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Theorem bcthlem2 22186
Description: Lemma for bcth 22190. 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 6312 . . . . . . . 8  |-  ( k  =  n  ->  (
k  +  1 )  =  ( n  + 
1 ) )
32fveq2d 5885 . . . . . . 7  |-  ( k  =  n  ->  (
g `  ( k  +  1 ) )  =  ( g `  ( n  +  1
) ) )
4 id 23 . . . . . . . 8  |-  ( k  =  n  ->  k  =  n )
5 fveq2 5881 . . . . . . . 8  |-  ( k  =  n  ->  (
g `  k )  =  ( g `  n ) )
64, 5oveq12d 6323 . . . . . . 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 3187 . . . . 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 473 . . . 4  |-  ( (
ph  /\  n  e.  NN )  ->  ( g `
 ( n  + 
1 ) )  e.  ( n F ( g `  n ) ) )
10 bcthlem.9 . . . . . 6  |-  ( ph  ->  g : NN --> ( X  X.  RR+ ) )
1110ffvelrnda 6037 . . . . 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 22185 . . . . . 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 618 . . . . 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 213 . . 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 22149 . . . . . . . . . . . . 13  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
2013, 19syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  ( Met `  X ) )
21 metxmet 21280 . . . . . . . . . . . 12  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
2220, 21syl 17 . . . . . . . . . . 11  |-  ( ph  ->  D  e.  ( *Met `  X ) )
2312mopntop 21386 . . . . . . . . . . 11  |-  ( D  e.  ( *Met `  X )  ->  J  e.  Top )
2422, 23syl 17 . . . . . . . . . 10  |-  ( ph  ->  J  e.  Top )
2524adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  J  e.  Top )
26 xp1st 6837 . . . . . . . . . . . 12  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( g `  (
n  +  1 ) ) )  e.  X
)
27 xp2nd 6838 . . . . . . . . . . . . 13  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( g `  (
n  +  1 ) ) )  e.  RR+ )
2827rpxrd 11342 . . . . . . . . . . . 12  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( g `  (
n  +  1 ) ) )  e.  RR* )
2926, 28jca 534 . . . . . . . . . . 11  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( ( 1st `  ( g `  ( n  +  1
) ) )  e.  X  /\  ( 2nd `  ( g `  (
n  +  1 ) ) )  e.  RR* ) )
30 blssm 21364 . . . . . . . . . . . 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 1206 . . . . . . . . . . 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 479 . . . . . . . . . 10  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( 1st `  ( g `  ( n  +  1
) ) ) (
ball `  D )
( 2nd `  (
g `  ( n  +  1 ) ) ) )  C_  X
)
33 1st2nd2 6844 . . . . . . . . . . . . 13  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( g `  ( n  +  1 ) )  =  <. ( 1st `  ( g `
 ( n  + 
1 ) ) ) ,  ( 2nd `  (
g `  ( n  +  1 ) ) ) >. )
3433fveq2d 5885 . . . . . . . . . . . 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 6308 . . . . . . . . . . . 12  |-  ( ( 1st `  ( g `
 ( n  + 
1 ) ) ) ( ball `  D
) ( 2nd `  (
g `  ( n  +  1 ) ) ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( g `  (
n  +  1 ) ) ) ,  ( 2nd `  ( g `
 ( n  + 
1 ) ) )
>. )
3634, 35syl6reqr 2489 . . . . . . . . . . 11  |-  ( ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )  ->  ( ( 1st `  ( g `  ( n  +  1
) ) ) (
ball `  D )
( 2nd `  (
g `  ( n  +  1 ) ) ) )  =  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) ) )
3736adantl 467 . . . . . . . . . 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 21387 . . . . . . . . . . . 12  |-  ( D  e.  ( *Met `  X )  ->  X  =  U. J )
3922, 38syl 17 . . . . . . . . . . 11  |-  ( ph  ->  X  =  U. J
)
4039adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  X  =  U. J )
4132, 37, 403sstr3d 3512 . . . . . . . . 9  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) )  C_  U. J
)
42 eqid 2429 . . . . . . . . . 10  |-  U. J  =  U. J
4342sscls 20002 . . . . . . . . 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 665 . . . . . . . 8  |-  ( (
ph  /\  ( g `  ( n  +  1 ) )  e.  ( X  X.  RR+ )
)  ->  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) )  C_  (
( cls `  J
) `  ( ( ball `  D ) `  ( g `  (
n  +  1 ) ) ) ) )
45 difss2 3600 . . . . . . . 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 3477 . . . . . . . 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 39 . . . . . . 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 26 . . . . . 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 435 . . . . 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 1219 . . . 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 466 . . 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 2846 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782    \ cdif 3439    C_ wss 3442   <.cop 4008   U.cuni 4222   class class class wbr 4426   {copab 4483    X. cxp 4852   -->wf 5597   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806   1c1 9539    + caddc 9541   RR*cxr 9673    < clt 9674    / cdiv 10268   NNcn 10609   RR+crp 11302   *Metcxmt 18890   Metcme 18891   ballcbl 18892   MetOpencmopn 18895   Topctop 19848   Clsdccld 19962   clsccl 19964   CMetcms 22117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-topgen 15301  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-top 19852  df-bases 19853  df-topon 19854  df-cld 19965  df-cls 19967  df-cmet 22120
This theorem is referenced by:  bcthlem3  22187  bcthlem4  22188
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