Theorem snmlfval 31312

Description: The function F from snmlval 31313 maps N to the relative density of B in the first N digits of the digit string of A in base R. (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
snmlfval	|- ( N e. NN -> ( F ` N ) = ( ( # ` { k e. ( 1 ... N ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) / N ) )

Distinct variable groups:   A, n   B, n   k, n, N   R, n

Allowed substitution hints:   A( k)   B( k)   R( k)   F( k, n)

Proof of Theorem snmlfval

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . 5 |- ( n = N -> ( 1 ... n ) = ( 1 ... N ) )
2		rabeq 3192	. . . . 5 |- ( ( 1 ... n ) = ( 1 ... N ) -> { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } = { k e. ( 1 ... N ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } )
3	1, 2	syl 17	. . . 4 |- ( n = N -> { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } = { k e. ( 1 ... N ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } )
4	3	fveq2d 6195	. . 3 |- ( n = N -> ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) = ( # ` { k e. ( 1 ... N ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) )
5		id 22	. . 3 |- ( n = N -> n = N )
6	4, 5	oveq12d 6668	. 2 |- ( n = N -> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) / n ) = ( ( # ` { k e. ( 1 ... N ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) / N ) )
7		snmlff.f	. 2 |- F = ( n e. NN |-> ( ( # ` { k e. ( 1 ... n ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) / n ) )
8		ovex 6678	. 2 |- ( ( # ` { k e. ( 1 ... N ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) / N ) e. _V
9	6, 7, 8	fvmpt 6282	1 |- ( N e. NN -> ( F ` N ) = ( ( # ` { k e. ( 1 ... N ) | ( |_ ` ( ( A x. ( R ^ k ) ) mod R ) ) = B } ) / N ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   1c1 9937   x. cmul 9941   / cdiv 10684   NNcn 11020   ...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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by: (None)