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Theorem bcthlem2 20795
Description: Lemma for bcth 20799. The balls in the sequence form an inclusion chain. (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
bcthlem2  |-  ( ph  ->  A. n  e.  NN  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) ) )
Distinct variable groups:    k, n, r, x, z    C, r, x    g, k, n, r, x, z, D   
g, F, k, n, r, x, z    g, J, k, n, r, x, z    g, M, k, n, r, x, z    ph, k, n, r, x, z    x, R    g, X, k, n, r, x, z
Allowed substitution hints:    ph( g)    C( z, g, k, n)    R( z, g, k, n, r)

Proof of Theorem bcthlem2
StepHypRef Expression
1 bcthlem.11 . . . . 5  |-  ( ph  ->  A. k  e.  NN  ( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) ) )
2 oveq1 6097 . . . . . . . 8  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
32fveq2d 5692 . . . . . . 7  |-  ( k  =  n  ->  (
g `  ( k  +  1 ) )  =  ( g `  ( n  +  1
) ) )
4 id 22 . . . . . . . 8  |-  ( k  =  n  ->  k  =  n )
5 fveq2 5688 . . . . . . . 8  |-  ( k  =  n  ->  (
g `  k )  =  ( g `  n ) )
64, 5oveq12d 6108 . . . . . . 7  |-  ( k  =  n  ->  (
k F ( g `
 k ) )  =  ( n F ( g `  n
) ) )
73, 6eleq12d 2509 . . . . . 6  |-  ( k  =  n  ->  (
( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) )  <->  ( g `  ( n  +  1 ) )  e.  ( n F ( g `
 n ) ) ) )
87rspccva 3069 . . . . 5  |-  ( ( A. k  e.  NN  ( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) )  /\  n  e.  NN )  ->  ( g `  (
n  +  1 ) )  e.  ( n F ( g `  n ) ) )
91, 8sylan 468 . . . 4  |-  ( (
ph  /\  n  e.  NN )  ->  ( g `
 ( n  + 
1 ) )  e.  ( n F ( g `  n ) ) )
10 bcthlem.9 . . . . . 6  |-  ( ph  ->  g : NN --> ( X  X.  RR+ ) )
1110ffvelrnda 5840 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( g `
 n )  e.  ( X  X.  RR+ ) )
12 bcth.2 . . . . . . 7  |-  J  =  ( MetOpen `  D )
13 bcthlem.4 . . . . . . 7  |-  ( ph  ->  D  e.  ( CMet `  X ) )
14 bcthlem.5 . . . . . . 7  |-  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 20794 . . . . . 6  |-  ( (
ph  /\  ( n  e.  NN  /\  ( g `
 n )  e.  ( X  X.  RR+ ) ) )  -> 
( ( g `  ( n  +  1
) )  e.  ( n F ( g `
 n ) )  <-> 
( ( g `  ( n  +  1
) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  (
g `  ( n  +  1 ) ) )  <  ( 1  /  n )  /\  ( ( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  n )
)  \  ( M `  n ) ) ) ) )
1615expr 612 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( g `  n )  e.  ( X  X.  RR+ )  ->  ( (
g `  ( n  +  1 ) )  e.  ( n F ( g `  n
) )  <->  ( (
g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  ( g `  (
n  +  1 ) ) )  <  (
1  /  n )  /\  ( ( cls `  J ) `  (
( ball `  D ) `  ( g `  (
n  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  n )
)  \  ( M `  n ) ) ) ) ) )
1711, 16mpd 15 . . . 4  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( g `  ( n  +  1 ) )  e.  ( n F ( g `  n
) )  <->  ( (
g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  ( g `  (
n  +  1 ) ) )  <  (
1  /  n )  /\  ( ( cls `  J ) `  (
( ball `  D ) `  ( g `  (
n  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  n )
)  \  ( M `  n ) ) ) ) )
189, 17mpbid 210 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  ( g `  (
n  +  1 ) ) )  <  (
1  /  n )  /\  ( ( cls `  J ) `  (
( ball `  D ) `  ( g `  (
n  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  n )
)  \  ( M `  n ) ) ) )
19 cmetmet 20756 . . . . . . . . . . . . 13  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
2013, 19syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  ( Met `  X ) )
21 metxmet 19868 . . . . . . . . . . . 12  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
2220, 21syl 16 . . . . . . . . . . 11  |-  ( ph  ->  D  e.  ( *Met `  X ) )
2312mopntop 19974 . . . . . . . . . . 11  |-  ( D  e.  ( *Met `  X )  ->  J  e.  Top )
2422, 23syl 16 . . . . . . . . . 10  |-  ( ph  ->  J  e.  Top )
2524adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  J  e.  Top )
26 xp1st 6605 . . . . . . . . . . . 12  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( g `  (
n  +  1 ) ) )  e.  X
)
27 xp2nd 6606 . . . . . . . . . . . . 13  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( g `  (
n  +  1 ) ) )  e.  RR+ )
2827rpxrd 11024 . . . . . . . . . . . 12  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( g `  (
n  +  1 ) ) )  e.  RR* )
2926, 28jca 529 . . . . . . . . . . 11  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( ( 1st `  ( g `  ( n  +  1
) ) )  e.  X  /\  ( 2nd `  ( g `  (
n  +  1 ) ) )  e.  RR* ) )
30 blssm 19952 . . . . . . . . . . . 12  |-  ( ( D  e.  ( *Met `  X )  /\  ( 1st `  (
g `  ( n  +  1 ) ) )  e.  X  /\  ( 2nd `  ( g `
 ( n  + 
1 ) ) )  e.  RR* )  ->  (
( 1st `  (
g `  ( n  +  1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( n  +  1 ) ) ) )  C_  X
)
31303expb 1183 . . . . . . . . . . 11  |-  ( ( D  e.  ( *Met `  X )  /\  ( ( 1st `  ( g `  (
n  +  1 ) ) )  e.  X  /\  ( 2nd `  (
g `  ( n  +  1 ) ) )  e.  RR* )
)  ->  ( ( 1st `  ( g `  ( n  +  1
) ) ) (
ball `  D )
( 2nd `  (
g `  ( n  +  1 ) ) ) )  C_  X
)
3222, 29, 31syl2an 474 . . . . . . . . . 10  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( 1st `  ( g `  ( n  +  1
) ) ) (
ball `  D )
( 2nd `  (
g `  ( n  +  1 ) ) ) )  C_  X
)
33 1st2nd2 6612 . . . . . . . . . . . . 13  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( g `  ( n  +  1 ) )  =  <. ( 1st `  ( g `
 ( n  + 
1 ) ) ) ,  ( 2nd `  (
g `  ( n  +  1 ) ) ) >. )
3433fveq2d 5692 . . . . . . . . . . . 12  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( g `  (
n  +  1 ) ) ) ,  ( 2nd `  ( g `
 ( n  + 
1 ) ) )
>. ) )
35 df-ov 6093 . . . . . . . . . . . 12  |-  ( ( 1st `  ( g `
 ( n  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( n  +  1 ) ) ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( g `  (
n  +  1 ) ) ) ,  ( 2nd `  ( g `
 ( n  + 
1 ) ) )
>. )
3634, 35syl6reqr 2492 . . . . . . . . . . 11  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( ( 1st `  ( g `  ( n  +  1
) ) ) (
ball `  D )
( 2nd `  (
g `  ( n  +  1 ) ) ) )  =  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) ) )
3736adantl 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( 1st `  ( g `  ( n  +  1
) ) ) (
ball `  D )
( 2nd `  (
g `  ( n  +  1 ) ) ) )  =  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) ) )
3812mopnuni 19975 . . . . . . . . . . . 12  |-  ( D  e.  ( *Met `  X )  ->  X  =  U. J )
3922, 38syl 16 . . . . . . . . . . 11  |-  ( ph  ->  X  =  U. J
)
4039adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  X  =  U. J )
4132, 37, 403sstr3d 3395 . . . . . . . . 9  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) )  C_  U. J
)
42 eqid 2441 . . . . . . . . . 10  |-  U. J  =  U. J
4342sscls 18619 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) )  C_  U. J )  ->  (
( ball `  D ) `  ( g `  (
n  +  1 ) ) )  C_  (
( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) ) ) )
4425, 41, 43syl2anc 656 . . . . . . . 8  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) )  C_  (
( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) ) ) )
45 difss2 3482 . . . . . . . 8  |-  ( ( ( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  n )
)  \  ( M `  n ) )  -> 
( ( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) ) )
46 sstr2 3360 . . . . . . . 8  |-  ( ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) )  C_  ( ( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) ) )  -> 
( ( ( cls `  J ) `  (
( ball `  D ) `  ( g `  (
n  +  1 ) ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) )  -> 
( ( ball `  D
) `  ( g `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) ) ) )
4744, 45, 46syl2im 38 . . . . . . 7  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( (
( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  n )
)  \  ( M `  n ) )  -> 
( ( ball `  D
) `  ( g `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) ) ) )
4847a1d 25 . . . . . 6  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( 2nd `  ( g `  ( n  +  1
) ) )  < 
( 1  /  n
)  ->  ( (
( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  n )
)  \  ( M `  n ) )  -> 
( ( ball `  D
) `  ( g `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) ) ) ) )
4948ex 434 . . . . 5  |-  ( ph  ->  ( ( g `  ( n  +  1
) )  e.  ( X  X.  RR+ )  ->  ( ( 2nd `  (
g `  ( n  +  1 ) ) )  <  ( 1  /  n )  -> 
( ( ( cls `  J ) `  (
( ball `  D ) `  ( g `  (
n  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  n )
)  \  ( M `  n ) )  -> 
( ( ball `  D
) `  ( g `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) ) ) ) ) )
50493impd 1196 . . . 4  |-  ( ph  ->  ( ( ( g `
 ( n  + 
1 ) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  (
g `  ( n  +  1 ) ) )  <  ( 1  /  n )  /\  ( ( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  n )
)  \  ( M `  n ) ) )  ->  ( ( ball `  D ) `  (
g `  ( n  +  1 ) ) )  C_  ( ( ball `  D ) `  ( g `  n
) ) ) )
5150adantr 462 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( g `  (
n  +  1 ) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  ( g `  ( n  +  1
) ) )  < 
( 1  /  n
)  /\  ( ( cls `  J ) `  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) ) ) 
C_  ( ( (
ball `  D ) `  ( g `  n
) )  \  ( M `  n )
) )  ->  (
( ball `  D ) `  ( g `  (
n  +  1 ) ) )  C_  (
( ball `  D ) `  ( g `  n
) ) ) )
5218, 51mpd 15 . 2  |-  ( (
ph  /\  n  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
n  +  1 ) ) )  C_  (
( ball `  D ) `  ( g `  n
) ) )
5352ralrimiva 2797 1  |-  ( ph  ->  A. n  e.  NN  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713    \ cdif 3322    C_ wss 3325   <.cop 3880   U.cuni 4088   class class class wbr 4289   {copab 4346    X. cxp 4834   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575   1c1 9279    + caddc 9281   RR*cxr 9413    < clt 9414    / cdiv 9989   NNcn 10318   RR+crp 10987   *Metcxmt 17760   Metcme 17761   ballcbl 17762   MetOpencmopn 17765   Topctop 18457   Clsdccld 18579   clsccl 18581   CMetcms 20724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-topgen 14378  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-top 18462  df-bases 18464  df-topon 18465  df-cld 18582  df-cls 18584  df-cmet 20727
This theorem is referenced by:  bcthlem3  20796  bcthlem4  20797
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