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Theorem bcthlem2 18747
Description: Lemma for bcth 18751. 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 5865 . . . . . . . 8  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
32fveq2d 5529 . . . . . . 7  |-  ( k  =  n  ->  (
g `  ( k  +  1 ) )  =  ( g `  ( n  +  1
) ) )
4 id 19 . . . . . . . 8  |-  ( k  =  n  ->  k  =  n )
5 fveq2 5525 . . . . . . . 8  |-  ( k  =  n  ->  (
g `  k )  =  ( g `  n ) )
64, 5oveq12d 5876 . . . . . . 7  |-  ( k  =  n  ->  (
k F ( g `
 k ) )  =  ( n F ( g `  n
) ) )
73, 6eleq12d 2351 . . . . . 6  |-  ( k  =  n  ->  (
( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) )  <->  ( g `  ( n  +  1 ) )  e.  ( n F ( g `
 n ) ) ) )
87rspccva 2883 . . . . 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 457 . . . 4  |-  ( (
ph  /\  n  e.  NN )  ->  ( g `
 ( n  + 
1 ) )  e.  ( n F ( g `  n ) ) )
10 bcthlem.9 . . . . . 6  |-  ( ph  ->  g : NN --> ( X  X.  RR+ ) )
11 ffvelrn 5663 . . . . . 6  |-  ( ( g : NN --> ( X  X.  RR+ )  /\  n  e.  NN )  ->  (
g `  n )  e.  ( X  X.  RR+ ) )
1210, 11sylan 457 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( g `
 n )  e.  ( X  X.  RR+ ) )
13 bcth.2 . . . . . . 7  |-  J  =  ( MetOpen `  D )
14 bcthlem.4 . . . . . . 7  |-  ( ph  ->  D  e.  ( CMet `  X ) )
15 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 )
) ) ) } )
1613, 14, 15bcthlem1 18746 . . . . . 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 ) ) ) ) )
1716expr 598 . . . . 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 ) ) ) ) ) )
1812, 17mpd 14 . . . 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 ) ) ) ) )
199, 18mpbid 201 . . 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 ) ) ) )
20 cmetmet 18712 . . . . . . . . . . . . 13  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
2114, 20syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  ( Met `  X ) )
22 metxmet 17899 . . . . . . . . . . . 12  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
2321, 22syl 15 . . . . . . . . . . 11  |-  ( ph  ->  D  e.  ( * Met `  X ) )
2413mopntop 17986 . . . . . . . . . . 11  |-  ( D  e.  ( * Met `  X )  ->  J  e.  Top )
2523, 24syl 15 . . . . . . . . . 10  |-  ( ph  ->  J  e.  Top )
2625adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  J  e.  Top )
27 xp1st 6149 . . . . . . . . . . . 12  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( g `  (
n  +  1 ) ) )  e.  X
)
28 xp2nd 6150 . . . . . . . . . . . . 13  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( g `  (
n  +  1 ) ) )  e.  RR+ )
2928rpxrd 10391 . . . . . . . . . . . 12  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( g `  (
n  +  1 ) ) )  e.  RR* )
3027, 29jca 518 . . . . . . . . . . 11  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( ( 1st `  ( g `  ( n  +  1
) ) )  e.  X  /\  ( 2nd `  ( g `  (
n  +  1 ) ) )  e.  RR* ) )
31 blssm 17968 . . . . . . . . . . . 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
)
32313expb 1152 . . . . . . . . . . 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
)
3323, 30, 32syl2an 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( 1st `  ( g `  ( n  +  1
) ) ) (
ball `  D )
( 2nd `  (
g `  ( n  +  1 ) ) ) )  C_  X
)
34 1st2nd2 6159 . . . . . . . . . . . . 13  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( g `  ( n  +  1 ) )  =  <. ( 1st `  ( g `
 ( n  + 
1 ) ) ) ,  ( 2nd `  (
g `  ( n  +  1 ) ) ) >. )
3534fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( g `  (
n  +  1 ) ) ) ,  ( 2nd `  ( g `
 ( n  + 
1 ) ) )
>. ) )
36 df-ov 5861 . . . . . . . . . . . 12  |-  ( ( 1st `  ( g `
 ( n  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( n  +  1 ) ) ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( g `  (
n  +  1 ) ) ) ,  ( 2nd `  ( g `
 ( n  + 
1 ) ) )
>. )
3735, 36syl6reqr 2334 . . . . . . . . . . 11  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( ( 1st `  ( g `  ( n  +  1
) ) ) (
ball `  D )
( 2nd `  (
g `  ( n  +  1 ) ) ) )  =  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) ) )
3837adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( 1st `  ( g `  ( n  +  1
) ) ) (
ball `  D )
( 2nd `  (
g `  ( n  +  1 ) ) ) )  =  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) ) )
3913mopnuni 17987 . . . . . . . . . . . 12  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
4023, 39syl 15 . . . . . . . . . . 11  |-  ( ph  ->  X  =  U. J
)
4140adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  X  =  U. J )
4233, 38, 413sstr3d 3220 . . . . . . . . 9  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) )  C_  U. J
)
43 eqid 2283 . . . . . . . . . 10  |-  U. J  =  U. J
4443sscls 16793 . . . . . . . . 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 ) ) ) ) )
4526, 42, 44syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) )  C_  (
( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) ) ) )
46 difss 3303 . . . . . . . . 9  |-  ( ( ( ball `  D
) `  ( g `  n ) )  \ 
( M `  n
) )  C_  (
( ball `  D ) `  ( g `  n
) )
47 sstr2 3186 . . . . . . . . 9  |-  ( ( ( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  n )
)  \  ( M `  n ) )  -> 
( ( ( (
ball `  D ) `  ( g `  n
) )  \  ( M `  n )
)  C_  ( ( ball `  D ) `  ( g `  n
) )  ->  (
( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) ) ) )
4846, 47mpi 16 . . . . . . . 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 ) ) )
49 sstr2 3186 . . . . . . . 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 ) ) ) )
5045, 48, 49syl2im 34 . . . . . . 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 ) ) ) )
5150a1d 22 . . . . . 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 ) ) ) ) )
5251ex 423 . . . . 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 ) ) ) ) ) )
53523impd 1165 . . . 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
) ) ) )
5453adantr 451 . . 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
) ) ) )
5519, 54mpd 14 . 2  |-  ( (
ph  /\  n  e.  NN )  ->  ( (
ball `  D ) `  ( g `  (
n  +  1 ) ) )  C_  (
( ball `  D ) `  ( g `  n
) ) )
5655ralrimiva 2626 1  |-  ( ph  ->  A. n  e.  NN  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    \ cdif 3149    C_ wss 3152   <.cop 3643   U.cuni 3827   class class class wbr 4023   {copab 4076    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   1c1 8738    + caddc 8740   RR*cxr 8866    < clt 8867    / cdiv 9423   NNcn 9746   RR+crp 10354   * Metcxmt 16369   Metcme 16370   ballcbl 16371   MetOpencmopn 16372   Topctop 16631   Clsdccld 16753   clsccl 16755   CMetcms 18680
This theorem is referenced by:  bcthlem3  18748  bcthlem4  18749
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-cls 16758  df-cmet 18683
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