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Theorem snmlval 31645
Description: The property "𝐴 is simply normal in base 𝑅". A number is simply normal if each digit 0 ≤ 𝑏 < 𝑅 occurs in the base- 𝑅 digit string of 𝐴 with frequency 1 / 𝑅 (which is consistent with the expectation in an infinite random string of numbers selected from 0...𝑅 − 1). (Contributed by Mario Carneiro, 6-Apr-2015.)
Hypothesis
Ref Expression
snml.s 𝑆 = (𝑟 ∈ (ℤ‘2) ↦ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})
Assertion
Ref Expression
snmlval (𝐴 ∈ (𝑆𝑅) ↔ (𝑅 ∈ (ℤ‘2) ∧ 𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))
Distinct variable groups:   𝑘,𝑏,𝑛,𝑥,𝐴   𝑟,𝑏,𝑅,𝑘,𝑛,𝑥
Allowed substitution hints:   𝐴(𝑟)   𝑆(𝑥,𝑘,𝑛,𝑟,𝑏)

Proof of Theorem snmlval
StepHypRef Expression
1 oveq1 6799 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑟 − 1) = (𝑅 − 1))
21oveq2d 6808 . . . . . . . 8 (𝑟 = 𝑅 → (0...(𝑟 − 1)) = (0...(𝑅 − 1)))
3 oveq1 6799 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑅 → (𝑟𝑘) = (𝑅𝑘))
43oveq2d 6808 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → (𝑥 · (𝑟𝑘)) = (𝑥 · (𝑅𝑘)))
5 id 22 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅𝑟 = 𝑅)
64, 5oveq12d 6810 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → ((𝑥 · (𝑟𝑘)) mod 𝑟) = ((𝑥 · (𝑅𝑘)) mod 𝑅))
76fveq2d 6336 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)))
87eqeq1d 2772 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → ((⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏 ↔ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏))
98rabbidv 3338 . . . . . . . . . . . 12 (𝑟 = 𝑅 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏} = {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏})
109fveq2d 6336 . . . . . . . . . . 11 (𝑟 = 𝑅 → (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) = (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}))
1110oveq1d 6807 . . . . . . . . . 10 (𝑟 = 𝑅 → ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) / 𝑛) = ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛))
1211mpteq2dv 4877 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) = (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)))
13 oveq2 6800 . . . . . . . . 9 (𝑟 = 𝑅 → (1 / 𝑟) = (1 / 𝑅))
1412, 13breq12d 4797 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟) ↔ (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))
152, 14raleqbidv 3300 . . . . . . 7 (𝑟 = 𝑅 → (∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟) ↔ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))
1615rabbidv 3338 . . . . . 6 (𝑟 = 𝑅 → {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)} = {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)})
17 snml.s . . . . . 6 𝑆 = (𝑟 ∈ (ℤ‘2) ↦ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})
18 reex 10228 . . . . . . 7 ℝ ∈ V
1918rabex 4943 . . . . . 6 {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)} ∈ V
2016, 17, 19fvmpt 6424 . . . . 5 (𝑅 ∈ (ℤ‘2) → (𝑆𝑅) = {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)})
2120eleq2d 2835 . . . 4 (𝑅 ∈ (ℤ‘2) → (𝐴 ∈ (𝑆𝑅) ↔ 𝐴 ∈ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)}))
22 oveq1 6799 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → (𝑥 · (𝑅𝑘)) = (𝐴 · (𝑅𝑘)))
2322fvoveq1d 6814 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)))
2423eqeq1d 2772 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏 ↔ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏))
2524rabbidv 3338 . . . . . . . . . 10 (𝑥 = 𝐴 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏} = {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏})
2625fveq2d 6336 . . . . . . . . 9 (𝑥 = 𝐴 → (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) = (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}))
2726oveq1d 6807 . . . . . . . 8 (𝑥 = 𝐴 → ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛) = ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛))
2827mpteq2dv 4877 . . . . . . 7 (𝑥 = 𝐴 → (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) = (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)))
2928breq1d 4794 . . . . . 6 (𝑥 = 𝐴 → ((𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅) ↔ (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))
3029ralbidv 3134 . . . . 5 (𝑥 = 𝐴 → (∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅) ↔ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))
3130elrab 3513 . . . 4 (𝐴 ∈ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)} ↔ (𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))
3221, 31syl6bb 276 . . 3 (𝑅 ∈ (ℤ‘2) → (𝐴 ∈ (𝑆𝑅) ↔ (𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅))))
3332pm5.32i 556 . 2 ((𝑅 ∈ (ℤ‘2) ∧ 𝐴 ∈ (𝑆𝑅)) ↔ (𝑅 ∈ (ℤ‘2) ∧ (𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅))))
3417dmmptss 5775 . . . 4 dom 𝑆 ⊆ (ℤ‘2)
35 elfvdm 6361 . . . 4 (𝐴 ∈ (𝑆𝑅) → 𝑅 ∈ dom 𝑆)
3634, 35sseldi 3748 . . 3 (𝐴 ∈ (𝑆𝑅) → 𝑅 ∈ (ℤ‘2))
3736pm4.71ri 542 . 2 (𝐴 ∈ (𝑆𝑅) ↔ (𝑅 ∈ (ℤ‘2) ∧ 𝐴 ∈ (𝑆𝑅)))
38 3anass 1079 . 2 ((𝑅 ∈ (ℤ‘2) ∧ 𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)) ↔ (𝑅 ∈ (ℤ‘2) ∧ (𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅))))
3933, 37, 383bitr4i 292 1 (𝐴 ∈ (𝑆𝑅) ↔ (𝑅 ∈ (ℤ‘2) ∧ 𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  {crab 3064   class class class wbr 4784  cmpt 4861  dom cdm 5249  cfv 6031  (class class class)co 6792  cr 10136  0cc0 10137  1c1 10138   · cmul 10142  cmin 10467   / cdiv 10885  cn 11221  2c2 11271  cuz 11887  ...cfz 12532  cfl 12798   mod cmo 12875  cexp 13066  chash 13320  cli 14422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-cnex 10193  ax-resscn 10194
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795
This theorem is referenced by:  snmlflim  31646
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