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Theorem snmlff 23914
Description: The function  F from snmlval 23916 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 11037 . . . . . . 7  |-  ( n  e.  NN  ->  (
1 ... n )  e. 
Fin )
3 ssrab2 3260 . . . . . . 7  |-  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  C_  ( 1 ... n
)
4 ssfi 7085 . . . . . . 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 643 . . . . . 6  |-  ( n  e.  NN  ->  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  e.  Fin )
6 hashcl 11352 . . . . . 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 15 . . . . 5  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  e. 
NN0 )
87nn0red 10021 . . . 4  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  e.  RR )
9 nndivre 9783 . . . 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 650 . . 3  |-  ( n  e.  NN  ->  (
( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n )  e.  RR )
117nn0ge0d 10023 . . . 4  |-  ( n  e.  NN  ->  0  <_  ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } ) )
12 nnre 9755 . . . 4  |-  ( n  e.  NN  ->  n  e.  RR )
13 nngt0 9777 . . . 4  |-  ( n  e.  NN  ->  0  <  n )
14 divge0 9627 . . . 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 1183 . . 3  |-  ( n  e.  NN  ->  0  <_  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n ) )
16 ssdomg 6909 . . . . . . . 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 1366 . . . . . . 7  |-  ( n  e.  NN  ->  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  ~<_  ( 1 ... n ) )
18 hashdom 11363 . . . . . . . 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 642 . . . . . . 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 223 . . . . . 6  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  <_ 
( # `  ( 1 ... n ) ) )
21 nnnn0 9974 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  NN0 )
22 hashfz1 11347 . . . . . . 7  |-  ( n  e.  NN0  ->  ( # `  ( 1 ... n
) )  =  n )
2321, 22syl 15 . . . . . 6  |-  ( n  e.  NN  ->  ( # `
 ( 1 ... n ) )  =  n )
2420, 23breqtrd 4049 . . . . 5  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  <_  n )
25 nncn 9756 . . . . . 6  |-  ( n  e.  NN  ->  n  e.  CC )
2625mulid1d 8854 . . . . 5  |-  ( n  e.  NN  ->  (
n  x.  1 )  =  n )
2724, 26breqtrrd 4051 . . . 4  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  <_ 
( n  x.  1 ) )
28 1re 8839 . . . . . 6  |-  1  e.  RR
2928a1i 10 . . . . 5  |-  ( n  e.  NN  ->  1  e.  RR )
30 ledivmul 9631 . . . . 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 1186 . . . 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 223 . . 3  |-  ( n  e.  NN  ->  (
( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n )  <_ 
1 )
33 0re 8840 . . . 4  |-  0  e.  RR
3433, 28elicc2i 10718 . . 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 1136 . 2  |-  ( n  e.  NN  ->  (
( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n )  e.  ( 0 [,] 1
) )
361, 35fmpti 5685 1  |-  F : NN
--> ( 0 [,] 1
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1625    e. wcel 1686   {crab 2549    C_ wss 3154   class class class wbr 4025    e. cmpt 4079   -->wf 5253   ` cfv 5257  (class class class)co 5860    ~<_ cdom 6863   Fincfn 6865   RRcr 8738   0cc0 8739   1c1 8740    x. cmul 8744    < clt 8869    <_ cle 8870    / cdiv 9425   NNcn 9748   NN0cn0 9967   [,]cicc 10661   ...cfz 10784   |_cfl 10926    mod cmo 10975   ^cexp 11106   #chash 11339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-card 7574  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-n0 9968  df-z 10027  df-uz 10233  df-icc 10665  df-fz 10785  df-hash 11340
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