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Theorem bcth 19235
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 19234 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 2464 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
x  e.  X  <->  y  e.  X ) )
4 eleq1 2464 . . . . . . . . . . 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 4182 . . . . . . . . . . 11  |-  ( ( x  =  y  /\  r  =  m )  ->  ( r  <  (
1  /  k )  <-> 
m  <  ( 1  /  k ) ) )
8 oveq12 6049 . . . . . . . . . . . . 13  |-  ( ( x  =  y  /\  r  =  m )  ->  ( x ( ball `  D ) r )  =  ( y (
ball `  D )
m ) )
98fveq2d 5691 . . . . . . . . . . . 12  |-  ( ( x  =  y  /\  r  =  m )  ->  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  =  ( ( cls `  J
) `  ( y
( ball `  D )
m ) ) )
109sseq1d 3335 . . . . . . . . . . 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 4237 . . . . . . . 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 6048 . . . . . . . . . . . 12  |-  ( k  =  n  ->  (
1  /  k )  =  ( 1  /  n ) )
1514breq2d 4184 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
m  <  ( 1  /  k )  <->  m  <  ( 1  /  n ) ) )
16 fveq2 5687 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  ( M `  k )  =  ( M `  n ) )
1716difeq2d 3425 . . . . . . . . . . . 12  |-  ( k  =  n  ->  (
( ( ball `  D
) `  z )  \  ( M `  k ) )  =  ( ( ( ball `  D ) `  z
)  \  ( M `  n ) ) )
1817sseq2d 3336 . . . . . . . . . . 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 4231 . . . . . . . 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 2448 . . . . . . 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 5687 . . . . . . . . . . . 12  |-  ( z  =  g  ->  (
( ball `  D ) `  z )  =  ( ( ball `  D
) `  g )
)
2423difeq1d 3424 . . . . . . . . . . 11  |-  ( z  =  g  ->  (
( ( ball `  D
) `  z )  \  ( M `  n ) )  =  ( ( ( ball `  D ) `  g
)  \  ( M `  n ) ) )
2524sseq2d 3336 . . . . . . . . . 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 4231 . . . . . . 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 6111 . . . . . 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 5691 . . . . . . . . 9  |-  ( k  =  n  ->  (
( int `  J
) `  ( M `  k ) )  =  ( ( int `  J
) `  ( M `  n ) ) )
3332eqeq1d 2412 . . . . . . . 8  |-  ( k  =  n  ->  (
( ( int `  J
) `  ( M `  k ) )  =  (/) 
<->  ( ( int `  J
) `  ( M `  n ) )  =  (/) ) )
3433cbvralv 2892 . . . . . . 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 19234 . . . . 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 2603 . . 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 2569 . . . 4  |-  ( ( ( int `  J
) `  ( M `  k ) )  =/=  (/) 
<->  -.  ( ( int `  J ) `  ( M `  k )
)  =  (/) )
4140rexbii 2691 . . 3  |-  ( E. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =/=  (/) 
<->  E. k  e.  NN  -.  ( ( int `  J
) `  ( M `  k ) )  =  (/) )
42 rexnal 2677 . . 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 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667    \ cdif 3277    C_ wss 3280   (/)c0 3588   U.cuni 3975   class class class wbr 4172   {copab 4225    X. cxp 4835   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1c1 8947    < clt 9076    / cdiv 9633   NNcn 9956   RR+crp 10568   ballcbl 16643   MetOpencmopn 16646   Clsdccld 17035   intcnt 17036   clsccl 17037   CMetcms 19160
This theorem is referenced by:  bcth2  19236  bcth3  19237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-dc 8282  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ico 10878  df-rest 13605  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-top 16918  df-bases 16920  df-topon 16921  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lm 17247  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-cfil 19161  df-cau 19162  df-cmet 19163
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