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Theorem heiborlem5 26216
Description: Lemma for heibor 26222. The function  M is a set of point-and-radius pairs suitable for application to caubl 19132. (Contributed by Jeff Madsen, 23-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 ) ) >.
)
Assertion
Ref Expression
heiborlem5  |-  ( ph  ->  M : NN --> ( X  X.  RR+ ) )
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    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)    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 heiborlem5
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 nnnn0 10161 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
2 inss1 3505 . . . . . . . . 9  |-  ( ~P X  i^i  Fin )  C_ 
~P X
3 heibor.7 . . . . . . . . . 10  |-  ( ph  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
43ffvelrnda 5810 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  ( ~P X  i^i  Fin ) )
52, 4sseldi 3290 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  ~P X )
65elpwid 3752 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  C_  X
)
7 heibor.1 . . . . . . . . 9  |-  J  =  ( MetOpen `  D )
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.6 . . . . . . . . 9  |-  ( ph  ->  D  e.  ( CMet `  X ) )
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 ) ) ) )
167, 8, 9, 10, 11, 3, 12, 13, 14, 15heiborlem4 26215 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( S `  k ) G k )
17 fvex 5683 . . . . . . . . . 10  |-  ( S `
 k )  e. 
_V
18 vex 2903 . . . . . . . . . 10  |-  k  e. 
_V
197, 8, 9, 17, 18heiborlem2 26213 . . . . . . . . 9  |-  ( ( S `  k ) G k  <->  ( k  e.  NN0  /\  ( S `
 k )  e.  ( F `  k
)  /\  ( ( S `  k ) B k )  e.  K ) )
2019simp2bi 973 . . . . . . . 8  |-  ( ( S `  k ) G k  ->  ( S `  k )  e.  ( F `  k
) )
2116, 20syl 16 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( S `  k )  e.  ( F `  k ) )
226, 21sseldd 3293 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( S `  k )  e.  X
)
231, 22sylan2 461 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( S `
 k )  e.  X )
2423ralrimiva 2733 . . . 4  |-  ( ph  ->  A. k  e.  NN  ( S `  k )  e.  X )
25 fveq2 5669 . . . . . 6  |-  ( k  =  n  ->  ( S `  k )  =  ( S `  n ) )
2625eleq1d 2454 . . . . 5  |-  ( k  =  n  ->  (
( S `  k
)  e.  X  <->  ( S `  n )  e.  X
) )
2726cbvralv 2876 . . . 4  |-  ( A. k  e.  NN  ( S `  k )  e.  X  <->  A. n  e.  NN  ( S `  n )  e.  X )
2824, 27sylib 189 . . 3  |-  ( ph  ->  A. n  e.  NN  ( S `  n )  e.  X )
29 3re 10004 . . . . . . 7  |-  3  e.  RR
30 3pos 10017 . . . . . . 7  |-  0  <  3
3129, 30elrpii 10548 . . . . . 6  |-  3  e.  RR+
32 2nn 10066 . . . . . . . 8  |-  2  e.  NN
33 nnnn0 10161 . . . . . . . 8  |-  ( n  e.  NN  ->  n  e.  NN0 )
34 nnexpcl 11322 . . . . . . . 8  |-  ( ( 2  e.  NN  /\  n  e.  NN0 )  -> 
( 2 ^ n
)  e.  NN )
3532, 33, 34sylancr 645 . . . . . . 7  |-  ( n  e.  NN  ->  (
2 ^ n )  e.  NN )
3635nnrpd 10580 . . . . . 6  |-  ( n  e.  NN  ->  (
2 ^ n )  e.  RR+ )
37 rpdivcl 10567 . . . . . 6  |-  ( ( 3  e.  RR+  /\  (
2 ^ n )  e.  RR+ )  ->  (
3  /  ( 2 ^ n ) )  e.  RR+ )
3831, 36, 37sylancr 645 . . . . 5  |-  ( n  e.  NN  ->  (
3  /  ( 2 ^ n ) )  e.  RR+ )
39 opelxpi 4851 . . . . . 6  |-  ( ( ( S `  n
)  e.  X  /\  ( 3  /  (
2 ^ n ) )  e.  RR+ )  -> 
<. ( S `  n
) ,  ( 3  /  ( 2 ^ n ) ) >.  e.  ( X  X.  RR+ ) )
4039expcom 425 . . . . 5  |-  ( ( 3  /  ( 2 ^ n ) )  e.  RR+  ->  ( ( S `  n )  e.  X  ->  <. ( S `  n ) ,  ( 3  / 
( 2 ^ n
) ) >.  e.  ( X  X.  RR+ )
) )
4138, 40syl 16 . . . 4  |-  ( n  e.  NN  ->  (
( S `  n
)  e.  X  ->  <. ( S `  n
) ,  ( 3  /  ( 2 ^ n ) ) >.  e.  ( X  X.  RR+ ) ) )
4241ralimia 2723 . . 3  |-  ( A. n  e.  NN  ( S `  n )  e.  X  ->  A. n  e.  NN  <. ( S `  n ) ,  ( 3  /  ( 2 ^ n ) )
>.  e.  ( X  X.  RR+ ) )
4328, 42syl 16 . 2  |-  ( ph  ->  A. n  e.  NN  <.
( S `  n
) ,  ( 3  /  ( 2 ^ n ) ) >.  e.  ( X  X.  RR+ ) )
44 heibor.12 . . 3  |-  M  =  ( n  e.  NN  |->  <. ( S `  n
) ,  ( 3  /  ( 2 ^ n ) ) >.
)
4544fmpt 5830 . 2  |-  ( A. n  e.  NN  <. ( S `  n ) ,  ( 3  / 
( 2 ^ n
) ) >.  e.  ( X  X.  RR+ )  <->  M : NN --> ( X  X.  RR+ ) )
4643, 45sylib 189 1  |-  ( ph  ->  M : NN --> ( X  X.  RR+ ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {cab 2374   A.wral 2650   E.wrex 2651    i^i cin 3263    C_ wss 3264   ifcif 3683   ~Pcpw 3743   <.cop 3761   U.cuni 3958   U_ciun 4036   class class class wbr 4154   {copab 4207    e. cmpt 4208    X. cxp 4817   -->wf 5391   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   2ndc2nd 6288   Fincfn 7046   0cc0 8924   1c1 8925    + caddc 8927    - cmin 9224    / cdiv 9610   NNcn 9933   2c2 9982   3c3 9983   NN0cn0 10154   RR+crp 10545    seq cseq 11251   ^cexp 11310   ballcbl 16615   MetOpencmopn 16618   CMetcms 19079
This theorem is referenced by:  heiborlem8  26219  heiborlem9  26220
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-seq 11252  df-exp 11311
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