MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bcthlem3 Structured version   Unicode version

Theorem bcthlem3 19281
Description: Lemma for bcth 19284. The limit point of the centers in the sequence is in the intersection of every ball in the sequence. (Contributed by Mario Carneiro, 7-Jan-2014.)
Hypotheses
Ref Expression
bcth.2  |-  J  =  ( MetOpen `  D )
bcthlem.4  |-  ( ph  ->  D  e.  ( CMet `  X ) )
bcthlem.5  |-  F  =  ( 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 )
) ) ) } )
bcthlem.6  |-  ( ph  ->  M : NN --> ( Clsd `  J ) )
bcthlem.7  |-  ( ph  ->  R  e.  RR+ )
bcthlem.8  |-  ( ph  ->  C  e.  X )
bcthlem.9  |-  ( ph  ->  g : NN --> ( X  X.  RR+ ) )
bcthlem.10  |-  ( ph  ->  ( g `  1
)  =  <. C ,  R >. )
bcthlem.11  |-  ( ph  ->  A. k  e.  NN  ( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) ) )
Assertion
Ref Expression
bcthlem3  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  A  e.  NN )  ->  x  e.  ( (
ball `  D ) `  ( g `  A
) ) )
Distinct variable groups:    k, r, x, z, A    C, r, x    g, k, r, x, z, D    g, F, k, r, x, z    g, J, k, r, x, z   
g, M, k, r, x, z    ph, k,
r, x, z    x, R    g, X, k, r, x, z
Allowed substitution hints:    ph( g)    A( g)    C( z, g, k)    R( z, g, k, r)

Proof of Theorem bcthlem3
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 bcthlem.11 . . . . . . 7  |-  ( ph  ->  A. k  e.  NN  ( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) ) )
2 oveq1 6090 . . . . . . . . . 10  |-  ( k  =  A  ->  (
k  +  1 )  =  ( A  + 
1 ) )
32fveq2d 5734 . . . . . . . . 9  |-  ( k  =  A  ->  (
g `  ( k  +  1 ) )  =  ( g `  ( A  +  1
) ) )
4 id 21 . . . . . . . . . 10  |-  ( k  =  A  ->  k  =  A )
5 fveq2 5730 . . . . . . . . . 10  |-  ( k  =  A  ->  (
g `  k )  =  ( g `  A ) )
64, 5oveq12d 6101 . . . . . . . . 9  |-  ( k  =  A  ->  (
k F ( g `
 k ) )  =  ( A F ( g `  A
) ) )
73, 6eleq12d 2506 . . . . . . . 8  |-  ( k  =  A  ->  (
( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) )  <->  ( g `  ( A  +  1 ) )  e.  ( A F ( g `
 A ) ) ) )
87rspccva 3053 . . . . . . 7  |-  ( ( A. k  e.  NN  ( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) )  /\  A  e.  NN )  ->  ( g `  ( A  +  1 ) )  e.  ( A F ( g `  A ) ) )
91, 8sylan 459 . . . . . 6  |-  ( (
ph  /\  A  e.  NN )  ->  ( g `
 ( A  + 
1 ) )  e.  ( A F ( g `  A ) ) )
10 bcthlem.9 . . . . . . . 8  |-  ( ph  ->  g : NN --> ( X  X.  RR+ ) )
1110ffvelrnda 5872 . . . . . . 7  |-  ( (
ph  /\  A  e.  NN )  ->  ( g `
 A )  e.  ( X  X.  RR+ ) )
12 bcth.2 . . . . . . . . 9  |-  J  =  ( MetOpen `  D )
13 bcthlem.4 . . . . . . . . 9  |-  ( ph  ->  D  e.  ( CMet `  X ) )
14 bcthlem.5 . . . . . . . . 9  |-  F  =  ( 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 )
) ) ) } )
1512, 13, 14bcthlem1 19279 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  NN  /\  ( g `
 A )  e.  ( X  X.  RR+ ) ) )  -> 
( ( g `  ( A  +  1
) )  e.  ( A F ( g `
 A ) )  <-> 
( ( g `  ( A  +  1
) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  (
g `  ( A  +  1 ) ) )  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( ( ball `  D ) `  ( g `  ( A  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  A )
)  \  ( M `  A ) ) ) ) )
1615expr 600 . . . . . . 7  |-  ( (
ph  /\  A  e.  NN )  ->  ( ( g `  A )  e.  ( X  X.  RR+ )  ->  ( (
g `  ( A  +  1 ) )  e.  ( A F ( g `  A
) )  <->  ( (
g `  ( A  +  1 ) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  ( g `  ( A  +  1 ) ) )  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
( ball `  D ) `  ( g `  ( A  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  A )
)  \  ( M `  A ) ) ) ) ) )
1711, 16mpd 15 . . . . . 6  |-  ( (
ph  /\  A  e.  NN )  ->  ( ( g `  ( A  +  1 ) )  e.  ( A F ( g `  A
) )  <->  ( (
g `  ( A  +  1 ) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  ( g `  ( A  +  1 ) ) )  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
( ball `  D ) `  ( g `  ( A  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  A )
)  \  ( M `  A ) ) ) ) )
189, 17mpbid 203 . . . . 5  |-  ( (
ph  /\  A  e.  NN )  ->  ( ( g `  ( A  +  1 ) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  ( g `  ( A  +  1 ) ) )  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
( ball `  D ) `  ( g `  ( A  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  A )
)  \  ( M `  A ) ) ) )
1918simp3d 972 . . . 4  |-  ( (
ph  /\  A  e.  NN )  ->  ( ( cls `  J ) `
 ( ( ball `  D ) `  (
g `  ( A  +  1 ) ) ) )  C_  (
( ( ball `  D
) `  ( g `  A ) )  \ 
( M `  A
) ) )
2019difss2d 3479 . . 3  |-  ( (
ph  /\  A  e.  NN )  ->  ( ( cls `  J ) `
 ( ( ball `  D ) `  (
g `  ( A  +  1 ) ) ) )  C_  (
( ball `  D ) `  ( g `  A
) ) )
21203adant2 977 . 2  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  A  e.  NN )  ->  ( ( cls `  J
) `  ( ( ball `  D ) `  ( g `  ( A  +  1 ) ) ) )  C_  ( ( ball `  D
) `  ( g `  A ) ) )
22 peano2nn 10014 . . 3  |-  ( A  e.  NN  ->  ( A  +  1 )  e.  NN )
23 cmetmet 19241 . . . . 5  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
24 metxmet 18366 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
2513, 23, 243syl 19 . . . 4  |-  ( ph  ->  D  e.  ( * Met `  X ) )
26 bcthlem.6 . . . . 5  |-  ( ph  ->  M : NN --> ( Clsd `  J ) )
27 bcthlem.7 . . . . 5  |-  ( ph  ->  R  e.  RR+ )
28 bcthlem.8 . . . . 5  |-  ( ph  ->  C  e.  X )
29 bcthlem.10 . . . . 5  |-  ( ph  ->  ( g `  1
)  =  <. C ,  R >. )
3012, 13, 14, 26, 27, 28, 10, 29, 1bcthlem2 19280 . . . 4  |-  ( ph  ->  A. n  e.  NN  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) ) )
3125, 10, 30, 12caublcls 19263 . . 3  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  ( A  +  1
)  e.  NN )  ->  x  e.  ( ( cls `  J
) `  ( ( ball `  D ) `  ( g `  ( A  +  1 ) ) ) ) )
3222, 31syl3an3 1220 . 2  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  A  e.  NN )  ->  x  e.  ( ( cls `  J ) `
 ( ( ball `  D ) `  (
g `  ( A  +  1 ) ) ) ) )
3321, 32sseldd 3351 1  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  A  e.  NN )  ->  x  e.  ( (
ball `  D ) `  ( g `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    \ cdif 3319    C_ wss 3322   <.cop 3819   class class class wbr 4214   {copab 4267    X. cxp 4878    o. ccom 4884   -->wf 5452   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1stc1st 6349   2ndc2nd 6350   1c1 8993    + caddc 8995    < clt 9122    / cdiv 9679   NNcn 10002   RR+crp 10614   * Metcxmt 16688   Metcme 16689   ballcbl 16690   MetOpencmopn 16693   Clsdccld 17082   clsccl 17084   ~~> tclm 17292   CMetcms 19209
This theorem is referenced by:  bcthlem4  19282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-topgen 13669  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-top 16965  df-bases 16967  df-topon 16968  df-cld 17085  df-ntr 17086  df-cls 17087  df-lm 17295  df-cmet 19212
  Copyright terms: Public domain W3C validator