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Theorem snmlff 23270
Description: The function  F from snmlval 23272 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 10987 . . . . . . 7  |-  ( n  e.  NN  ->  (
1 ... n )  e. 
Fin )
3 ssrab2 3219 . . . . . . 7  |-  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  C_  ( 1 ... n
)
4 ssfi 7037 . . . . . . 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 646 . . . . . 6  |-  ( n  e.  NN  ->  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  e.  Fin )
6 hashcl 11302 . . . . . 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 17 . . . . 5  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  e. 
NN0 )
87nn0red 9972 . . . 4  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  e.  RR )
9 nndivre 9735 . . . 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 653 . . 3  |-  ( n  e.  NN  ->  (
( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n )  e.  RR )
117nn0ge0d 9974 . . . 4  |-  ( n  e.  NN  ->  0  <_  ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } ) )
12 nnre 9707 . . . 4  |-  ( n  e.  NN  ->  n  e.  RR )
13 nngt0 9729 . . . 4  |-  ( n  e.  NN  ->  0  <  n )
14 divge0 9579 . . . 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 1188 . . 3  |-  ( n  e.  NN  ->  0  <_  ( ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n ) )
16 ssdomg 6861 . . . . . . . 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 1372 . . . . . . 7  |-  ( n  e.  NN  ->  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  ~<_  ( 1 ... n ) )
18 hashdom 11313 . . . . . . . 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 645 . . . . . . 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 225 . . . . . 6  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  <_ 
( # `  ( 1 ... n ) ) )
21 nnnn0 9925 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  NN0 )
22 hashfz1 11297 . . . . . . 7  |-  ( n  e.  NN0  ->  ( # `  ( 1 ... n
) )  =  n )
2321, 22syl 17 . . . . . 6  |-  ( n  e.  NN  ->  ( # `
 ( 1 ... n ) )  =  n )
2420, 23breqtrd 4007 . . . . 5  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  <_  n )
25 nncn 9708 . . . . . 6  |-  ( n  e.  NN  ->  n  e.  CC )
2625mulid1d 8806 . . . . 5  |-  ( n  e.  NN  ->  (
n  x.  1 )  =  n )
2724, 26breqtrrd 4009 . . . 4  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  <_ 
( n  x.  1 ) )
28 1re 8791 . . . . . 6  |-  1  e.  RR
2928a1i 12 . . . . 5  |-  ( n  e.  NN  ->  1  e.  RR )
30 ledivmul 9583 . . . . 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 1191 . . . 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 225 . . 3  |-  ( n  e.  NN  ->  (
( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n )  <_ 
1 )
33 0re 8792 . . . 4  |-  0  e.  RR
3433, 28elicc2i 10668 . . 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 1141 . 2  |-  ( n  e.  NN  ->  (
( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B } )  /  n )  e.  ( 0 [,] 1
) )
361, 35fmpti 5603 1  |-  F : NN
--> ( 0 [,] 1
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1619    e. wcel 1621   {crab 2520    C_ wss 3113   class class class wbr 3983    e. cmpt 4037   -->wf 4655   ` cfv 4659  (class class class)co 5778    ~<_ cdom 6815   Fincfn 6817   RRcr 8690   0cc0 8691   1c1 8692    x. cmul 8696    < clt 8821    <_ cle 8822    / cdiv 9377   NNcn 9700   NN0cn0 9918   [,]cicc 10611   ...cfz 10734   |_cfl 10876    mod cmo 10925   ^cexp 11056   #chash 11289
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-oadd 6437  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-card 7526  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-n0 9919  df-z 9978  df-uz 10184  df-icc 10615  df-fz 10735  df-hash 11290
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