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Theorem bcthlem2 21515
Description: Lemma for bcth 21519. 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 6290 . . . . . . . 8  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
32fveq2d 5869 . . . . . . 7  |-  ( k  =  n  ->  (
g `  ( k  +  1 ) )  =  ( g `  ( n  +  1
) ) )
4 id 22 . . . . . . . 8  |-  ( k  =  n  ->  k  =  n )
5 fveq2 5865 . . . . . . . 8  |-  ( k  =  n  ->  (
g `  k )  =  ( g `  n ) )
64, 5oveq12d 6301 . . . . . . 7  |-  ( k  =  n  ->  (
k F ( g `
 k ) )  =  ( n F ( g `  n
) ) )
73, 6eleq12d 2549 . . . . . 6  |-  ( k  =  n  ->  (
( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) )  <->  ( g `  ( n  +  1 ) )  e.  ( n F ( g `
 n ) ) ) )
87rspccva 3213 . . . . 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 6020 . . . . 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 21514 . . . . . 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 21476 . . . . . . . . . . . . 13  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
2013, 19syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  ( Met `  X ) )
21 metxmet 20588 . . . . . . . . . . . 12  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
2220, 21syl 16 . . . . . . . . . . 11  |-  ( ph  ->  D  e.  ( *Met `  X ) )
2312mopntop 20694 . . . . . . . . . . 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 6814 . . . . . . . . . . . 12  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( g `  (
n  +  1 ) ) )  e.  X
)
27 xp2nd 6815 . . . . . . . . . . . . 13  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( g `  (
n  +  1 ) ) )  e.  RR+ )
2827rpxrd 11256 . . . . . . . . . . . 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 20672 . . . . . . . . . . . 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 1197 . . . . . . . . . . 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 6821 . . . . . . . . . . . . 13  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( g `  ( n  +  1 ) )  =  <. ( 1st `  ( g `
 ( n  + 
1 ) ) ) ,  ( 2nd `  (
g `  ( n  +  1 ) ) ) >. )
3433fveq2d 5869 . . . . . . . . . . . 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 6286 . . . . . . . . . . . 12  |-  ( ( 1st `  ( g `
 ( n  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( n  +  1 ) ) ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( g `  (
n  +  1 ) ) ) ,  ( 2nd `  ( g `
 ( n  + 
1 ) ) )
>. )
3634, 35syl6reqr 2527 . . . . . . . . . . 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 20695 . . . . . . . . . . . 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 3546 . . . . . . . . 9  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) )  C_  U. J
)
42 eqid 2467 . . . . . . . . . 10  |-  U. J  =  U. J
4342sscls 19339 . . . . . . . . 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 3633 . . . . . . . 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 3511 . . . . . . . 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 1210 . . . 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 2878 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 973    = wceq 1379    e. wcel 1767   A.wral 2814    \ cdif 3473    C_ wss 3476   <.cop 4033   U.cuni 4245   class class class wbr 4447   {copab 4504    X. cxp 4997   -->wf 5583   ` cfv 5587  (class class class)co 6283    |-> cmpt2 6285   1stc1st 6782   2ndc2nd 6783   1c1 9492    + caddc 9494   RR*cxr 9626    < clt 9627    / cdiv 10205   NNcn 10535   RR+crp 11219   *Metcxmt 18190   Metcme 18191   ballcbl 18192   MetOpencmopn 18195   Topctop 19177   Clsdccld 19299   clsccl 19301   CMetcms 21444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7900  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-n0 10795  df-z 10864  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-topgen 14698  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-top 19182  df-bases 19184  df-topon 19185  df-cld 19302  df-cls 19304  df-cmet 21447
This theorem is referenced by:  bcthlem3  21516  bcthlem4  21517
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