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Theorem bcthlem3 18764
Description: Lemma for bcth 18767. The limit point of the centers in the sequence is in the intersection of every ball in the sequence. (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
bcthlem3  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  A  e.  NN )  ->  x  e.  ( (
ball `  D ) `  ( g `  A
) ) )
Distinct variable groups:    k, r, x, z, A    C, r, x    g, k, r, x, z, D    g, F, k, r, x, z    g, J, k, r, x, z   
g, M, k, r, x, z    ph, k,
r, x, z    x, R    g, X, k, r, x, z
Allowed substitution hints:    ph( g)    A( g)    C( z, g, k)    R( z, g, k, r)

Proof of Theorem bcthlem3
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 bcthlem.11 . . . . . . 7  |-  ( ph  ->  A. k  e.  NN  ( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) ) )
2 oveq1 5881 . . . . . . . . . 10  |-  ( k  =  A  ->  (
k  +  1 )  =  ( A  + 
1 ) )
32fveq2d 5545 . . . . . . . . 9  |-  ( k  =  A  ->  (
g `  ( k  +  1 ) )  =  ( g `  ( A  +  1
) ) )
4 id 19 . . . . . . . . . 10  |-  ( k  =  A  ->  k  =  A )
5 fveq2 5541 . . . . . . . . . 10  |-  ( k  =  A  ->  (
g `  k )  =  ( g `  A ) )
64, 5oveq12d 5892 . . . . . . . . 9  |-  ( k  =  A  ->  (
k F ( g `
 k ) )  =  ( A F ( g `  A
) ) )
73, 6eleq12d 2364 . . . . . . . 8  |-  ( k  =  A  ->  (
( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) )  <->  ( g `  ( A  +  1 ) )  e.  ( A F ( g `
 A ) ) ) )
87rspccva 2896 . . . . . . 7  |-  ( ( A. k  e.  NN  ( g `  (
k  +  1 ) )  e.  ( k F ( g `  k ) )  /\  A  e.  NN )  ->  ( g `  ( A  +  1 ) )  e.  ( A F ( g `  A ) ) )
91, 8sylan 457 . . . . . 6  |-  ( (
ph  /\  A  e.  NN )  ->  ( g `
 ( A  + 
1 ) )  e.  ( A F ( g `  A ) ) )
10 bcthlem.9 . . . . . . . 8  |-  ( ph  ->  g : NN --> ( X  X.  RR+ ) )
11 ffvelrn 5679 . . . . . . . 8  |-  ( ( g : NN --> ( X  X.  RR+ )  /\  A  e.  NN )  ->  (
g `  A )  e.  ( X  X.  RR+ ) )
1210, 11sylan 457 . . . . . . 7  |-  ( (
ph  /\  A  e.  NN )  ->  ( g `
 A )  e.  ( X  X.  RR+ ) )
13 bcth.2 . . . . . . . . 9  |-  J  =  ( MetOpen `  D )
14 bcthlem.4 . . . . . . . . 9  |-  ( ph  ->  D  e.  ( CMet `  X ) )
15 bcthlem.5 . . . . . . . . 9  |-  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 18762 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  NN  /\  ( g `
 A )  e.  ( X  X.  RR+ ) ) )  -> 
( ( g `  ( A  +  1
) )  e.  ( A F ( g `
 A ) )  <-> 
( ( g `  ( A  +  1
) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  (
g `  ( A  +  1 ) ) )  <  ( 1  /  A )  /\  ( ( cls `  J
) `  ( ( ball `  D ) `  ( g `  ( A  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  A )
)  \  ( M `  A ) ) ) ) )
1716expr 598 . . . . . . 7  |-  ( (
ph  /\  A  e.  NN )  ->  ( ( g `  A )  e.  ( X  X.  RR+ )  ->  ( (
g `  ( A  +  1 ) )  e.  ( A F ( g `  A
) )  <->  ( (
g `  ( A  +  1 ) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  ( g `  ( A  +  1 ) ) )  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
( ball `  D ) `  ( g `  ( A  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  A )
)  \  ( M `  A ) ) ) ) ) )
1812, 17mpd 14 . . . . . 6  |-  ( (
ph  /\  A  e.  NN )  ->  ( ( g `  ( A  +  1 ) )  e.  ( A F ( g `  A
) )  <->  ( (
g `  ( A  +  1 ) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  ( g `  ( A  +  1 ) ) )  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
( ball `  D ) `  ( g `  ( A  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  A )
)  \  ( M `  A ) ) ) ) )
199, 18mpbid 201 . . . . 5  |-  ( (
ph  /\  A  e.  NN )  ->  ( ( g `  ( A  +  1 ) )  e.  ( X  X.  RR+ )  /\  ( 2nd `  ( g `  ( A  +  1 ) ) )  <  (
1  /  A )  /\  ( ( cls `  J ) `  (
( ball `  D ) `  ( g `  ( A  +  1 ) ) ) )  C_  ( ( ( ball `  D ) `  (
g `  A )
)  \  ( M `  A ) ) ) )
2019simp3d 969 . . . 4  |-  ( (
ph  /\  A  e.  NN )  ->  ( ( cls `  J ) `
 ( ( ball `  D ) `  (
g `  ( A  +  1 ) ) ) )  C_  (
( ( ball `  D
) `  ( g `  A ) )  \ 
( M `  A
) ) )
21 difss 3316 . . . 4  |-  ( ( ( ball `  D
) `  ( g `  A ) )  \ 
( M `  A
) )  C_  (
( ball `  D ) `  ( g `  A
) )
2220, 21syl6ss 3204 . . 3  |-  ( (
ph  /\  A  e.  NN )  ->  ( ( cls `  J ) `
 ( ( ball `  D ) `  (
g `  ( A  +  1 ) ) ) )  C_  (
( ball `  D ) `  ( g `  A
) ) )
23223adant2 974 . 2  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  A  e.  NN )  ->  ( ( cls `  J
) `  ( ( ball `  D ) `  ( g `  ( A  +  1 ) ) ) )  C_  ( ( ball `  D
) `  ( g `  A ) ) )
24 peano2nn 9774 . . 3  |-  ( A  e.  NN  ->  ( A  +  1 )  e.  NN )
25 cmetmet 18728 . . . . 5  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
26 metxmet 17915 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
2714, 25, 263syl 18 . . . 4  |-  ( ph  ->  D  e.  ( * Met `  X ) )
28 bcthlem.6 . . . . 5  |-  ( ph  ->  M : NN --> ( Clsd `  J ) )
29 bcthlem.7 . . . . 5  |-  ( ph  ->  R  e.  RR+ )
30 bcthlem.8 . . . . 5  |-  ( ph  ->  C  e.  X )
31 bcthlem.10 . . . . 5  |-  ( ph  ->  ( g `  1
)  =  <. C ,  R >. )
3213, 14, 15, 28, 29, 30, 10, 31, 1bcthlem2 18763 . . . 4  |-  ( ph  ->  A. n  e.  NN  ( ( ball `  D
) `  ( g `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( g `  n ) ) )
3327, 10, 32, 13caublcls 18750 . . 3  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  ( A  +  1
)  e.  NN )  ->  x  e.  ( ( cls `  J
) `  ( ( ball `  D ) `  ( g `  ( A  +  1 ) ) ) ) )
3424, 33syl3an3 1217 . 2  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  A  e.  NN )  ->  x  e.  ( ( cls `  J ) `
 ( ( ball `  D ) `  (
g `  ( A  +  1 ) ) ) ) )
3523, 34sseldd 3194 1  |-  ( (
ph  /\  ( 1st  o.  g ) ( ~~> t `  J ) x  /\  A  e.  NN )  ->  x  e.  ( (
ball `  D ) `  ( g `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    \ cdif 3162    C_ wss 3165   <.cop 3656   class class class wbr 4039   {copab 4092    X. cxp 4703    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   1c1 8754    + caddc 8756    < clt 8883    / cdiv 9439   NNcn 9762   RR+crp 10370   * Metcxmt 16385   Metcme 16386   ballcbl 16387   MetOpencmopn 16388   Clsdccld 16769   clsccl 16771   ~~> tclm 16972   CMetcms 18696
This theorem is referenced by:  bcthlem4  18765
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-cld 16772  df-ntr 16773  df-cls 16774  df-lm 16975  df-cmet 18699
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