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

Theorem heiborlem9 28716
Description: Lemma for heibor 28718. Discharge the hypotheses of heiborlem8 28715 by applying caubl 20817 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 20796 . . . . . . 7  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
3 metxmet 19908 . . . . . . 7  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( *Met `  X
) )
41, 2, 33syl 20 . . . . . 6  |-  ( ph  ->  D  e.  ( *Met `  X ) )
5 heibor.1 . . . . . . 7  |-  J  =  ( MetOpen `  D )
65mopntopon 20013 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  J  e.  (TopOn `  X )
)
74, 6syl 16 . . . . 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 28712 . . . . . . . 8  |-  ( ph  ->  M : NN --> ( X  X.  RR+ ) )
185, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16heiborlem6 28713 . . . . . . . 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 28714 . . . . . . . . 9  |-  A. r  e.  RR+  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  r
2019a1i 11 . . . . . . . 8  |-  ( ph  ->  A. r  e.  RR+  E. k  e.  NN  ( 2nd `  ( M `  k ) )  < 
r )
214, 17, 18, 20caubl 20817 . . . . . . 7  |-  ( ph  ->  ( 1st  o.  M
)  e.  ( Cau `  D ) )
225cmetcau 20799 . . . . . . 7  |-  ( ( D  e.  ( CMet `  X )  /\  ( 1st  o.  M )  e.  ( Cau `  D
) )  ->  ( 1st  o.  M )  e. 
dom  ( ~~> t `  J ) )
231, 21, 22syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 1st  o.  M
)  e.  dom  ( ~~> t `  J )
)
245methaus 20094 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  J  e.  Haus )
254, 24syl 16 . . . . . . 7  |-  ( ph  ->  J  e.  Haus )
26 lmfun 18984 . . . . . . 7  |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J )
)
27 funfvbrb 5815 . . . . . . 7  |-  ( Fun  ( ~~> t `  J
)  ->  ( ( 1st  o.  M )  e. 
dom  ( ~~> t `  J )  <->  ( 1st  o.  M ) ( ~~> t `  J ) ( ( ~~> t `  J ) `
 ( 1st  o.  M ) ) ) )
2825, 26, 273syl 20 . . . . . 6  |-  ( ph  ->  ( ( 1st  o.  M )  e.  dom  (
~~> t `  J )  <-> 
( 1st  o.  M
) ( ~~> t `  J ) ( ( ~~> t `  J ) `
 ( 1st  o.  M ) ) ) )
2923, 28mpbid 210 . . . . 5  |-  ( ph  ->  ( 1st  o.  M
) ( ~~> t `  J ) ( ( ~~> t `  J ) `
 ( 1st  o.  M ) ) )
30 lmcl 18900 . . . . 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 661 . . . 4  |-  ( ph  ->  ( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  X )
32 heiborlem9.14 . . . 4  |-  ( ph  ->  U. U  =  X )
3331, 32eleqtrrd 2519 . . 3  |-  ( ph  ->  ( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  U. U )
34 eluni2 4094 . . 3  |-  ( ( ( ~~> t `  J
) `  ( 1st  o.  M ) )  e. 
U. U  <->  E. t  e.  U  ( ( ~~> t `  J ) `  ( 1st  o.  M
) )  e.  t )
3533, 34sylib 196 . 2  |-  ( ph  ->  E. t  e.  U  ( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  t )
361adantr 465 . . 3  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  D  e.  ( CMet `  X ) )
3711adantr 465 . . 3  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
3812adantr 465 . . 3  |-  ( (
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 465 . . 3  |-  ( (
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 465 . . 3  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  C G 0 )
41 heibor.13 . . . 4  |-  ( ph  ->  U  C_  J )
4241adantr 465 . . 3  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  U  C_  J )
43 fvex 5700 . . 3  |-  ( ( ~~> t `  J ) `
 ( 1st  o.  M ) )  e. 
_V
44 simprr 756 . . 3  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  -> 
( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  t )
45 simprl 755 . . 3  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  -> 
t  e.  U )
4629adantr 465 . . 3  |-  ( (
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 28715 . 2  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  ps )
4835, 47rexlimddv 2844 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {cab 2428   A.wral 2714   E.wrex 2715    i^i cin 3326    C_ wss 3327   ifcif 3790   ~Pcpw 3859   <.cop 3882   U.cuni 4090   U_ciun 4170   class class class wbr 4291   {copab 4348    e. cmpt 4349   dom cdm 4839    o. ccom 4843   Fun wfun 5411   -->wf 5413   ` cfv 5417  (class class class)co 6090    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575   Fincfn 7309   0cc0 9281   1c1 9282    + caddc 9284    < clt 9417    - cmin 9594    / cdiv 9992   NNcn 10321   2c2 10370   3c3 10371   NN0cn0 10578   RR+crp 10990    seqcseq 11805   ^cexp 11864   *Metcxmt 17800   Metcme 17801   ballcbl 17802   MetOpencmopn 17805  TopOnctopon 18498   ~~> tclm 18829   Hauscha 18911   Caucca 20763   CMetcms 20764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-er 7100  df-map 7215  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-q 10953  df-rp 10991  df-xneg 11088  df-xadd 11089  df-xmul 11090  df-ico 11305  df-icc 11306  df-fl 11641  df-seq 11806  df-exp 11865  df-rest 14360  df-topgen 14381  df-psmet 17808  df-xmet 17809  df-met 17810  df-bl 17811  df-mopn 17812  df-fbas 17813  df-fg 17814  df-top 18502  df-bases 18504  df-topon 18505  df-cld 18622  df-ntr 18623  df-cls 18624  df-nei 18701  df-lm 18832  df-haus 18918  df-fil 19418  df-fm 19510  df-flim 19511  df-flf 19512  df-cfil 20765  df-cau 20766  df-cmet 20767
This theorem is referenced by:  heiborlem10  28717
  Copyright terms: Public domain W3C validator