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Theorem snmlval 31693
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 6849 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑟 − 1) = (𝑅 − 1))
21oveq2d 6858 . . . . . . . 8 (𝑟 = 𝑅 → (0...(𝑟 − 1)) = (0...(𝑅 − 1)))
3 oveq1 6849 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → (𝑟𝑘) = (𝑅𝑘))
43oveq2d 6858 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → (𝑥 · (𝑟𝑘)) = (𝑥 · (𝑅𝑘)))
5 id 22 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅𝑟 = 𝑅)
64, 5oveq12d 6860 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → ((𝑥 · (𝑟𝑘)) mod 𝑟) = ((𝑥 · (𝑅𝑘)) mod 𝑅))
76fveqeq2d 6383 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → ((⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏 ↔ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏))
87rabbidv 3338 . . . . . . . . . . . 12 (𝑟 = 𝑅 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏} = {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏})
98fveq2d 6379 . . . . . . . . . . 11 (𝑟 = 𝑅 → (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) = (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}))
109oveq1d 6857 . . . . . . . . . 10 (𝑟 = 𝑅 → ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) / 𝑛) = ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛))
1110mpteq2dv 4904 . . . . . . . . 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)))
12 oveq2 6850 . . . . . . . . 9 (𝑟 = 𝑅 → (1 / 𝑟) = (1 / 𝑅))
1311, 12breq12d 4822 . . . . . . . 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)))
142, 13raleqbidv 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)))
1514rabbidv 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)})
16 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)})
17 reex 10280 . . . . . . 7 ℝ ∈ V
1817rabex 4973 . . . . . 6 {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)} ∈ V
1915, 16, 18fvmpt 6471 . . . . 5 (𝑅 ∈ (ℤ‘2) → (𝑆𝑅) = {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)})
2019eleq2d 2830 . . . 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)}))
21 oveq1 6849 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → (𝑥 · (𝑅𝑘)) = (𝐴 · (𝑅𝑘)))
2221fvoveq1d 6864 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)))
2322eqeq1d 2767 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏 ↔ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏))
2423rabbidv 3338 . . . . . . . . . 10 (𝑥 = 𝐴 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏} = {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏})
2524fveq2d 6379 . . . . . . . . 9 (𝑥 = 𝐴 → (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) = (♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}))
2625oveq1d 6857 . . . . . . . 8 (𝑥 = 𝐴 → ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛) = ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛))
2726mpteq2dv 4904 . . . . . . 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)))
2827breq1d 4819 . . . . . 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)))
2928ralbidv 3133 . . . . 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)))
3029elrab 3519 . . . 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)))
3120, 30syl6bb 278 . . 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))))
3231pm5.32i 570 . 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))))
3316dmmptss 5817 . . . 4 dom 𝑆 ⊆ (ℤ‘2)
34 elfvdm 6407 . . . 4 (𝐴 ∈ (𝑆𝑅) → 𝑅 ∈ dom 𝑆)
3533, 34sseldi 3759 . . 3 (𝐴 ∈ (𝑆𝑅) → 𝑅 ∈ (ℤ‘2))
3635pm4.71ri 556 . 2 (𝐴 ∈ (𝑆𝑅) ↔ (𝑅 ∈ (ℤ‘2) ∧ 𝐴 ∈ (𝑆𝑅)))
37 3anass 1116 . 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))))
3832, 36, 373bitr4i 294 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  {crab 3059   class class class wbr 4809  cmpt 4888  dom cdm 5277  cfv 6068  (class class class)co 6842  cr 10188  0cc0 10189  1c1 10190   · cmul 10194  cmin 10520   / cdiv 10938  cn 11274  2c2 11327  cuz 11886  ...cfz 12533  cfl 12799   mod cmo 12876  cexp 13067  chash 13321  cli 14502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-cnex 10245  ax-resscn 10246
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845
This theorem is referenced by:  snmlflim  31694
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