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Theorem heiborlem9 26422
Description: Lemma for heibor 26424. Discharge the hypotheses of heiborlem8 26421 by applying caubl 19217 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 19196 . . . . . . 7  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
3 metxmet 18321 . . . . . . 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 18426 . . . . . 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 26418 . . . . . . . 8  |-  ( ph  ->  M : NN --> ( X  X.  RR+ ) )
185, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16heiborlem6 26419 . . . . . . . 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 26420 . . . . . . . . 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 19217 . . . . . . 7  |-  ( ph  ->  ( 1st  o.  M
)  e.  ( Cau `  D ) )
225cmetcau 19199 . . . . . . 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 18507 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  J  e.  Haus )
254, 24syl 16 . . . . . . 7  |-  ( ph  ->  J  e.  Haus )
26 lmfun 17403 . . . . . . 7  |-  ( J  e.  Haus  ->  Fun  ( ~~> t `  J )
)
27 funfvbrb 5806 . . . . . . 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 17319 . . . . 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 2485 . . 3  |-  ( ph  ->  ( ( ~~> t `  J ) `  ( 1st  o.  M ) )  e.  U. U )
34 eluni2 3983 . . 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 5705 . . 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 26421 . 2  |-  ( (
ph  /\  ( t  e.  U  /\  (
( ~~> t `  J
) `  ( 1st  o.  M ) )  e.  t ) )  ->  ps )
4835, 47rexlimddv 2798 1  |-  ( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {cab 2394   A.wral 2670   E.wrex 2671    i^i cin 3283    C_ wss 3284   ifcif 3703   ~Pcpw 3763   <.cop 3781   U.cuni 3979   U_ciun 4057   class class class wbr 4176   {copab 4229    e. cmpt 4230   dom cdm 4841    o. ccom 4845   Fun wfun 5411   -->wf 5413   ` cfv 5417  (class class class)co 6044    e. cmpt2 6046   1stc1st 6310   2ndc2nd 6311   Fincfn 7072   0cc0 8950   1c1 8951    + caddc 8953    < clt 9080    - cmin 9251    / cdiv 9637   NNcn 9960   2c2 10009   3c3 10010   NN0cn0 10181   RR+crp 10572    seq cseq 11282   ^cexp 11341   * Metcxmt 16645   Metcme 16646   ballcbl 16647   MetOpencmopn 16650  TopOnctopon 16918   ~~> tclm 17248   Hauscha 17330   Caucca 19163   CMetcms 19164
This theorem is referenced by:  heiborlem10  26423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-er 6868  df-map 6983  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-q 10535  df-rp 10573  df-xneg 10670  df-xadd 10671  df-xmul 10672  df-ico 10882  df-icc 10883  df-fl 11161  df-seq 11283  df-exp 11342  df-rest 13609  df-topgen 13626  df-psmet 16653  df-xmet 16654  df-met 16655  df-bl 16656  df-mopn 16657  df-fbas 16658  df-fg 16659  df-top 16922  df-bases 16924  df-topon 16925  df-cld 17042  df-ntr 17043  df-cls 17044  df-nei 17121  df-lm 17251  df-haus 17337  df-fil 17835  df-fm 17927  df-flim 17928  df-flf 17929  df-cfil 19165  df-cau 19166  df-cmet 19167
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