Theorem snmlfval

Description: The function F from snmlval 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)

	Ref	Expression
Hypothesis	snmlff.f	$$⊢F=\left(n\in \mathbb{N}⟼\frac{\left|\left\{k\in \left(1\dots n\right)|⌊A{R}^k\mathrm{mod}R⌋=B\right\}\right|}{n}\right)$$
Assertion	snmlfval	$$⊢N\in \mathbb{N}\to F\left(N\right)=\frac{\left|\left\{k\in \left(1\dots N\right)|⌊A{R}^k\mathrm{mod}R⌋=B\right\}\right|}{N}$$

Proof

Step	Hyp	Ref	Expression
1		snmlff.f	$$⊢F=\left(n\in \mathbb{N}⟼\frac{\left|\left\{k\in \left(1\dots n\right)|⌊A{R}^k\mathrm{mod}R⌋=B\right\}\right|}{n}\right)$$
2		oveq2	$$⊢n=N\to \left(1\dots n\right)=\left(1\dots N\right)$$
3	2	rabeqdv	$$⊢n=N\to \left\{k\in \left(1\dots n\right)|⌊A{R}^k\mathrm{mod}R⌋=B\right\}=\left\{k\in \left(1\dots N\right)|⌊A{R}^k\mathrm{mod}R⌋=B\right\}$$
4	3	fveq2d	$$⊢n=N\to \left|\left\{k\in \left(1\dots n\right)|⌊A{R}^k\mathrm{mod}R⌋=B\right\}\right|=\left|\left\{k\in \left(1\dots N\right)|⌊A{R}^k\mathrm{mod}R⌋=B\right\}\right|$$
5		id	$$⊢n=N\to n=N$$
6	4 5	oveq12d	$$⊢n=N\to \frac{\left|\left\{k\in \left(1\dots n\right)|⌊A{R}^k\mathrm{mod}R⌋=B\right\}\right|}{n}=\frac{\left|\left\{k\in \left(1\dots N\right)|⌊A{R}^k\mathrm{mod}R⌋=B\right\}\right|}{N}$$
7		ovex	$$⊢\frac{\left|\left\{k\in \left(1\dots N\right)|⌊A{R}^k\mathrm{mod}R⌋=B\right\}\right|}{N}\in \mathrm{V}$$
8	6 1 7	fvmpt	$$⊢N\in \mathbb{N}\to F\left(N\right)=\frac{\left|\left\{k\in \left(1\dots N\right)|⌊A{R}^k\mathrm{mod}R⌋=B\right\}\right|}{N}$$