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Theorem heiborlem9 30477
Description: Lemma for heibor 30479. Discharge the hypotheses of heiborlem8 30476 by applying caubl 21871 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 21850 . . . . . . 7  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
3 metxmet 20962 . . . . . . 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 21067 . . . . . 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 30473 . . . . . . . 8  |-  ( ph  ->  M : NN --> ( X  X.  RR+ ) )
185, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16heiborlem6 30474 . . . . . . . 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 30475 . . . . . . . . 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 21871 . . . . . . 7  |-  ( ph  ->  ( 1st  o.  M
)  e.  ( Cau `  D ) )
225cmetcau 21853 . . . . . . 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 21148 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  J  e.  Haus )
254, 24syl 16 . . . . . . 7  |-  ( ph  ->  J  e.  Haus )
26 lmfun 20008 . . . . . . 7  |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J )
)
27 funfvbrb 6001 . . . . . . 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 19924 . . . . 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 2548 . . 3  |-  ( ph  ->  ( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  U. U )
34 eluni2 4255 . . 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 5882 . . 3  |-  ( ( ~~> t `  J ) `
 ( 1st  o.  M ) )  e. 
_V
44 simprr 757 . . 3  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  -> 
( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  t )
45 simprl 756 . . 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 30476 . 2  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  ps )
4835, 47rexlimddv 2953 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {cab 2442   A.wral 2807   E.wrex 2808    i^i cin 3470    C_ wss 3471   ifcif 3944   ~Pcpw 4015   <.cop 4038   U.cuni 4251   U_ciun 4332   class class class wbr 4456   {copab 4514    |-> cmpt 4515   dom cdm 5008    o. ccom 5012   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798   Fincfn 7535   0cc0 9509   1c1 9510    + caddc 9512    < clt 9645    - cmin 9824    / cdiv 10227   NNcn 10556   2c2 10606   3c3 10607   NN0cn0 10816   RR+crp 11245    seqcseq 12109   ^cexp 12168   *Metcxmt 18529   Metcme 18530   ballcbl 18531   MetOpencmopn 18534  TopOnctopon 19521   ~~> tclm 19853   Hauscha 19935   Caucca 21817   CMetcms 21818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ico 11560  df-icc 11561  df-fl 11931  df-seq 12110  df-exp 12169  df-rest 14839  df-topgen 14860  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-fbas 18542  df-fg 18543  df-top 19525  df-bases 19527  df-topon 19528  df-cld 19646  df-ntr 19647  df-cls 19648  df-nei 19725  df-lm 19856  df-haus 19942  df-fil 20472  df-fm 20564  df-flim 20565  df-flf 20566  df-cfil 21819  df-cau 21820  df-cmet 21821
This theorem is referenced by:  heiborlem10  30478
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