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Theorem snmlff 24797
Description: The function  F from snmlval 24799 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 11241 . . . . . . 7  |-  ( n  e.  NN  ->  (
1 ... n )  e. 
Fin )
3 ssrab2 3373 . . . . . . 7  |-  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  C_  ( 1 ... n
)
4 ssfi 7267 . . . . . . 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 11568 . . . . . 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 10209 . . . 4  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  e.  RR )
9 nndivre 9969 . . . 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 10211 . . . 4  |-  ( n  e.  NN  ->  0  <_  ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } ) )
12 nnre 9941 . . . 4  |-  ( n  e.  NN  ->  n  e.  RR )
13 nngt0 9963 . . . 4  |-  ( n  e.  NN  ->  0  <  n )
14 divge0 9813 . . . 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 7091 . . . . . . . 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 1382 . . . . . . 7  |-  ( n  e.  NN  ->  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  ~<_  ( 1 ... n ) )
18 hashdom 11582 . . . . . . . 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 10162 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  NN0 )
22 hashfz1 11559 . . . . . . 7  |-  ( n  e.  NN0  ->  ( # `  ( 1 ... n
) )  =  n )
2321, 22syl 16 . . . . . 6  |-  ( n  e.  NN  ->  ( # `
 ( 1 ... n ) )  =  n )
2420, 23breqtrd 4179 . . . . 5  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  <_  n )
25 nncn 9942 . . . . . 6  |-  ( n  e.  NN  ->  n  e.  CC )
2625mulid1d 9040 . . . . 5  |-  ( n  e.  NN  ->  (
n  x.  1 )  =  n )
2724, 26breqtrrd 4181 . . . 4  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  <_ 
( n  x.  1 ) )
28 1re 9025 . . . . . 6  |-  1  e.  RR
2928a1i 11 . . . . 5  |-  ( n  e.  NN  ->  1  e.  RR )
30 ledivmul 9817 . . . . 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 9026 . . . 4  |-  0  e.  RR
3433, 28elicc2i 10910 . . 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 5833 1  |-  F : NN
--> ( 0 [,] 1
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1717   {crab 2655    C_ wss 3265   class class class wbr 4155    e. cmpt 4209   -->wf 5392   ` cfv 5396  (class class class)co 6022    ~<_ cdom 7045   Fincfn 7047   RRcr 8924   0cc0 8925   1c1 8926    x. cmul 8930    < clt 9055    <_ cle 9056    / cdiv 9611   NNcn 9934   NN0cn0 10155   [,]cicc 10853   ...cfz 10977   |_cfl 11130    mod cmo 11179   ^cexp 11311   #chash 11547
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 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-n0 10156  df-z 10217  df-uz 10423  df-icc 10857  df-fz 10978  df-hash 11548
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