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Theorem heiborlem5 26642
Description: Lemma for heibor 26648. The function  M is a set of point-and-radius pairs suitable for application to caubl 18749. (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 9988 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
2 inss1 3402 . . . . . . . . 9  |-  ( ~P X  i^i  Fin )  C_ 
~P X
3 heibor.7 . . . . . . . . . 10  |-  ( ph  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
4 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( F : NN0 --> ( ~P X  i^i  Fin )  /\  k  e.  NN0 )  ->  ( F `  k )  e.  ( ~P X  i^i  Fin ) )
53, 4sylan 457 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  ( ~P X  i^i  Fin ) )
62, 5sseldi 3191 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  ~P X )
7 elpwi 3646 . . . . . . . 8  |-  ( ( F `  k )  e.  ~P X  -> 
( F `  k
)  C_  X )
86, 7syl 15 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  C_  X
)
9 heibor.1 . . . . . . . . 9  |-  J  =  ( MetOpen `  D )
10 heibor.3 . . . . . . . . 9  |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin )
u  C_  U. v }
11 heibor.4 . . . . . . . . 9  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
12 heibor.5 . . . . . . . . 9  |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D ) ( 1  /  ( 2 ^ m ) ) ) )
13 heibor.6 . . . . . . . . 9  |-  ( ph  ->  D  e.  ( CMet `  X ) )
14 heibor.8 . . . . . . . . 9  |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n
) ( y B n ) )
15 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
) )
16 heibor.10 . . . . . . . . 9  |-  ( ph  ->  C G 0 )
17 heibor.11 . . . . . . . . 9  |-  S  =  seq  0 ( T ,  ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) )
189, 10, 11, 12, 13, 3, 14, 15, 16, 17heiborlem4 26641 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( S `  k ) G k )
19 fvex 5555 . . . . . . . . . 10  |-  ( S `
 k )  e. 
_V
20 vex 2804 . . . . . . . . . 10  |-  k  e. 
_V
219, 10, 11, 19, 20heiborlem2 26639 . . . . . . . . 9  |-  ( ( S `  k ) G k  <->  ( k  e.  NN0  /\  ( S `
 k )  e.  ( F `  k
)  /\  ( ( S `  k ) B k )  e.  K ) )
2221simp2bi 971 . . . . . . . 8  |-  ( ( S `  k ) G k  ->  ( S `  k )  e.  ( F `  k
) )
2318, 22syl 15 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( S `  k )  e.  ( F `  k ) )
248, 23sseldd 3194 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( S `  k )  e.  X
)
251, 24sylan2 460 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( S `
 k )  e.  X )
2625ralrimiva 2639 . . . 4  |-  ( ph  ->  A. k  e.  NN  ( S `  k )  e.  X )
27 fveq2 5541 . . . . . 6  |-  ( k  =  n  ->  ( S `  k )  =  ( S `  n ) )
2827eleq1d 2362 . . . . 5  |-  ( k  =  n  ->  (
( S `  k
)  e.  X  <->  ( S `  n )  e.  X
) )
2928cbvralv 2777 . . . 4  |-  ( A. k  e.  NN  ( S `  k )  e.  X  <->  A. n  e.  NN  ( S `  n )  e.  X )
3026, 29sylib 188 . . 3  |-  ( ph  ->  A. n  e.  NN  ( S `  n )  e.  X )
31 3re 9833 . . . . . . 7  |-  3  e.  RR
32 3pos 9846 . . . . . . 7  |-  0  <  3
3331, 32elrpii 10373 . . . . . 6  |-  3  e.  RR+
34 2nn 9893 . . . . . . . 8  |-  2  e.  NN
35 nnnn0 9988 . . . . . . . 8  |-  ( n  e.  NN  ->  n  e.  NN0 )
36 nnexpcl 11132 . . . . . . . 8  |-  ( ( 2  e.  NN  /\  n  e.  NN0 )  -> 
( 2 ^ n
)  e.  NN )
3734, 35, 36sylancr 644 . . . . . . 7  |-  ( n  e.  NN  ->  (
2 ^ n )  e.  NN )
3837nnrpd 10405 . . . . . 6  |-  ( n  e.  NN  ->  (
2 ^ n )  e.  RR+ )
39 rpdivcl 10392 . . . . . 6  |-  ( ( 3  e.  RR+  /\  (
2 ^ n )  e.  RR+ )  ->  (
3  /  ( 2 ^ n ) )  e.  RR+ )
4033, 38, 39sylancr 644 . . . . 5  |-  ( n  e.  NN  ->  (
3  /  ( 2 ^ n ) )  e.  RR+ )
41 opelxpi 4737 . . . . . 6  |-  ( ( ( S `  n
)  e.  X  /\  ( 3  /  (
2 ^ n ) )  e.  RR+ )  -> 
<. ( S `  n
) ,  ( 3  /  ( 2 ^ n ) ) >.  e.  ( X  X.  RR+ ) )
4241expcom 424 . . . . 5  |-  ( ( 3  /  ( 2 ^ n ) )  e.  RR+  ->  ( ( S `  n )  e.  X  ->  <. ( S `  n ) ,  ( 3  / 
( 2 ^ n
) ) >.  e.  ( X  X.  RR+ )
) )
4340, 42syl 15 . . . 4  |-  ( n  e.  NN  ->  (
( S `  n
)  e.  X  ->  <. ( S `  n
) ,  ( 3  /  ( 2 ^ n ) ) >.  e.  ( X  X.  RR+ ) ) )
4443ralimia 2629 . . 3  |-  ( A. n  e.  NN  ( S `  n )  e.  X  ->  A. n  e.  NN  <. ( S `  n ) ,  ( 3  /  ( 2 ^ n ) )
>.  e.  ( X  X.  RR+ ) )
4530, 44syl 15 . 2  |-  ( ph  ->  A. n  e.  NN  <.
( S `  n
) ,  ( 3  /  ( 2 ^ n ) ) >.  e.  ( X  X.  RR+ ) )
46 heibor.12 . . 3  |-  M  =  ( n  e.  NN  |->  <. ( S `  n
) ,  ( 3  /  ( 2 ^ n ) ) >.
)
4746fmpt 5697 . 2  |-  ( A. n  e.  NN  <. ( S `  n ) ,  ( 3  / 
( 2 ^ n
) ) >.  e.  ( X  X.  RR+ )  <->  M : NN --> ( X  X.  RR+ ) )
4845, 47sylib 188 1  |-  ( ph  ->  M : NN --> ( X  X.  RR+ ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   ifcif 3578   ~Pcpw 3638   <.cop 3656   U.cuni 3843   U_ciun 3921   class class class wbr 4039   {copab 4092    e. cmpt 4093    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   2ndc2nd 6137   Fincfn 6879   0cc0 8753   1c1 8754    + caddc 8756    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   3c3 9812   NN0cn0 9981   RR+crp 10370    seq cseq 11062   ^cexp 11120   ballcbl 16387   MetOpencmopn 16388   CMetcms 18696
This theorem is referenced by:  heiborlem8  26645  heiborlem9  26646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121
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