P. 345 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34401-34500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eulerpartlemgu 34401*	Lemma for eulerpart 34406: Rewriting the 𝑈 set for an odd partition Note that interestingly, this proof reuses marypha2lem2 9330. (Contributed by Thierry Arnoux, 10-Aug-2018.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    &   ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))    &   ⊢ 𝑈 = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))    ⇒   ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝑈 = {⟨𝑡, 𝑛⟩ ∣ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)‘𝑡))})
Theorem	eulerpartlemgh 34402*	Lemma for eulerpart 34406: The 𝐹 function is a bijection on the 𝑈 subsets. (Contributed by Thierry Arnoux, 15-Aug-2018.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    &   ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))    &   ⊢ 𝑈 = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡)))    ⇒   ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐹 ↾ 𝑈):𝑈–1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚})
Theorem	eulerpartlemgf 34403*	Lemma for eulerpart 34406: Images under 𝐺 have finite support. (Contributed by Thierry Arnoux, 29-Aug-2018.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    &   ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))    ⇒   ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡(𝐺‘𝐴) “ ℕ) ∈ Fin)
Theorem	eulerpartlemgs2 34404*	Lemma for eulerpart 34406: The 𝐺 function also preserves partition sums. (Contributed by Thierry Arnoux, 10-Sep-2017.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    &   ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))    &   ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑆‘(𝐺‘𝐴)) = (𝑆‘𝐴))
Theorem	eulerpartlemn 34405*	Lemma for eulerpart 34406. (Contributed by Thierry Arnoux, 30-Aug-2018.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    &   ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥))    &   ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin}    &   ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})    &   ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽}    &   ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽))))))    &   ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘))    ⇒   ⊢ (𝐺 ↾ 𝑂):𝑂–1-1-onto→𝐷
Theorem	eulerpart 34406*	Euler's theorem on partitions, also known as a special case of Glaisher's theorem. Let 𝑃 be the set of all partitions of 𝑁, represented as multisets of positive integers, which is to say functions from ℕ to ℕ0 where the value of the function represents the number of repetitions of an individual element, and the sum of all the elements with repetition equals 𝑁. Then the set 𝑂 of all partitions that only consist of odd numbers and the set 𝐷 of all partitions which have no repeated elements have the same cardinality. This is Metamath 100 proof #45. (Contributed by Thierry Arnoux, 14-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.)
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)}    &   ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛}    &   ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1}    ⇒   ⊢ (♯‘𝑂) = (♯‘𝐷)
21.3.21  Sequences defined by strong recursion
Syntax	csseq 34407	Sequences defined by strong recursion.
class seqstr
Definition	df-sseq 34408*	Define a builder for sequences by strong recursion, i.e., by computing the value of the n-th element of the sequence from all preceding elements and not just the previous one. (Contributed by Thierry Arnoux, 21-Apr-2019.)
⊢ seqstr = (𝑚 ∈ V, 𝑓 ∈ V ↦ (𝑚 ∪ (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝑓‘𝑥)”⟩)), (ℕ0 × {(𝑚 ++ ⟨“(𝑓‘𝑚)”⟩)})))))
Theorem	subiwrd 34409	Lemma for sseqp1 34419. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝐹:ℕ0⟶𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐹 ↾ (0..^𝑁)) ∈ Word 𝑆)
Theorem	subiwrdlen 34410	Length of a subword of an infinite word. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝐹:ℕ0⟶𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (♯‘(𝐹 ↾ (0..^𝑁))) = 𝑁)
Theorem	iwrdsplit 34411	Lemma for sseqp1 34419. (Contributed by Thierry Arnoux, 25-Apr-2019.) (Proof shortened by AV, 14-Oct-2022.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝐹:ℕ0⟶𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐹 ↾ (0..^(𝑁 + 1))) = ((𝐹 ↾ (0..^𝑁)) ++ ⟨“(𝐹‘𝑁)”⟩))
Theorem	sseqval 34412*	Value of the strong sequence builder function. The set 𝑊 represents here the words of length greater than or equal to the lenght of the initial sequence 𝑀. (Contributed by Thierry Arnoux, 21-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    ⇒   ⊢ (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)})))))
Theorem	sseqfv1 34413	Value of the strong sequence builder function at one of its initial values. (Contributed by Thierry Arnoux, 21-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝑀)))    ⇒   ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (𝑀‘𝑁))
Theorem	sseqfn 34414	A strong recursive sequence is a function over the nonnegative integers. (Contributed by Thierry Arnoux, 23-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    ⇒   ⊢ (𝜑 → (𝑀seqstr𝐹) Fn ℕ0)
Theorem	sseqmw 34415	Lemma for sseqf 34416 amd sseqp1 34419. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    ⇒   ⊢ (𝜑 → 𝑀 ∈ 𝑊)
Theorem	sseqf 34416	A strong recursive sequence is a function over the nonnegative integers. (Contributed by Thierry Arnoux, 23-Apr-2019.) (Proof shortened by AV, 7-Mar-2022.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    ⇒   ⊢ (𝜑 → (𝑀seqstr𝐹):ℕ0⟶𝑆)
Theorem	sseqfres 34417	The first elements in the strong recursive sequence are the sequence initializer. (Contributed by Thierry Arnoux, 23-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    ⇒   ⊢ (𝜑 → ((𝑀seqstr𝐹) ↾ (0..^(♯‘𝑀))) = 𝑀)
Theorem	sseqfv2 34418*	Value of the strong sequence builder function. (Contributed by Thierry Arnoux, 21-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(♯‘𝑀)))    ⇒   ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (lastS‘(seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ ⟨“(𝐹‘𝑥)”⟩)), (ℕ0 × {(𝑀 ++ ⟨“(𝐹‘𝑀)”⟩)}))‘𝑁)))
Theorem	sseqp1 34419	Value of the strong sequence builder function at a successor. (Contributed by Thierry Arnoux, 24-Apr-2019.)
⊢ (𝜑 → 𝑆 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ Word 𝑆)    &   ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀))))    &   ⊢ (𝜑 → 𝐹:𝑊⟶𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(♯‘𝑀)))    ⇒   ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (𝐹‘((𝑀seqstr𝐹) ↾ (0..^𝑁))))
21.3.22  Fibonacci Numbers
Syntax	cfib 34420	The Fibonacci sequence.
class Fibci
Definition	df-fib 34421	Define the Fibonacci sequence, where that each element is the sum of the two preceding ones, starting from 0 and 1. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ Fibci = (⟨“01”⟩seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))))
Theorem	fiblem 34422	Lemma for fib0 34423, fib1 34424 and fibp1 34425. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))):(Word ℕ0 ∩ (◡♯ “ (ℤ≥‘(♯‘⟨“01”⟩))))⟶ℕ0
Theorem	fib0 34423	Value of the Fibonacci sequence at index 0. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (Fibci‘0) = 0
Theorem	fib1 34424	Value of the Fibonacci sequence at index 1. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (Fibci‘1) = 1
Theorem	fibp1 34425	Value of the Fibonacci sequence at higher indices. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (𝑁 ∈ ℕ → (Fibci‘(𝑁 + 1)) = ((Fibci‘(𝑁 − 1)) + (Fibci‘𝑁)))
Theorem	fib2 34426	Value of the Fibonacci sequence at index 2. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (Fibci‘2) = 1
Theorem	fib3 34427	Value of the Fibonacci sequence at index 3. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (Fibci‘3) = 2
Theorem	fib4 34428	Value of the Fibonacci sequence at index 4. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (Fibci‘4) = 3
Theorem	fib5 34429	Value of the Fibonacci sequence at index 5. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (Fibci‘5) = 5
Theorem	fib6 34430	Value of the Fibonacci sequence at index 6. (Contributed by Thierry Arnoux, 25-Apr-2019.)
⊢ (Fibci‘6) = 8
21.3.23  Probability
21.3.23.1  Probability Theory
Syntax	cprb 34431	Extend class notation to include the class of probability measures.
class Prob
Definition	df-prob 34432	Define the class of probability measures as the set of measures with total measure 1. (Contributed by Thierry Arnoux, 14-Sep-2016.)
⊢ Prob = {𝑝 ∈ ∪ ran measures ∣ (𝑝‘∪ dom 𝑝) = 1}
Theorem	elprob 34433	The property of being a probability measure. (Contributed by Thierry Arnoux, 8-Dec-2016.)
⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1))
Theorem	domprobmeas 34434	A probability measure is a measure on its domain. (Contributed by Thierry Arnoux, 23-Dec-2016.)
⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
Theorem	domprobsiga 34435	The domain of a probability measure is a sigma-algebra. (Contributed by Thierry Arnoux, 23-Dec-2016.)
⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
Theorem	probtot 34436	The probability of the universe set is 1. Second axiom of Kolmogorov. (Contributed by Thierry Arnoux, 8-Dec-2016.)
⊢ (𝑃 ∈ Prob → (𝑃‘∪ dom 𝑃) = 1)
Theorem	prob01 34437	A probability is an element of [ 0 , 1 ]. First axiom of Kolmogorov. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃) → (𝑃‘𝐴) ∈ (0[,]1))
Theorem	probnul 34438	The probability of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ (𝑃 ∈ Prob → (𝑃‘∅) = 0)
Theorem	unveldomd 34439	The universe is an element of the domain of the probability, the universe (entire probability space) being ∪ dom 𝑃 in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    ⇒   ⊢ (𝜑 → ∪ dom 𝑃 ∈ dom 𝑃)
Theorem	unveldom 34440	The universe is an element of the domain of the probability, the universe (entire probability space) being ∪ dom 𝑃 in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
⊢ (𝑃 ∈ Prob → ∪ dom 𝑃 ∈ dom 𝑃)
Theorem	nuleldmp 34441	The empty set is an element of the domain of the probability. (Contributed by Thierry Arnoux, 22-Jan-2017.)
⊢ (𝑃 ∈ Prob → ∅ ∈ dom 𝑃)
Theorem	probcun 34442*	The probability of the union of a countable disjoint set of events is the sum of their probabilities. (Third axiom of Kolmogorov) Here, the Σ construct cannot be used as it can handle infinite indexing set only if they are subsets of ℤ, which is not the case here. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 ≼ ω ∧ Disj 𝑥 ∈ 𝐴 𝑥)) → (𝑃‘∪ 𝐴) = Σ*𝑥 ∈ 𝐴(𝑃‘𝑥))
Theorem	probun 34443	The probability of the union two incompatible events is the sum of their probabilities. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ((𝐴 ∩ 𝐵) = ∅ → (𝑃‘(𝐴 ∪ 𝐵)) = ((𝑃‘𝐴) + (𝑃‘𝐵))))
Theorem	probdif 34444	The probability of the difference of two event sets. (Contributed by Thierry Arnoux, 12-Dec-2016.)
⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → (𝑃‘(𝐴 ∖ 𝐵)) = ((𝑃‘𝐴) − (𝑃‘(𝐴 ∩ 𝐵))))
Theorem	probinc 34445	A probability law is increasing with regard to event set inclusion. (Contributed by Thierry Arnoux, 10-Feb-2017.)
⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ 𝐴 ⊆ 𝐵) → (𝑃‘𝐴) ≤ (𝑃‘𝐵))
Theorem	probdsb 34446	The probability of the complement of a set. That is, the probability that the event 𝐴 does not occur. (Contributed by Thierry Arnoux, 15-Dec-2016.)
⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃) → (𝑃‘(∪ dom 𝑃 ∖ 𝐴)) = (1 − (𝑃‘𝐴)))
Theorem	probmeasd 34447	A probability measure is a measure. (Contributed by Thierry Arnoux, 2-Feb-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    ⇒   ⊢ (𝜑 → 𝑃 ∈ ∪ ran measures)
Theorem	probvalrnd 34448	The value of a probability is a real number. (Contributed by Thierry Arnoux, 2-Feb-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝑃)    ⇒   ⊢ (𝜑 → (𝑃‘𝐴) ∈ ℝ)
Theorem	probtotrnd 34449	The probability of the universe set is finite. (Contributed by Thierry Arnoux, 2-Feb-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    ⇒   ⊢ (𝜑 → (𝑃‘∪ dom 𝑃) ∈ ℝ)
Theorem	totprobd 34450*	Law of total probability, deduction form. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝒫 dom 𝑃)    &   ⊢ (𝜑 → ∪ 𝐵 = ∪ dom 𝑃)    &   ⊢ (𝜑 → 𝐵 ≼ ω)    &   ⊢ (𝜑 → Disj 𝑏 ∈ 𝐵 𝑏)    ⇒   ⊢ (𝜑 → (𝑃‘𝐴) = Σ*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴)))
Theorem	totprob 34451*	Law of total probability. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ (∪ 𝐵 = ∪ dom 𝑃 ∧ 𝐵 ∈ 𝒫 dom 𝑃 ∧ (𝐵 ≼ ω ∧ Disj 𝑏 ∈ 𝐵 𝑏))) → (𝑃‘𝐴) = Σ*𝑏 ∈ 𝐵(𝑃‘(𝑏 ∩ 𝐴)))
Theorem	probfinmeasb 34452	Build a probability measure from a finite measure. (Contributed by Thierry Arnoux, 31-Jan-2017.)
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑀 ∘f/c /𝑒 (𝑀‘∪ 𝑆)) ∈ Prob)
Theorem	probfinmeasbALTV 34453*	Alternate version of probfinmeasb 34452. (Contributed by Thierry Arnoux, 17-Dec-2016.) (New usage is discouraged.)
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝑀‘∪ 𝑆) ∈ ℝ+) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) ∈ Prob)
Theorem	probmeasb 34454*	Build a probability from a measure and a set with finite measure. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆 ∧ (𝑀‘𝐴) ∈ ℝ+) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘(𝑥 ∩ 𝐴)) / (𝑀‘𝐴))) ∈ Prob)
21.3.23.2  Conditional Probabilities
Syntax	ccprob 34455	Extends class notation with the conditional probability builder.
class cprob
Definition	df-cndprob 34456*	Define the conditional probability. (Contributed by Thierry Arnoux, 14-Sep-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
⊢ cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎 ∩ 𝑏)) / (𝑝‘𝑏))))
Theorem	cndprobval 34457	The value of the conditional probability , i.e. the probability for the event 𝐴, given 𝐵, under the probability law 𝑃. (Contributed by Thierry Arnoux, 21-Jan-2017.)
⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) → ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩) = ((𝑃‘(𝐴 ∩ 𝐵)) / (𝑃‘𝐵)))
Theorem	cndprobin 34458	An identity linking conditional probability and intersection. (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑃‘𝐵) ≠ 0) → (((cprob‘𝑃)‘⟨𝐴, 𝐵⟩) · (𝑃‘𝐵)) = (𝑃‘(𝐴 ∩ 𝐵)))
Theorem	cndprob01 34459	The conditional probability has values in [0, 1]. (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑃‘𝐵) ≠ 0) → ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩) ∈ (0[,]1))
Theorem	cndprobtot 34460	The conditional probability given a certain event is one. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ (𝑃‘𝐴) ≠ 0) → ((cprob‘𝑃)‘⟨∪ dom 𝑃, 𝐴⟩) = 1)
Theorem	cndprobnul 34461	The conditional probability given empty event is zero. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
⊢ ((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ (𝑃‘𝐴) ≠ 0) → ((cprob‘𝑃)‘⟨∅, 𝐴⟩) = 0)
Theorem	cndprobprob 34462*	The conditional probability defines a probability law. (Contributed by Thierry Arnoux, 23-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
⊢ ((𝑃 ∈ Prob ∧ 𝐵 ∈ dom 𝑃 ∧ (𝑃‘𝐵) ≠ 0) → (𝑎 ∈ dom 𝑃 ↦ ((cprob‘𝑃)‘⟨𝑎, 𝐵⟩)) ∈ Prob)
Theorem	bayesth 34463	Bayes Theorem. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
⊢ (((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ 𝐵 ∈ dom 𝑃) ∧ (𝑃‘𝐴) ≠ 0 ∧ (𝑃‘𝐵) ≠ 0) → ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩) = ((((cprob‘𝑃)‘⟨𝐵, 𝐴⟩) · (𝑃‘𝐴)) / (𝑃‘𝐵)))
21.3.23.3  Real-valued Random Variables
Syntax	crrv 34464	Extend class notation with the class of real-valued random variables.
class rRndVar
Definition	df-rrv 34465	In its generic definition, a random variable is a measurable function from a probability space to a Borel set. Here, we specifically target real-valued random variables, i.e. measurable function from a probability space to the Borel sigma-algebra on the set of real numbers. (Contributed by Thierry Arnoux, 20-Sep-2016.) (Revised by Thierry Arnoux, 25-Jan-2017.)
⊢ rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅ℝ))
Theorem	rrvmbfm 34466	A real-valued random variable is a measurable function from its sample space to the Borel sigma-algebra. (Contributed by Thierry Arnoux, 25-Jan-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ)))
Theorem	isrrvv 34467*	Elementhood to the set of real-valued random variables with respect to the probability 𝑃. (Contributed by Thierry Arnoux, 25-Jan-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ (𝑋:∪ dom 𝑃⟶ℝ ∧ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃)))
Theorem	rrvvf 34468	A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    ⇒   ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ)
Theorem	rrvfn 34469	A real-valued random variable is a function over the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    ⇒   ⊢ (𝜑 → 𝑋 Fn ∪ dom 𝑃)
Theorem	rrvdm 34470	The domain of a random variable is the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    ⇒   ⊢ (𝜑 → dom 𝑋 = ∪ dom 𝑃)
Theorem	rrvrnss 34471	The range of a random variable as a subset of ℝ. (Contributed by Thierry Arnoux, 6-Feb-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    ⇒   ⊢ (𝜑 → ran 𝑋 ⊆ ℝ)
Theorem	rrvf2 34472	A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    ⇒   ⊢ (𝜑 → 𝑋:dom 𝑋⟶ℝ)
Theorem	rrvdmss 34473	The domain of a random variable. This is useful to shorten proofs. (Contributed by Thierry Arnoux, 25-Jan-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    ⇒   ⊢ (𝜑 → ∪ dom 𝑃 ⊆ dom 𝑋)
Theorem	rrvfinvima 34474*	For a real-value random variable 𝑋, any open interval in ℝ is the image of a measurable set. (Contributed by Thierry Arnoux, 25-Jan-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    ⇒   ⊢ (𝜑 → ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃)
Theorem	0rrv 34475*	The constant function equal to zero is a random variable. (Contributed by Thierry Arnoux, 16-Jan-2017.) (Revised by Thierry Arnoux, 30-Jan-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    ⇒   ⊢ (𝜑 → (𝑥 ∈ ∪ dom 𝑃 ↦ 0) ∈ (rRndVar‘𝑃))
Theorem	rrvadd 34476	The sum of two random variables is a random variable. (Contributed by Thierry Arnoux, 4-Jun-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝑌 ∈ (rRndVar‘𝑃))    ⇒   ⊢ (𝜑 → (𝑋 ∘f + 𝑌) ∈ (rRndVar‘𝑃))
Theorem	rrvmulc 34477	A random variable multiplied by a constant is a random variable. (Contributed by Thierry Arnoux, 17-Jan-2017.) (Revised by Thierry Arnoux, 22-May-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝑋 ∘f/c · 𝐶) ∈ (rRndVar‘𝑃))
Theorem	rrvsum 34478	An indexed sum of random variables is a random variable. (Contributed by Thierry Arnoux, 22-May-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋:ℕ⟶(rRndVar‘𝑃))    &   ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑆 = (seq1( ∘f + , 𝑋)‘𝑁))    ⇒   ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑆 ∈ (rRndVar‘𝑃))
Theorem	boolesineq 34479*	Boole's inequality (union bound). For any finite or countable collection of events, the probability of their union is at most the sum of their probabilities. (Suggested by DeepSeek R1.) (Contributed by Ender Ting, 30-Apr-2025.)
⊢ ((𝑃 ∈ Prob ∧ 𝐴:ℕ⟶dom 𝑃) → (𝑃‘∪ 𝑛 ∈ ℕ (𝐴‘𝑛)) ≤ Σ*𝑛 ∈ ℕ(𝑃‘(𝐴‘𝑛)))
21.3.23.4  Preimage set mapping operator
Syntax	corvc 34480	Extend class notation to include the preimage set mapping operator.
class ∘RV/𝑐𝑅
Definition	df-orvc 34481*	Define the preimage set mapping operator. In probability theory, the notation 𝑃(𝑋 = 𝐴) denotes the probability that a random variable 𝑋 takes the value 𝐴. We introduce here an operator which enables to write this in Metamath as (𝑃‘(𝑋∘RV/𝑐 I 𝐴)), and keep a similar notation. Because with this notation (𝑋∘RV/𝑐 I 𝐴) is a set, we can also apply it to conditional probabilities, like in (𝑃‘(𝑋∘RV/𝑐 I 𝐴) ∣ (𝑌∘RV/𝑐 I 𝐵))). The oRVC operator transforms a relation 𝑅 into an operation taking a random variable 𝑋 and a constant 𝐶, and returning the preimage through 𝑋 of the equivalence class of 𝐶. The most commonly used relations are: - equality: {𝑋 = 𝐴} as (𝑋∘RV/𝑐 I 𝐴) cf. ideq 5799- elementhood: {𝑋 ∈ 𝐴} as (𝑋∘RV/𝑐 E 𝐴) cf. epel 5524- less-than: {𝑋 ≤ 𝐴} as (𝑋∘RV/𝑐 ≤ 𝐴) Even though it is primarily designed to be used within probability theory and with random variables, this operator is defined on generic functions, and could be used in other fields, e.g., for continuous functions. (Contributed by Thierry Arnoux, 15-Jan-2017.)
⊢ ∘RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}))
Theorem	orvcval 34482*	Value of the preimage mapping operator applied on a given random variable and constant. (Contributed by Thierry Arnoux, 19-Jan-2017.)
⊢ (𝜑 → Fun 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))
Theorem	orvcval2 34483*	Another way to express the value of the preimage mapping operator. (Contributed by Thierry Arnoux, 19-Jan-2017.)
⊢ (𝜑 → Fun 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴})
Theorem	elorvc 34484*	Elementhood of a preimage. (Contributed by Thierry Arnoux, 21-Jan-2017.)
⊢ (𝜑 → Fun 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    ⇒   ⊢ ((𝜑 ∧ 𝑧 ∈ dom 𝑋) → (𝑧 ∈ (𝑋∘RV/𝑐𝑅𝐴) ↔ (𝑋‘𝑧)𝑅𝐴))
Theorem	orvcval4 34485*	The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 34482. (Contributed by Thierry Arnoux, 21-Jan-2017.)
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}))
Theorem	orvcoel 34486*	If the relation produces open sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} ∈ 𝐽)    ⇒   ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) ∈ 𝑆)
Theorem	orvccel 34487*	If the relation produces closed sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} ∈ (Clsd‘𝐽))    ⇒   ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) ∈ 𝑆)
Theorem	elorrvc 34488*	Elementhood of a preimage for a real-valued random variable. (Contributed by Thierry Arnoux, 21-Jan-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ dom 𝑃) → (𝑧 ∈ (𝑋∘RV/𝑐𝑅𝐴) ↔ (𝑋‘𝑧)𝑅𝐴))
Theorem	orrvcval4 34489*	The value of the preimage mapping operator can be restricted to preimages of subsets of ℝ. (Contributed by Thierry Arnoux, 21-Jan-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴}))
Theorem	orrvcoel 34490*	If the relation produces open sets, preimage maps of a random variable are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴} ∈ (topGen‘ran (,)))    ⇒   ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) ∈ dom 𝑃)
Theorem	orrvccel 34491*	If the relation produces closed sets, preimage maps are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴} ∈ (Clsd‘(topGen‘ran (,))))    ⇒   ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) ∈ dom 𝑃)
Theorem	orvcgteel 34492	Preimage maps produced by the "greater than or equal to" relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝑋∘RV/𝑐◡ ≤ 𝐴) ∈ dom 𝑃)
21.3.23.5  Distribution Functions
Theorem	orvcelval 34493	Preimage maps produced by the membership relation. (Contributed by Thierry Arnoux, 6-Feb-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝐴 ∈ 𝔅ℝ)    ⇒   ⊢ (𝜑 → (𝑋∘RV/𝑐 E 𝐴) = (◡𝑋 “ 𝐴))
Theorem	orvcelel 34494	Preimage maps produced by the membership relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝐴 ∈ 𝔅ℝ)    ⇒   ⊢ (𝜑 → (𝑋∘RV/𝑐 E 𝐴) ∈ dom 𝑃)
Theorem	dstrvval 34495*	The value of the distribution of a random variable. (Contributed by Thierry Arnoux, 9-Feb-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝐷 = (𝑎 ∈ 𝔅ℝ ↦ (𝑃‘(𝑋∘RV/𝑐 E 𝑎))))    &   ⊢ (𝜑 → 𝐴 ∈ 𝔅ℝ)    ⇒   ⊢ (𝜑 → (𝐷‘𝐴) = (𝑃‘(◡𝑋 “ 𝐴)))
Theorem	dstrvprob 34496*	The distribution of a random variable is a probability law. (TODO: could be shortened using dstrvval 34495). (Contributed by Thierry Arnoux, 10-Feb-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝐷 = (𝑎 ∈ 𝔅ℝ ↦ (𝑃‘(𝑋∘RV/𝑐 E 𝑎))))    ⇒   ⊢ (𝜑 → 𝐷 ∈ Prob)
21.3.23.6  Cumulative Distribution Functions
Theorem	orvclteel 34497	Preimage maps produced by the "less than or equal to" relation are measurable sets. (Contributed by Thierry Arnoux, 4-Feb-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) ∈ dom 𝑃)
Theorem	dstfrvel 34498	Elementhood of preimage maps produced by the "less than or equal to" relation. (Contributed by Thierry Arnoux, 13-Feb-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ∪ dom 𝑃)    &   ⊢ (𝜑 → (𝑋‘𝐵) ≤ 𝐴)    ⇒   ⊢ (𝜑 → 𝐵 ∈ (𝑋∘RV/𝑐 ≤ 𝐴))
Theorem	dstfrvunirn 34499*	The limit of all preimage maps by the "less than or equal to" relation is the universe. (Contributed by Thierry Arnoux, 12-Feb-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    ⇒   ⊢ (𝜑 → ∪ ran (𝑛 ∈ ℕ ↦ (𝑋∘RV/𝑐 ≤ 𝑛)) = ∪ dom 𝑃)
Theorem	orvclteinc 34500	Preimage maps produced by the "less than or equal to" relation are increasing. (Contributed by Thierry Arnoux, 11-Feb-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) ⊆ (𝑋∘RV/𝑐 ≤ 𝐵))