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Theorem snmlff 23083
Description: The function  F from snmlval 23085 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 10913 . . . . . . 7  |-  ( n  e.  NN  ->  (
1 ... n )  e. 
Fin )
3 ssrab2 3179 . . . . . . 7  |-  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }  C_  ( 1 ... n
)
4 ssfi 6968 . . . . . . 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 11228 . . . . . 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 9898 . . . 4  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  e.  RR )
9 nndivre 9661 . . . 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 9900 . . . 4  |-  ( n  e.  NN  ->  0  <_  ( # `  {
k  e.  ( 1 ... n )  |  ( |_ `  (
( A  x.  ( R ^ k ) )  mod  R ) )  =  B } ) )
12 nnre 9633 . . . 4  |-  ( n  e.  NN  ->  n  e.  RR )
13 nngt0 9655 . . . 4  |-  ( n  e.  NN  ->  0  <  n )
14 divge0 9505 . . . 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 6793 . . . . . . . 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 11239 . . . . . . . 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 9851 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  NN0 )
22 hashfz1 11223 . . . . . . 7  |-  ( n  e.  NN0  ->  ( # `  ( 1 ... n
) )  =  n )
2321, 22syl 17 . . . . . 6  |-  ( n  e.  NN  ->  ( # `
 ( 1 ... n ) )  =  n )
2420, 23breqtrd 3944 . . . . 5  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  <_  n )
25 nncn 9634 . . . . . 6  |-  ( n  e.  NN  ->  n  e.  CC )
2625mulid1d 8732 . . . . 5  |-  ( n  e.  NN  ->  (
n  x.  1 )  =  n )
2724, 26breqtrrd 3946 . . . 4  |-  ( n  e.  NN  ->  ( # `
 { k  e.  ( 1 ... n
)  |  ( |_
`  ( ( A  x.  ( R ^
k ) )  mod 
R ) )  =  B } )  <_ 
( n  x.  1 ) )
28 1re 8717 . . . . . 6  |-  1  e.  RR
2928a1i 12 . . . . 5  |-  ( n  e.  NN  ->  1  e.  RR )
30 ledivmul 9509 . . . . 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 8718 . . . 4  |-  0  e.  RR
3433, 28elicc2i 10594 . . 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 5535 1  |-  F : NN
--> ( 0 [,] 1
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1619    e. wcel 1621   {crab 2512    C_ wss 3078   class class class wbr 3920    e. cmpt 3974   -->wf 4588   ` cfv 4592  (class class class)co 5710    ~<_ cdom 6747   Fincfn 6749   RRcr 8616   0cc0 8617   1c1 8618    x. cmul 8622    < clt 8747    <_ cle 8748    / cdiv 9303   NNcn 9626   NN0cn0 9844   [,]cicc 10537   ...cfz 10660   |_cfl 10802    mod cmo 10851   ^cexp 10982   #chash 11215
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-n0 9845  df-z 9904  df-uz 10110  df-icc 10541  df-fz 10661  df-hash 11216
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