Theorem snmlff 31311

Description: The function F from snmlval 31313 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

Step	Hyp	Ref	Expression
1		snmlff.f	. 2 |- F = ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) / n ) )
2		fzfid 12772	. . . . . . 7 |- ( n e. NN -> ( 1 ... n ) e. Fin )
3		ssrab2 3687	. . . . . . 7 |- { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } C_ ( 1 ... n )
4		ssfi 8180	. . . . . . 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 )
5	2, 3, 4	sylancl 694	. . . . . 6 |- ( n e. NN -> { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } e. Fin )
6		hashcl 13147	. . . . . 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 )
7	5, 6	syl 17	. . . . 5 |- ( n e. NN -> ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) e. NN0 )
8	7	nn0red 11352	. . . 4 |- ( n e. NN -> ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) e. RR )
9		nndivre 11056	. . . 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 )
10	8, 9	mpancom 703	. . 3 |- ( n e. NN -> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) / n ) e. RR )
11	7	nn0ge0d 11354	. . . 4 |- ( n e. NN -> 0 <_ ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) )
12		nnre 11027	. . . 4 |- ( n e. NN -> n e. RR )
13		nngt0 11049	. . . 4 |- ( n e. NN -> 0 < n )
14		divge0 10892	. . . 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 ) )
15	8, 11, 12, 13, 14	syl22anc 1327	. . 3 |- ( n e. NN -> 0 <_ ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) / n ) )
16		ssdomg 8001	. . . . . . . 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 ) ) )
17	2, 3, 16	mpisyl 21	. . . . . . 7 |- ( n e. NN -> { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ~<_ ( 1 ... n ) )
18		hashdom 13168	. . . . . . . 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 ) ) )
19	5, 2, 18	syl2anc 693	. . . . . . 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 ) ) )
20	17, 19	mpbird 247	. . . . . 6 |- ( n e. NN -> ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) <_ ( # ` ( 1 ... n ) ) )
21		nnnn0 11299	. . . . . . 7 |- ( n e. NN -> n e. NN0 )
22		hashfz1 13134	. . . . . . 7 |- ( n e. NN0 -> ( # ` ( 1 ... n ) ) = n )
23	21, 22	syl 17	. . . . . 6 |- ( n e. NN -> ( # ` ( 1 ... n ) ) = n )
24	20, 23	breqtrd 4679	. . . . 5 |- ( n e. NN -> ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) <_ n )
25		nncn 11028	. . . . . 6 |- ( n e. NN -> n e. CC )
26	25	mulid1d 10057	. . . . 5 |- ( n e. NN -> ( n x. 1 ) = n )
27	24, 26	breqtrrd 4681	. . . 4 |- ( n e. NN -> ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) <_ ( n x. 1 ) )
28		1red 10055	. . . . 5 |- ( n e. NN -> 1 e. RR )
29		ledivmul 10899	. . . . 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 ) ) )
30	8, 28, 12, 13, 29	syl112anc 1330	. . . 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 ) ) )
31	27, 30	mpbird 247	. . 3 |- ( n e. NN -> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) / n ) <_ 1 )
32		0re 10040	. . . 4 |- 0 e. RR
33		1re 10039	. . . 4 |- 1 e. RR
34	32, 33	elicc2i 12239	. . 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 ) )
35	10, 15, 31, 34	syl3anbrc 1246	. 2 |- ( n e. NN -> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) / n ) e. ( 0 [,] 1 ) )
36	1, 35	fmpti 6383	1 |- F : NN --> ( 0 [,] 1 )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ~<_ cdom 7953   Fincfn 7955   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020   NN0cn0 11292   [,]cicc 12178   ...cfz 12326   |_cfl 12591   mod cmo 12668   ^cexp 12860   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-icc 12182  df-fz 12327  df-hash 13118

This theorem is referenced by: (None)