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Theorem snmlval 31021
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 6611 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑟 − 1) = (𝑅 − 1))
21oveq2d 6620 . . . . . . . 8 (𝑟 = 𝑅 → (0...(𝑟 − 1)) = (0...(𝑅 − 1)))
3 oveq1 6611 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑅 → (𝑟𝑘) = (𝑅𝑘))
43oveq2d 6620 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → (𝑥 · (𝑟𝑘)) = (𝑥 · (𝑅𝑘)))
5 id 22 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅𝑟 = 𝑅)
64, 5oveq12d 6622 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → ((𝑥 · (𝑟𝑘)) mod 𝑟) = ((𝑥 · (𝑅𝑘)) mod 𝑅))
76fveq2d 6152 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)))
87eqeq1d 2623 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → ((⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏 ↔ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏))
98rabbidv 3177 . . . . . . . . . . . 12 (𝑟 = 𝑅 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏} = {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏})
109fveq2d 6152 . . . . . . . . . . 11 (𝑟 = 𝑅 → (#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) = (#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}))
1110oveq1d 6619 . . . . . . . . . 10 (𝑟 = 𝑅 → ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) / 𝑛) = ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛))
1211mpteq2dv 4705 . . . . . . . . 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 6612 . . . . . . . . 9 (𝑟 = 𝑅 → (1 / 𝑟) = (1 / 𝑅))
1412, 13breq12d 4626 . . . . . . . 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 3141 . . . . . . 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 3177 . . . . . 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 9971 . . . . . . 7 ℝ ∈ V
1918rabex 4773 . . . . . 6 {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)} ∈ V
2016, 17, 19fvmpt 6239 . . . . 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 2684 . . . 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 6611 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → (𝑥 · (𝑅𝑘)) = (𝐴 · (𝑅𝑘)))
2322oveq1d 6619 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → ((𝑥 · (𝑅𝑘)) mod 𝑅) = ((𝐴 · (𝑅𝑘)) mod 𝑅))
2423fveq2d 6152 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)))
2524eqeq1d 2623 . . . . . . . . . . 11 (𝑥 = 𝐴 → ((⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏 ↔ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏))
2625rabbidv 3177 . . . . . . . . . 10 (𝑥 = 𝐴 → {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏} = {𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏})
2726fveq2d 6152 . . . . . . . . 9 (𝑥 = 𝐴 → (#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) = (#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}))
2827oveq1d 6619 . . . . . . . 8 (𝑥 = 𝐴 → ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛) = ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛))
2928mpteq2dv 4705 . . . . . . 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)))
3029breq1d 4623 . . . . . 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)))
3130ralbidv 2980 . . . . 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)))
3231elrab 3346 . . . 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)))
3321, 32syl6bb 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))))
3433pm5.32i 668 . 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))))
3517dmmptss 5590 . . . 4 dom 𝑆 ⊆ (ℤ‘2)
36 elfvdm 6177 . . . 4 (𝐴 ∈ (𝑆𝑅) → 𝑅 ∈ dom 𝑆)
3735, 36sseldi 3581 . . 3 (𝐴 ∈ (𝑆𝑅) → 𝑅 ∈ (ℤ‘2))
3837pm4.71ri 664 . 2 (𝐴 ∈ (𝑆𝑅) ↔ (𝑅 ∈ (ℤ‘2) ∧ 𝐴 ∈ (𝑆𝑅)))
39 3anass 1040 . 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))))
4034, 38, 393bitr4i 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911   class class class wbr 4613  cmpt 4673  dom cdm 5074  cfv 5847  (class class class)co 6604  cr 9879  0cc0 9880  1c1 9881   · cmul 9885  cmin 10210   / cdiv 10628  cn 10964  2c2 11014  cuz 11631  ...cfz 12268  cfl 12531   mod cmo 12608  cexp 12800  #chash 13057  cli 14149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-cnex 9936  ax-resscn 9937
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607
This theorem is referenced by:  snmlflim  31022
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