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Theorem bcth 20971
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 a closure with 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 20970 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 753 . . . . . 6  |-  ( ( ( D  e.  (
CMet `  X )  /\  M : NN --> ( Clsd `  J ) )  /\  A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) )  ->  D  e.  ( CMet `  X
) )
3 eleq1 2526 . . . . . . . . . . 11  |-  ( x  =  y  ->  (
x  e.  X  <->  y  e.  X ) )
4 eleq1 2526 . . . . . . . . . . 11  |-  ( r  =  m  ->  (
r  e.  RR+  <->  m  e.  RR+ ) )
53, 4bi2anan9 868 . . . . . . . . . 10  |-  ( ( x  =  y  /\  r  =  m )  ->  ( ( x  e.  X  /\  r  e.  RR+ )  <->  ( y  e.  X  /\  m  e.  RR+ ) ) )
6 simpr 461 . . . . . . . . . . . 12  |-  ( ( x  =  y  /\  r  =  m )  ->  r  =  m )
76breq1d 4409 . . . . . . . . . . 11  |-  ( ( x  =  y  /\  r  =  m )  ->  ( r  <  (
1  /  k )  <-> 
m  <  ( 1  /  k ) ) )
8 oveq12 6208 . . . . . . . . . . . . 13  |-  ( ( x  =  y  /\  r  =  m )  ->  ( x ( ball `  D ) r )  =  ( y (
ball `  D )
m ) )
98fveq2d 5802 . . . . . . . . . . . 12  |-  ( ( x  =  y  /\  r  =  m )  ->  ( ( cls `  J
) `  ( x
( ball `  D )
r ) )  =  ( ( cls `  J
) `  ( y
( ball `  D )
m ) ) )
109sseq1d 3490 . . . . . . . . . . 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 710 . . . . . . . . . 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 710 . . . . . . . . 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 4468 . . . . . . . 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 6207 . . . . . . . . . . . 12  |-  ( k  =  n  ->  (
1  /  k )  =  ( 1  /  n ) )
1514breq2d 4411 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
m  <  ( 1  /  k )  <->  m  <  ( 1  /  n ) ) )
16 fveq2 5798 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  ( M `  k )  =  ( M `  n ) )
1716difeq2d 3581 . . . . . . . . . . . 12  |-  ( k  =  n  ->  (
( ( ball `  D
) `  z )  \  ( M `  k ) )  =  ( ( ( ball `  D ) `  z
)  \  ( M `  n ) ) )
1817sseq2d 3491 . . . . . . . . . . 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 710 . . . . . . . . . 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 703 . . . . . . . . 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 4462 . . . . . . . 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 2507 . . . . . . 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 5798 . . . . . . . . . . . 12  |-  ( z  =  g  ->  (
( ball `  D ) `  z )  =  ( ( ball `  D
) `  g )
)
2423difeq1d 3580 . . . . . . . . . . 11  |-  ( z  =  g  ->  (
( ( ball `  D
) `  z )  \  ( M `  n ) )  =  ( ( ( ball `  D ) `  g
)  \  ( M `  n ) ) )
2524sseq2d 3491 . . . . . . . . . 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 703 . . . . . . . . 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 703 . . . . . . . 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 4462 . . . . . . 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 6274 . . . . . 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 754 . . . . . 6  |-  ( ( ( D  e.  (
CMet `  X )  /\  M : NN --> ( Clsd `  J ) )  /\  A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) )  ->  M : NN
--> ( Clsd `  J
) )
31 simpr 461 . . . . . . 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 5802 . . . . . . . . 9  |-  ( k  =  n  ->  (
( int `  J
) `  ( M `  k ) )  =  ( ( int `  J
) `  ( M `  n ) ) )
3332eqeq1d 2456 . . . . . . . 8  |-  ( k  =  n  ->  (
( ( int `  J
) `  ( M `  k ) )  =  (/) 
<->  ( ( int `  J
) `  ( M `  n ) )  =  (/) ) )
3433cbvralv 3051 . . . . . . 7  |-  ( A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) 
<-> 
A. n  e.  NN  ( ( int `  J
) `  ( M `  n ) )  =  (/) )
3531, 34sylib 196 . . . . . 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 20970 . . . . 5  |-  ( ( ( D  e.  (
CMet `  X )  /\  M : NN --> ( Clsd `  J ) )  /\  A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) )  ->  ( ( int `  J ) `
 U. ran  M
)  =  (/) )
3736ex 434 . . . 4  |-  ( ( D  e.  ( CMet `  X )  /\  M : NN --> ( Clsd `  J
) )  ->  ( A. k  e.  NN  ( ( int `  J
) `  ( M `  k ) )  =  (/)  ->  ( ( int `  J ) `  U. ran  M )  =  (/) ) )
3837necon3ad 2661 . . 3  |-  ( ( D  e.  ( CMet `  X )  /\  M : NN --> ( Clsd `  J
) )  ->  (
( ( int `  J
) `  U. ran  M
)  =/=  (/)  ->  -.  A. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =  (/) ) )
39383impia 1185 . 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 2649 . . . 4  |-  ( ( ( int `  J
) `  ( M `  k ) )  =/=  (/) 
<->  -.  ( ( int `  J ) `  ( M `  k )
)  =  (/) )
4140rexbii 2858 . . 3  |-  ( E. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =/=  (/) 
<->  E. k  e.  NN  -.  ( ( int `  J
) `  ( M `  k ) )  =  (/) )
42 rexnal 2853 . . 3  |-  ( E. k  e.  NN  -.  ( ( int `  J
) `  ( M `  k ) )  =  (/) 
<->  -.  A. k  e.  NN  ( ( int `  J ) `  ( M `  k )
)  =  (/) )
4341, 42bitri 249 . 2  |-  ( E. k  e.  NN  (
( int `  J
) `  ( M `  k ) )  =/=  (/) 
<->  -.  A. k  e.  NN  ( ( int `  J ) `  ( M `  k )
)  =  (/) )
4439, 43sylibr 212 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   A.wral 2798   E.wrex 2799    \ cdif 3432    C_ wss 3435   (/)c0 3744   U.cuni 4198   class class class wbr 4399   {copab 4456    X. cxp 4945   ran crn 4948   -->wf 5521   ` cfv 5525  (class class class)co 6199    |-> cmpt2 6201   1c1 9393    < clt 9528    / cdiv 10103   NNcn 10432   RR+crp 11101   ballcbl 17927   MetOpencmopn 17930   Clsdccld 18751   intcnt 18752   clsccl 18753   CMetcms 20896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-dc 8725  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-er 7210  df-map 7325  df-pm 7326  df-en 7420  df-dom 7421  df-sdom 7422  df-sup 7801  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-n0 10690  df-z 10757  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ico 11416  df-rest 14479  df-topgen 14500  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-fbas 17938  df-fg 17939  df-top 18634  df-bases 18636  df-topon 18637  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-lm 18964  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-cfil 20897  df-cau 20898  df-cmet 20899
This theorem is referenced by:  bcth2  20972  bcth3  20973
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