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Theorem bcthlem2 20836
Description: Lemma for bcth 20840. 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 6098 . . . . . . . 8  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
32fveq2d 5695 . . . . . . 7  |-  ( k  =  n  ->  (
g `  ( k  +  1 ) )  =  ( g `  ( n  +  1
) ) )
4 id 22 . . . . . . . 8  |-  ( k  =  n  ->  k  =  n )
5 fveq2 5691 . . . . . . . 8  |-  ( k  =  n  ->  (
g `  k )  =  ( g `  n ) )
64, 5oveq12d 6109 . . . . . . 7  |-  ( k  =  n  ->  (
k F ( g `
 k ) )  =  ( n F ( g `  n
) ) )
73, 6eleq12d 2511 . . . . . 6  |-  ( k  =  n  ->  (
( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) )  <->  ( g `  ( n  +  1 ) )  e.  ( n F ( g `
 n ) ) ) )
87rspccva 3072 . . . . 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 471 . . . 4  |-  ( (
ph  /\  n  e.  NN )  ->  ( g `
 ( n  + 
1 ) )  e.  ( n F ( g `  n ) ) )
10 bcthlem.9 . . . . . 6  |-  ( ph  ->  g : NN --> ( X  X.  RR+ ) )
1110ffvelrnda 5843 . . . . 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 20835 . . . . . 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 615 . . . . 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 20797 . . . . . . . . . . . . 13  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
2013, 19syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  ( Met `  X ) )
21 metxmet 19909 . . . . . . . . . . . 12  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
2220, 21syl 16 . . . . . . . . . . 11  |-  ( ph  ->  D  e.  ( *Met `  X ) )
2312mopntop 20015 . . . . . . . . . . 11  |-  ( D  e.  ( *Met `  X )  ->  J  e.  Top )
2422, 23syl 16 . . . . . . . . . 10  |-  ( ph  ->  J  e.  Top )
2524adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  J  e.  Top )
26 xp1st 6606 . . . . . . . . . . . 12  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( g `  (
n  +  1 ) ) )  e.  X
)
27 xp2nd 6607 . . . . . . . . . . . . 13  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( g `  (
n  +  1 ) ) )  e.  RR+ )
2827rpxrd 11028 . . . . . . . . . . . 12  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( g `  (
n  +  1 ) ) )  e.  RR* )
2926, 28jca 532 . . . . . . . . . . 11  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( ( 1st `  ( g `  ( n  +  1
) ) )  e.  X  /\  ( 2nd `  ( g `  (
n  +  1 ) ) )  e.  RR* ) )
30 blssm 19993 . . . . . . . . . . . 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 1188 . . . . . . . . . . 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 477 . . . . . . . . . 10  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( 1st `  ( g `  ( n  +  1
) ) ) (
ball `  D )
( 2nd `  (
g `  ( n  +  1 ) ) ) )  C_  X
)
33 1st2nd2 6613 . . . . . . . . . . . . 13  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( g `  ( n  +  1 ) )  =  <. ( 1st `  ( g `
 ( n  + 
1 ) ) ) ,  ( 2nd `  (
g `  ( n  +  1 ) ) ) >. )
3433fveq2d 5695 . . . . . . . . . . . 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 6094 . . . . . . . . . . . 12  |-  ( ( 1st `  ( g `
 ( n  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( n  +  1 ) ) ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( g `  (
n  +  1 ) ) ) ,  ( 2nd `  ( g `
 ( n  + 
1 ) ) )
>. )
3634, 35syl6reqr 2494 . . . . . . . . . . 11  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( ( 1st `  ( g `  ( n  +  1
) ) ) (
ball `  D )
( 2nd `  (
g `  ( n  +  1 ) ) ) )  =  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) ) )
3736adantl 466 . . . . . . . . . 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 20016 . . . . . . . . . . . 12  |-  ( D  e.  ( *Met `  X )  ->  X  =  U. J )
3922, 38syl 16 . . . . . . . . . . 11  |-  ( ph  ->  X  =  U. J
)
4039adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  X  =  U. J )
4132, 37, 403sstr3d 3398 . . . . . . . . 9  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) )  C_  U. J
)
42 eqid 2443 . . . . . . . . . 10  |-  U. J  =  U. J
4342sscls 18660 . . . . . . . . 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 661 . . . . . . . 8  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) )  C_  (
( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) ) ) )
45 difss2 3485 . . . . . . . 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 3363 . . . . . . . 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 1201 . . . 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 465 . . 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 2799 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 965    = wceq 1369    e. wcel 1756   A.wral 2715    \ cdif 3325    C_ wss 3328   <.cop 3883   U.cuni 4091   class class class wbr 4292   {copab 4349    X. cxp 4838   -->wf 5414   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   1stc1st 6575   2ndc2nd 6576   1c1 9283    + caddc 9285   RR*cxr 9417    < clt 9418    / cdiv 9993   NNcn 10322   RR+crp 10991   *Metcxmt 17801   Metcme 17802   ballcbl 17803   MetOpencmopn 17806   Topctop 18498   Clsdccld 18620   clsccl 18622   CMetcms 20765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-topgen 14382  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-top 18503  df-bases 18505  df-topon 18506  df-cld 18623  df-cls 18625  df-cmet 20768
This theorem is referenced by:  bcthlem3  20837  bcthlem4  20838
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