3225    C_ wss 3228   (/)c0 3531   U.cuni 3906   class class class wbr 4102   {copab 4155    X. cxp 4766   ran crn 4769   -->wf 5330   ` cfv 5334  (class class class)co 5942    e. cmpt2 5944   1c1 8825    < clt 8954    / cdiv 9510   NNcn 9833   RR+crp 10443   ballcbl 16464   MetOpencmopn 16467   Clsdccld 16853   intcnt 16854   clsccl 16855   CMetcms 18778
This theorem is referenced by:  bcth2  18850  bcth3  18851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-dc 8159  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-er 6744  df-map 6859  df-pm 6860  df-en 6949  df-dom 6950  df-sdom 6951  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-n0 10055  df-z 10114  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-ico 10751  df-rest 13420  df-topgen 13437  df-xmet 16469  df-met 16470  df-bl 16471  df-mopn 16472  df-fbas 16473  df-fg 16474  df-top 16736  df-bases 16738  df-topon 16739  df-cld 16856  df-ntr 16857  df-cls 16858  df-nei 16935  df-lm 17059  df-fil 17637  df-fm 17729  df-flim 17730  df-flf 17731  df-cfil 18779  df-cau 18780  df-cmet 18781
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