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Theorem snmlff 25006
Description: The function  F from snmlval 25008 is a mapping from positive integers to real numbers in the range 
[ 0 ,  1 ]. (Contributed by Mario Carneiro, 6-Apr-2015.)
Hypothesis
Ref Expression
snmlff.f  |-  F  =  ( n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n ) )
Assertion
Ref Expression
snmlff  |-  F : NN
--> ( 0 [,] 1
)
Distinct variable groups:    A, n    B, n    k, n    R, n
Allowed substitution hints:    A( k)    B( k)    R( k)    F( k, n)

Proof of Theorem snmlff
StepHypRef Expression
1 snmlff.f . 2  |-  F  =  ( n  e.  NN  |->  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n ) )
2 fzfid 11302 . . . . . . 7  |-  ( n  e.  NN  ->  (
1 ... n )  e. 
Fin )
3 ssrab2 3420 . . . . . . 7  |-  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  C_  ( 1 ... n
)
4 ssfi 7321 . . . . . . 7  |-  ( ( ( 1 ... n
)  e.  Fin  /\  { k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  C_  ( 1 ... n
) )  ->  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  e.  Fin )
52, 3, 4sylancl 644 . . . . . 6  |-  ( n  e.  NN  ->  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  e.  Fin )
6 hashcl 11629 . . . . . 6  |-  ( { k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  e.  Fin  ->  ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  e.  NN0 )
75, 6syl 16 . . . . 5  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  e. 
NN0 )
87nn0red 10265 . . . 4  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  e.  RR )
9 nndivre 10025 . . . 4  |-  ( ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  e.  RR  /\  n  e.  NN )  ->  (
( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n )  e.  RR )
108, 9mpancom 651 . . 3  |-  ( n  e.  NN  ->  (
( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n )  e.  RR )
117nn0ge0d 10267 . . . 4  |-  ( n  e.  NN  ->  0  <_  ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } ) )
12 nnre 9997 . . . 4  |-  ( n  e.  NN  ->  n  e.  RR )
13 nngt0 10019 . . . 4  |-  ( n  e.  NN  ->  0  <  n )
14 divge0 9869 . . . 4  |-  ( ( ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  e.  RR  /\  0  <_  ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } ) )  /\  ( n  e.  RR  /\  0  <  n ) )  -> 
0  <_  ( ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  /  n ) )
158, 11, 12, 13, 14syl22anc 1185 . . 3  |-  ( n  e.  NN  ->  0  <_  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n ) )
16 ssdomg 7145 . . . . . . . 8  |-  ( ( 1 ... n )  e.  Fin  ->  ( { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  C_  ( 1 ... n
)  ->  { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B }  ~<_  ( 1 ... n ) ) )
172, 3, 16ee10 1385 . . . . . . 7  |-  ( n  e.  NN  ->  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  ~<_  ( 1 ... n ) )
18 hashdom 11643 . . . . . . . 8  |-  ( ( { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  e.  Fin  /\  ( 1 ... n )  e. 
Fin )  ->  (
( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  <_  ( # `  (
1 ... n ) )  <->  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  ~<_  ( 1 ... n
) ) )
195, 2, 18syl2anc 643 . . . . . . 7  |-  ( n  e.  NN  ->  (
( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  <_  ( # `  (
1 ... n ) )  <->  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  ~<_  ( 1 ... n
) ) )
2017, 19mpbird 224 . . . . . 6  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  <_ 
( # `  ( 1 ... n ) ) )
21 nnnn0 10218 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  NN0 )
22 hashfz1 11620 . . . . . . 7  |-  ( n  e.  NN0  ->  ( # `  ( 1 ... n
) )  =  n )
2321, 22syl 16 . . . . . 6  |-  ( n  e.  NN  ->  ( # `
 ( 1 ... n ) )  =  n )
2420, 23breqtrd 4228 . . . . 5  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  <_  n )
25 nncn 9998 . . . . . 6  |-  ( n  e.  NN  ->  n  e.  CC )
2625mulid1d 9095 . . . . 5  |-  ( n  e.  NN  ->  (
n  x.  1 )  =  n )
2724, 26breqtrrd 4230 . . . 4  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  <_ 
( n  x.  1 ) )
28 1re 9080 . . . . . 6  |-  1  e.  RR
2928a1i 11 . . . . 5  |-  ( n  e.  NN  ->  1  e.  RR )
30 ledivmul 9873 . . . . 5  |-  ( ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  e.  RR  /\  1  e.  RR  /\  ( n  e.  RR  /\  0  <  n ) )  -> 
( ( ( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
)  /  n )  <_  1  <->  ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  <_  ( n  x.  1 ) ) )
318, 29, 12, 13, 30syl112anc 1188 . . . 4  |-  ( n  e.  NN  ->  (
( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n )  <_ 
1  <->  ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  <_  ( n  x.  1 ) ) )
3227, 31mpbird 224 . . 3  |-  ( n  e.  NN  ->  (
( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n )  <_ 
1 )
33 0re 9081 . . . 4  |-  0  e.  RR
3433, 28elicc2i 10966 . . 3  |-  ( ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n )  e.  ( 0 [,] 1
)  <->  ( ( (
# `  { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  /  n )  e.  RR  /\  0  <_  ( ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  /  n )  /\  (
( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n )  <_ 
1 ) )
3510, 15, 32, 34syl3anbrc 1138 . 2  |-  ( n  e.  NN  ->  (
( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n )  e.  ( 0 [,] 1
) )
361, 35fmpti 5884 1  |-  F : NN
--> ( 0 [,] 1
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   {crab 2701    C_ wss 3312   class class class wbr 4204    e. cmpt 4258   -->wf 5442   ` cfv 5446  (class class class)co 6073    ~<_ cdom 7099   Fincfn 7101   RRcr 8979   0cc0 8980   1c1 8981    x. cmul 8985    < clt 9110    <_ cle 9111    / cdiv 9667   NNcn 9990   NN0cn0 10211   [,]cicc 10909   ...cfz 11033   |_cfl 11191    mod cmo 11240   ^cexp 11372   #chash 11608
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7816  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-n0 10212  df-z 10273  df-uz 10479  df-icc 10913  df-fz 11034  df-hash 11609
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