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Theorem heiborlem9 26528
Description: Lemma for heibor 26530. Discharge the hypotheses of heiborlem8 26527 by applying caubl 19260 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 19239 . . . . . . 7  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
3 metxmet 18364 . . . . . . 7  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
41, 2, 33syl 19 . . . . . 6  |-  ( ph  ->  D  e.  ( * Met `  X ) )
5 heibor.1 . . . . . . 7  |-  J  =  ( MetOpen `  D )
65mopntopon 18469 . . . . . 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 26524 . . . . . . . 8  |-  ( ph  ->  M : NN --> ( X  X.  RR+ ) )
185, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16heiborlem6 26525 . . . . . . . 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 26526 . . . . . . . . 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 19260 . . . . . . 7  |-  ( ph  ->  ( 1st  o.  M
)  e.  ( Cau `  D ) )
225cmetcau 19242 . . . . . . 7  |-  ( ( D  e.  ( CMet `  X )  /\  ( 1st  o.  M )  e.  ( Cau `  D
) )  ->  ( 1st  o.  M )  e. 
dom  ( ~~> t `  J ) )
231, 21, 22syl2anc 643 . . . . . 6  |-  ( ph  ->  ( 1st  o.  M
)  e.  dom  ( ~~> t `  J )
)
245methaus 18550 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  J  e.  Haus )
254, 24syl 16 . . . . . . 7  |-  ( ph  ->  J  e.  Haus )
26 lmfun 17445 . . . . . . 7  |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J )
)
27 funfvbrb 5843 . . . . . . 7  |-  ( Fun  ( ~~> t `  J
)  ->  ( ( 1st  o.  M )  e. 
dom  ( ~~> t `  J )  <->  ( 1st  o.  M ) ( ~~> t `  J ) ( ( ~~> t `  J ) `
 ( 1st  o.  M ) ) ) )
2825, 26, 273syl 19 . . . . . 6  |-  ( ph  ->  ( ( 1st  o.  M )  e.  dom  (
~~> t `  J )  <-> 
( 1st  o.  M
) ( ~~> t `  J ) ( ( ~~> t `  J ) `
 ( 1st  o.  M ) ) ) )
2923, 28mpbid 202 . . . . 5  |-  ( ph  ->  ( 1st  o.  M
) ( ~~> t `  J ) ( ( ~~> t `  J ) `
 ( 1st  o.  M ) ) )
30 lmcl 17361 . . . . 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 643 . . . 4  |-  ( ph  ->  ( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  X )
32 heiborlem9.14 . . . 4  |-  ( ph  ->  U. U  =  X )
3331, 32eleqtrrd 2513 . . 3  |-  ( ph  ->  ( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  U. U )
34 eluni2 4019 . . 3  |-  ( ( ( ~~> t `  J
) `  ( 1st  o.  M ) )  e. 
U. U  <->  E. t  e.  U  ( ( ~~> t `  J ) `  ( 1st  o.  M
) )  e.  t )
3533, 34sylib 189 . 2  |-  ( ph  ->  E. t  e.  U  ( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  t )
361adantr 452 . . 3  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  D  e.  ( CMet `  X ) )
3711adantr 452 . . 3  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
3812adantr 452 . . 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 452 . . 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 452 . . 3  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  C G 0 )
41 heibor.13 . . . 4  |-  ( ph  ->  U  C_  J )
4241adantr 452 . . 3  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  U  C_  J )
43 fvex 5742 . . 3  |-  ( ( ~~> t `  J ) `
 ( 1st  o.  M ) )  e. 
_V
44 simprr 734 . . 3  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  -> 
( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  t )
45 simprl 733 . . 3  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  -> 
t  e.  U )
4629adantr 452 . . 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 26527 . 2  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  ps )
4835, 47rexlimddv 2834 1  |-  ( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2422   A.wral 2705   E.wrex 2706    i^i cin 3319    C_ wss 3320   ifcif 3739   ~Pcpw 3799   <.cop 3817   U.cuni 4015   U_ciun 4093   class class class wbr 4212   {copab 4265    e. cmpt 4266   dom cdm 4878    o. ccom 4882   Fun wfun 5448   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348   Fincfn 7109   0cc0 8990   1c1 8991    + caddc 8993    < clt 9120    - cmin 9291    / cdiv 9677   NNcn 10000   2c2 10049   3c3 10050   NN0cn0 10221   RR+crp 10612    seq cseq 11323   ^cexp 11382   * Metcxmt 16686   Metcme 16687   ballcbl 16688   MetOpencmopn 16691  TopOnctopon 16959   ~~> tclm 17290   Hauscha 17372   Caucca 19206   CMetcms 19207
This theorem is referenced by:  heiborlem10  26529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ico 10922  df-icc 10923  df-fl 11202  df-seq 11324  df-exp 11383  df-rest 13650  df-topgen 13667  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-top 16963  df-bases 16965  df-topon 16966  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lm 17293  df-haus 17379  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-cfil 19208  df-cau 19209  df-cmet 19210
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