Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  heiborlem9 Unicode version

Theorem heiborlem9 26543
Description: Lemma for heibor 26545. Discharge the hypotheses of heiborlem8 26542 by applying caubl 18733 to get a convergent point and adding the open cover assumption. (Contributed by Jeff Madsen, 20-Jan-2014.)
Hypotheses
Ref Expression
heibor.1  |-  J  =  ( MetOpen `  D )
heibor.3  |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin )
u  C_  U. v }
heibor.4  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
heibor.5  |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D ) ( 1  /  ( 2 ^ m ) ) ) )
heibor.6  |-  ( ph  ->  D  e.  ( CMet `  X ) )
heibor.7  |-  ( ph  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
heibor.8  |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n
) ( y B n ) )
heibor.9  |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( T `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
heibor.10  |-  ( ph  ->  C G 0 )
heibor.11  |-  S  =  seq  0 ( T ,  ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) )
heibor.12  |-  M  =  ( n  e.  NN  |->  <. ( S `  n
) ,  ( 3  /  ( 2 ^ n ) ) >.
)
heibor.13  |-  ( ph  ->  U  C_  J )
heiborlem9.14  |-  ( ph  ->  U. U  =  X )
Assertion
Ref Expression
heiborlem9  |-  ( ph  ->  ps )
Distinct variable groups:    x, n, y, u, F    x, G    ph, x    m, n, u, v, x, y, z, D    m, M, u, x, y, z    T, m, n, x, y, z    B, n, u, v, y   
m, J, n, u, v, x, y, z    U, n, u, v, x, y, z    ps, y,
z    S, m, n, u, v, x, y, z   
m, X, n, u, v, x, y, z    C, m, n, u, v, y    n, K, x, y, z    x, B
Allowed substitution hints:    ph( y, z, v, u, m, n)    ps( x, v, u, m, n)    B( z, m)    C( x, z)    T( v, u)    U( m)    F( z, v, m)    G( y, z, v, u, m, n)    K( v, u, m)    M( v, n)

Proof of Theorem heiborlem9
Dummy variables  t 
k  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 heibor.6 . . . . . . 7  |-  ( ph  ->  D  e.  ( CMet `  X ) )
2 cmetmet 18712 . . . . . . 7  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
3 metxmet 17899 . . . . . . 7  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
41, 2, 33syl 18 . . . . . 6  |-  ( ph  ->  D  e.  ( * Met `  X ) )
5 heibor.1 . . . . . . 7  |-  J  =  ( MetOpen `  D )
65mopntopon 17985 . . . . . 6  |-  ( D  e.  ( * Met `  X )  ->  J  e.  (TopOn `  X )
)
74, 6syl 15 . . . . 5  |-  ( ph  ->  J  e.  (TopOn `  X ) )
8 heibor.3 . . . . . . . . 9  |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin )
u  C_  U. v }
9 heibor.4 . . . . . . . . 9  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
10 heibor.5 . . . . . . . . 9  |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D ) ( 1  /  ( 2 ^ m ) ) ) )
11 heibor.7 . . . . . . . . 9  |-  ( ph  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
12 heibor.8 . . . . . . . . 9  |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n
) ( y B n ) )
13 heibor.9 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( T `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
14 heibor.10 . . . . . . . . 9  |-  ( ph  ->  C G 0 )
15 heibor.11 . . . . . . . . 9  |-  S  =  seq  0 ( T ,  ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) )
16 heibor.12 . . . . . . . . 9  |-  M  =  ( n  e.  NN  |->  <. ( S `  n
) ,  ( 3  /  ( 2 ^ n ) ) >.
)
175, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16heiborlem5 26539 . . . . . . . 8  |-  ( ph  ->  M : NN --> ( X  X.  RR+ ) )
185, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16heiborlem6 26540 . . . . . . . 8  |-  ( ph  ->  A. k  e.  NN  ( ( ball `  D
) `  ( M `  ( k  +  1 ) ) )  C_  ( ( ball `  D
) `  ( M `  k ) ) )
195, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16heiborlem7 26541 . . . . . . . . 9  |-  A. r  e.  RR+  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  r
2019a1i 10 . . . . . . . 8  |-  ( ph  ->  A. r  e.  RR+  E. k  e.  NN  ( 2nd `  ( M `  k ) )  < 
r )
214, 17, 18, 20caubl 18733 . . . . . . 7  |-  ( ph  ->  ( 1st  o.  M
)  e.  ( Cau `  D ) )
225cmetcau 18715 . . . . . . 7  |-  ( ( D  e.  ( CMet `  X )  /\  ( 1st  o.  M )  e.  ( Cau `  D
) )  ->  ( 1st  o.  M )  e. 
dom  ( ~~> t `  J ) )
231, 21, 22syl2anc 642 . . . . . 6  |-  ( ph  ->  ( 1st  o.  M
)  e.  dom  ( ~~> t `  J )
)
245methaus 18066 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  J  e.  Haus )
254, 24syl 15 . . . . . . 7  |-  ( ph  ->  J  e.  Haus )
26 lmfun 17109 . . . . . . 7  |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J )
)
27 funfvbrb 5638 . . . . . . 7  |-  ( Fun  ( ~~> t `  J
)  ->  ( ( 1st  o.  M )  e. 
dom  ( ~~> t `  J )  <->  ( 1st  o.  M ) ( ~~> t `  J ) ( ( ~~> t `  J ) `
 ( 1st  o.  M ) ) ) )
2825, 26, 273syl 18 . . . . . 6  |-  ( ph  ->  ( ( 1st  o.  M )  e.  dom  (
~~> t `  J )  <-> 
( 1st  o.  M
) ( ~~> t `  J ) ( ( ~~> t `  J ) `
 ( 1st  o.  M ) ) ) )
2923, 28mpbid 201 . . . . 5  |-  ( ph  ->  ( 1st  o.  M
) ( ~~> t `  J ) ( ( ~~> t `  J ) `
 ( 1st  o.  M ) ) )
30 lmcl 17025 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  ( 1st  o.  M ) ( ~~> t `  J ) ( ( ~~> t `  J ) `  ( 1st  o.  M ) ) )  ->  ( ( ~~> t `  J ) `  ( 1st  o.  M
) )  e.  X
)
317, 29, 30syl2anc 642 . . . 4  |-  ( ph  ->  ( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  X )
32 heiborlem9.14 . . . 4  |-  ( ph  ->  U. U  =  X )
3331, 32eleqtrrd 2360 . . 3  |-  ( ph  ->  ( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  U. U )
34 eluni2 3831 . . 3  |-  ( ( ( ~~> t `  J
) `  ( 1st  o.  M ) )  e. 
U. U  <->  E. t  e.  U  ( ( ~~> t `  J ) `  ( 1st  o.  M
) )  e.  t )
3533, 34sylib 188 . 2  |-  ( ph  ->  E. t  e.  U  ( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  t )
361adantr 451 . . . . 5  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  D  e.  ( CMet `  X ) )
3711adantr 451 . . . . 5  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
3812adantr 451 . . . . 5  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n
) ( y B n ) )
3913adantr 451 . . . . 5  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( T `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
4014adantr 451 . . . . 5  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  C G 0 )
41 heibor.13 . . . . . 6  |-  ( ph  ->  U  C_  J )
4241adantr 451 . . . . 5  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  U  C_  J )
43 fvex 5539 . . . . 5  |-  ( ( ~~> t `  J ) `
 ( 1st  o.  M ) )  e. 
_V
44 simprr 733 . . . . 5  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  -> 
( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  t )
45 simprl 732 . . . . 5  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  -> 
t  e.  U )
4629adantr 451 . . . . 5  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  -> 
( 1st  o.  M
) ( ~~> t `  J ) ( ( ~~> t `  J ) `
 ( 1st  o.  M ) ) )
475, 8, 9, 10, 36, 37, 38, 39, 40, 15, 16, 42, 43, 44, 45, 46heiborlem8 26542 . . . 4  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  ps )
4847expr 598 . . 3  |-  ( (
ph  /\  t  e.  U )  ->  (
( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  t  ->  ps ) )
4948rexlimdva 2667 . 2  |-  ( ph  ->  ( E. t  e.  U  ( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  t  ->  ps ) )
5035, 49mpd 14 1  |-  ( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   ifcif 3565   ~Pcpw 3625   <.cop 3643   U.cuni 3827   U_ciun 3905   class class class wbr 4023   {copab 4076    e. cmpt 4077   dom cdm 4689    o. ccom 4693   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   Fincfn 6863   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   3c3 9796   NN0cn0 9965   RR+crp 10354    seq cseq 11046   ^cexp 11104   * Metcxmt 16369   Metcme 16370   ballcbl 16371   MetOpencmopn 16372  TopOnctopon 16632   ~~> tclm 16956   Hauscha 17036   Caucca 18679   CMetcms 18680
This theorem is referenced by:  heiborlem10  26544
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-iin 3908  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-1o 6479  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  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-3 9805  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-ico 10662  df-icc 10663  df-fl 10925  df-seq 11047  df-exp 11105  df-rest 13327  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-ntr 16757  df-cls 16758  df-nei 16835  df-lm 16959  df-haus 17043  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-cfil 18681  df-cau 18682  df-cmet 18683
  Copyright terms: Public domain W3C validator