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Theorem List for Metamath Proof Explorer - 29601-29700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdomprobsiga 29601 The domain of a probability measure is a sigma-algebra. (Contributed by Thierry Arnoux, 23-Dec-2016.)
(𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)

Theoremprobtot 29602 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)

Theoremprob01 29603 A probability is an element of [ 0 , 1 ]. First axiom of Kolmogorov. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃) → (𝑃𝐴) ∈ (0[,]1))

Theoremprobnul 29604 The probability of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝑃 ∈ Prob → (𝑃‘∅) = 0)

Theoremunveldomd 29605 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.)
(𝜑𝑃 ∈ Prob)       (𝜑 dom 𝑃 ∈ dom 𝑃)

Theoremunveldom 29606 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)

Theoremnuleldmp 29607 The empty set is an element of the domain of the probability. (Contributed by Thierry Arnoux, 22-Jan-2017.)
(𝑃 ∈ Prob → ∅ ∈ dom 𝑃)

Theoremprobcun 29608* 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 , which is not the case here. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑃 ∈ Prob ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝑃 𝐴) = Σ*𝑥𝐴(𝑃𝑥))

Theoremprobun 29609 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 𝑃𝐵 ∈ dom 𝑃) → ((𝐴𝐵) = ∅ → (𝑃‘(𝐴𝐵)) = ((𝑃𝐴) + (𝑃𝐵))))

Theoremprobdif 29610 The probability of the difference of two event sets. (Contributed by Thierry Arnoux, 12-Dec-2016.)
((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) → (𝑃‘(𝐴𝐵)) = ((𝑃𝐴) − (𝑃‘(𝐴𝐵))))

Theoremprobinc 29611 A probability law is increasing with regard to event set inclusion. (Contributed by Thierry Arnoux, 10-Feb-2017.)
(((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ 𝐴𝐵) → (𝑃𝐴) ≤ (𝑃𝐵))

Theoremprobdsb 29612 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 𝑃𝐴)) = (1 − (𝑃𝐴)))

Theoremprobmeasd 29613 A probability measure is a measure. (Contributed by Thierry Arnoux, 2-Feb-2017.)
(𝜑𝑃 ∈ Prob)       (𝜑𝑃 ran measures)

Theoremprobvalrnd 29614 The value of a probability is a real number. (Contributed by Thierry Arnoux, 2-Feb-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝐴 ∈ dom 𝑃)       (𝜑 → (𝑃𝐴) ∈ ℝ)

Theoremprobtotrnd 29615 The probability of the universe set is finite. (Contributed by Thierry Arnoux, 2-Feb-2017.)
(𝜑𝑃 ∈ Prob)       (𝜑 → (𝑃 dom 𝑃) ∈ ℝ)

Theoremtotprobd 29616* Law of total probability, deduction form. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝐴 ∈ dom 𝑃)    &   (𝜑𝐵 ∈ 𝒫 dom 𝑃)    &   (𝜑 𝐵 = dom 𝑃)    &   (𝜑𝐵 ≼ ω)    &   (𝜑Disj 𝑏𝐵 𝑏)       (𝜑 → (𝑃𝐴) = Σ*𝑏𝐵(𝑃‘(𝑏𝐴)))

Theoremtotprob 29617* Law of total probability. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ∧ ( 𝐵 = dom 𝑃𝐵 ∈ 𝒫 dom 𝑃 ∧ (𝐵 ≼ ω ∧ Disj 𝑏𝐵 𝑏))) → (𝑃𝐴) = Σ*𝑏𝐵(𝑃‘(𝑏𝐴)))

TheoremprobfinmeasbOLD 29618* Build a probability measure from a finite measure. (Contributed by Thierry Arnoux, 17-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑀 ∈ (measures‘𝑆) ∧ (𝑀 𝑆) ∈ ℝ+) → (𝑥𝑆 ↦ ((𝑀𝑥) /𝑒 (𝑀 𝑆))) ∈ Prob)

Theoremprobfinmeasb 29619 Build a probability measure from a finite measure. (Contributed by Thierry Arnoux, 31-Jan-2017.)
((𝑀 ∈ (measures‘𝑆) ∧ (𝑀 𝑆) ∈ ℝ+) → (𝑀𝑓/𝑐 /𝑒 (𝑀 𝑆)) ∈ Prob)

Theoremprobmeasb 29620* Build a probability from a measure and a set with finite measure. (Contributed by Thierry Arnoux, 25-Dec-2016.)
((𝑀 ∈ (measures‘𝑆) ∧ 𝐴𝑆 ∧ (𝑀𝐴) ∈ ℝ+) → (𝑥𝑆 ↦ ((𝑀‘(𝑥𝐴)) / (𝑀𝐴))) ∈ Prob)

20.3.20.2  Conditional Probabilities

Syntaxccprob 29621 Extends class notation with the conditional probability builder.
class cprob

Definitiondf-cndprob 29622* 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‘(𝑎𝑏)) / (𝑝𝑏))))

Theoremcndprobval 29623 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 𝑃𝐵 ∈ dom 𝑃) → ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩) = ((𝑃‘(𝐴𝐵)) / (𝑃𝐵)))

Theoremcndprobin 29624 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 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑃𝐵) ≠ 0) → (((cprob‘𝑃)‘⟨𝐴, 𝐵⟩) · (𝑃𝐵)) = (𝑃‘(𝐴𝐵)))

Theoremcndprob01 29625 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 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑃𝐵) ≠ 0) → ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩) ∈ (0[,]1))

Theoremcndprobtot 29626 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 ∧ (𝑃𝐴) ≠ 0) → ((cprob‘𝑃)‘⟨ dom 𝑃, 𝐴⟩) = 1)

Theoremcndprobnul 29627 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 ∧ (𝑃𝐴) ≠ 0) → ((cprob‘𝑃)‘⟨∅, 𝐴⟩) = 0)

Theoremcndprobprob 29628* 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 ∧ (𝑃𝐵) ≠ 0) → (𝑎 ∈ dom 𝑃 ↦ ((cprob‘𝑃)‘⟨𝑎, 𝐵⟩)) ∈ Prob)

Theorembayesth 29629 Bayes Theorem. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
(((𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃𝐵 ∈ dom 𝑃) ∧ (𝑃𝐴) ≠ 0 ∧ (𝑃𝐵) ≠ 0) → ((cprob‘𝑃)‘⟨𝐴, 𝐵⟩) = ((((cprob‘𝑃)‘⟨𝐵, 𝐴⟩) · (𝑃𝐴)) / (𝑃𝐵)))

20.3.20.3  Real Valued Random Variables

Syntaxcrrv 29630 Extend class notation with the class of real valued random variables.
class rRndVar

Definitiondf-rrv 29631 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 𝑝MblFnM𝔅))

Theoremrrvmbfm 29632 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.)
(𝜑𝑃 ∈ Prob)       (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅)))

Theoremisrrvv 29633* Elementhood to the set of real-valued random variables with respect to the probability 𝑃. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(𝜑𝑃 ∈ Prob)       (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ (𝑋: dom 𝑃⟶ℝ ∧ ∀𝑦 ∈ 𝔅 (𝑋𝑦) ∈ dom 𝑃)))

Theoremrrvvf 29634 A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))       (𝜑𝑋: dom 𝑃⟶ℝ)

Theoremrrvfn 29635 A real-valued random variable is a function over the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))       (𝜑𝑋 Fn dom 𝑃)

Theoremrrvdm 29636 The domain of a random variable is the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))       (𝜑 → dom 𝑋 = dom 𝑃)

Theoremrrvrnss 29637 The range of a random variable as a subset of . (Contributed by Thierry Arnoux, 6-Feb-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))       (𝜑 → ran 𝑋 ⊆ ℝ)

Theoremrrvf2 29638 A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))       (𝜑𝑋:dom 𝑋⟶ℝ)

Theoremrrvdmss 29639 The domain of a random variable. This is useful to shorten proofs. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))       (𝜑 dom 𝑃 ⊆ dom 𝑋)

Theoremrrvfinvima 29640* For a real-value random variable 𝑋, any open interval in is the image of a measurable set. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))       (𝜑 → ∀𝑦 ∈ 𝔅 (𝑋𝑦) ∈ dom 𝑃)

Theorem0rrv 29641* The constant function equal to zero is a random variable. (Contributed by Thierry Arnoux, 16-Jan-2017.) (Revised by Thierry Arnoux, 30-Jan-2017.)
(𝜑𝑃 ∈ Prob)       (𝜑 → (𝑥 dom 𝑃 ↦ 0) ∈ (rRndVar‘𝑃))

Theoremrrvadd 29642 The sum of two random variables is a random variable. (Contributed by Thierry Arnoux, 4-Jun-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝑌 ∈ (rRndVar‘𝑃))       (𝜑 → (𝑋𝑓 + 𝑌) ∈ (rRndVar‘𝑃))

Theoremrrvmulc 29643 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.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝑋𝑓/𝑐 · 𝐶) ∈ (rRndVar‘𝑃))

Theoremrrvsum 29644 An indexed sum of random variables is a random variable. (Contributed by Thierry Arnoux, 22-May-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋:ℕ⟶(rRndVar‘𝑃))    &   ((𝜑𝑁 ∈ ℕ) → 𝑆 = (seq1( ∘𝑓 + , 𝑋)‘𝑁))       ((𝜑𝑁 ∈ ℕ) → 𝑆 ∈ (rRndVar‘𝑃))

20.3.20.4  Preimage set mapping operator

Syntaxcorvc 29645 Extend class notation to include the preimage set mapping operator.
class RV/𝑐𝑅

Definitiondf-orvc 29646* 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‘(𝑋RV/𝑐 I 𝐴)), and keep a similar notation. Because with this notation (𝑋RV/𝑐 I 𝐴) is a set, we can also apply it to conditional probabilities, like in (𝑃‘(𝑋RV/𝑐 I 𝐴) ∣ (𝑌RV/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 (𝑋RV/𝑐 I 𝐴) cf. ideq 5088- elementhood: {𝑋𝐴} as (𝑋RV/𝑐 E 𝐴) cf. epel 4846- less-than: {𝑋𝐴} as (𝑋RV/𝑐𝐴)

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 “ {𝑦𝑦𝑅𝑎}))

Theoremorvcval 29647* Value of the preimage mapping operator applied on a given random variable and constant. (Contributed by Thierry Arnoux, 19-Jan-2017.)
(𝜑 → Fun 𝑋)    &   (𝜑𝑋𝑉)    &   (𝜑𝐴𝑊)       (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦𝑦𝑅𝐴}))

Theoremorvcval2 29648* Another way to express the value of the preimage mapping operator. (Contributed by Thierry Arnoux, 19-Jan-2017.)
(𝜑 → Fun 𝑋)    &   (𝜑𝑋𝑉)    &   (𝜑𝐴𝑊)       (𝜑 → (𝑋RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋𝑧)𝑅𝐴})

Theoremelorvc 29649* Elementhood of a preimage. (Contributed by Thierry Arnoux, 21-Jan-2017.)
(𝜑 → Fun 𝑋)    &   (𝜑𝑋𝑉)    &   (𝜑𝐴𝑊)       ((𝜑𝑧 ∈ dom 𝑋) → (𝑧 ∈ (𝑋RV/𝑐𝑅𝐴) ↔ (𝑋𝑧)𝑅𝐴))

Theoremorvcval4 29650* The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 29647. (Contributed by Thierry Arnoux, 21-Jan-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))    &   (𝜑𝐴𝑉)       (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦 𝐽𝑦𝑅𝐴}))

Theoremorvcoel 29651* If the relation produces open sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))    &   (𝜑𝐴𝑉)    &   (𝜑 → {𝑦 𝐽𝑦𝑅𝐴} ∈ 𝐽)       (𝜑 → (𝑋RV/𝑐𝑅𝐴) ∈ 𝑆)

Theoremorvccel 29652* If the relation produces closed sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
(𝜑𝑆 ran sigAlgebra)    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))    &   (𝜑𝐴𝑉)    &   (𝜑 → {𝑦 𝐽𝑦𝑅𝐴} ∈ (Clsd‘𝐽))       (𝜑 → (𝑋RV/𝑐𝑅𝐴) ∈ 𝑆)

Theoremelorrvc 29653* Elementhood of a preimage for a real-valued random variable. (Contributed by Thierry Arnoux, 21-Jan-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝐴𝑉)       ((𝜑𝑧 dom 𝑃) → (𝑧 ∈ (𝑋RV/𝑐𝑅𝐴) ↔ (𝑋𝑧)𝑅𝐴))

Theoremorrvcval4 29654* The value of the preimage mapping operator can be restricted to preimages of subsets of RR. (Contributed by Thierry Arnoux, 21-Jan-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝐴𝑉)       (𝜑 → (𝑋RV/𝑐𝑅𝐴) = (𝑋 “ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴}))

Theoremorrvcoel 29655* If the relation produces open sets, preimage maps of a random variable are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝐴𝑉)    &   (𝜑 → {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴} ∈ (topGen‘ran (,)))       (𝜑 → (𝑋RV/𝑐𝑅𝐴) ∈ dom 𝑃)

Theoremorrvccel 29656* If the relation produces closed sets, preimage maps are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝐴𝑉)    &   (𝜑 → {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴} ∈ (Clsd‘(topGen‘ran (,))))       (𝜑 → (𝑋RV/𝑐𝑅𝐴) ∈ dom 𝑃)

Theoremorvcgteel 29657 Preimage maps produced by the "greater than or equal" relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → (𝑋RV/𝑐𝐴) ∈ dom 𝑃)

20.3.20.5  Distribution Functions

Theoremorvcelval 29658 Preimage maps produced by the membership relation. (Contributed by Thierry Arnoux, 6-Feb-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝐴 ∈ 𝔅)       (𝜑 → (𝑋RV/𝑐 E 𝐴) = (𝑋𝐴))

Theoremorvcelel 29659 Preimage maps produced by the membership relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝐴 ∈ 𝔅)       (𝜑 → (𝑋RV/𝑐 E 𝐴) ∈ dom 𝑃)

Theoremdstrvval 29660* The value of the distribution of a random variable. (Contributed by Thierry Arnoux, 9-Feb-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝐷 = (𝑎 ∈ 𝔅 ↦ (𝑃‘(𝑋RV/𝑐 E 𝑎))))    &   (𝜑𝐴 ∈ 𝔅)       (𝜑 → (𝐷𝐴) = (𝑃‘(𝑋𝐴)))

Theoremdstrvprob 29661* The distribution of a random variable is a probability law. (TODO: could be shortened using dstrvval 29660) (Contributed by Thierry Arnoux, 10-Feb-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝐷 = (𝑎 ∈ 𝔅 ↦ (𝑃‘(𝑋RV/𝑐 E 𝑎))))       (𝜑𝐷 ∈ Prob)

20.3.20.6  Cumulative Distribution Functions

Theoremorvclteel 29662 Preimage maps produced by the "lower than or equal" relation are measurable sets. (Contributed by Thierry Arnoux, 4-Feb-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → (𝑋RV/𝑐𝐴) ∈ dom 𝑃)

Theoremdstfrvel 29663 Elementhood of preimage maps produced by the "lower than or equal" relation. (Contributed by Thierry Arnoux, 13-Feb-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 dom 𝑃)    &   (𝜑 → (𝑋𝐵) ≤ 𝐴)       (𝜑𝐵 ∈ (𝑋RV/𝑐𝐴))

Theoremdstfrvunirn 29664* The limit of all preimage maps by the "lower than or equal" relation is the universe. (Contributed by Thierry Arnoux, 12-Feb-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))       (𝜑 ran (𝑛 ∈ ℕ ↦ (𝑋RV/𝑐𝑛)) = dom 𝑃)

Theoremorvclteinc 29665 Preimage maps produced by the "lower than or equal" relation are increasing. (Contributed by Thierry Arnoux, 11-Feb-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝑋RV/𝑐𝐴) ⊆ (𝑋RV/𝑐𝐵))

Theoremdstfrvinc 29666* A cumulative distribution function is non-decreasing. (Contributed by Thierry Arnoux, 11-Feb-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝐹 = (𝑥 ∈ ℝ ↦ (𝑃‘(𝑋RV/𝑐𝑥))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐹𝐴) ≤ (𝐹𝐵))

Theoremdstfrvclim1 29667* The limit of the cumulative distribution function is one. (Contributed by Thierry Arnoux, 12-Feb-2017.) (Revised by Thierry Arnoux, 11-Jul-2017.)
(𝜑𝑃 ∈ Prob)    &   (𝜑𝑋 ∈ (rRndVar‘𝑃))    &   (𝜑𝐹 = (𝑥 ∈ ℝ ↦ (𝑃‘(𝑋RV/𝑐𝑥))))       (𝜑𝐹 ⇝ 1)

20.3.20.7  Probabilities - example

Theoremcoinfliplem 29668 Division in the extended real numbers can be used for the coin-flip example. (Contributed by Thierry Arnoux, 15-Jan-2017.)
𝐻 ∈ V    &   𝑇 ∈ V    &   𝐻𝑇    &   𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)    &   𝑋 = {⟨𝐻, 1⟩, ⟨𝑇, 0⟩}       𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 /𝑒 2)

Theoremcoinflipprob 29669 The 𝑃 we defined for coin-flip is a probability law. (Contributed by Thierry Arnoux, 15-Jan-2017.)
𝐻 ∈ V    &   𝑇 ∈ V    &   𝐻𝑇    &   𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)    &   𝑋 = {⟨𝐻, 1⟩, ⟨𝑇, 0⟩}       𝑃 ∈ Prob

Theoremcoinflipspace 29670 The space of our coin-flip probability. (Contributed by Thierry Arnoux, 15-Jan-2017.)
𝐻 ∈ V    &   𝑇 ∈ V    &   𝐻𝑇    &   𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)    &   𝑋 = {⟨𝐻, 1⟩, ⟨𝑇, 0⟩}       dom 𝑃 = 𝒫 {𝐻, 𝑇}

Theoremcoinflipuniv 29671 The universe of our coin-flip probability is {𝐻, 𝑇}. (Contributed by Thierry Arnoux, 15-Jan-2017.)
𝐻 ∈ V    &   𝑇 ∈ V    &   𝐻𝑇    &   𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)    &   𝑋 = {⟨𝐻, 1⟩, ⟨𝑇, 0⟩}        dom 𝑃 = {𝐻, 𝑇}

Theoremcoinfliprv 29672 The 𝑋 we defined for coin-flip is a random variable. (Contributed by Thierry Arnoux, 12-Jan-2017.)
𝐻 ∈ V    &   𝑇 ∈ V    &   𝐻𝑇    &   𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)    &   𝑋 = {⟨𝐻, 1⟩, ⟨𝑇, 0⟩}       𝑋 ∈ (rRndVar‘𝑃)

Theoremcoinflippv 29673 The probability of heads is one-half. (Contributed by Thierry Arnoux, 15-Jan-2017.)
𝐻 ∈ V    &   𝑇 ∈ V    &   𝐻𝑇    &   𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)    &   𝑋 = {⟨𝐻, 1⟩, ⟨𝑇, 0⟩}       (𝑃‘{𝐻}) = (1 / 2)

Theoremcoinflippvt 29674 The probability of tails is one-half. (Contributed by Thierry Arnoux, 5-Feb-2017.)
𝐻 ∈ V    &   𝑇 ∈ V    &   𝐻𝑇    &   𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)    &   𝑋 = {⟨𝐻, 1⟩, ⟨𝑇, 0⟩}       (𝑃‘{𝑇}) = (1 / 2)

20.3.20.8  Bertrand's Ballot Problem

Theoremballotlemoex 29675* 𝑂 is a set. (Contributed by Thierry Arnoux, 7-Dec-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}       𝑂 ∈ V

Theoremballotlem1 29676* The size of the universe is a binomial coefficient. (Contributed by Thierry Arnoux, 23-Nov-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}       (#‘𝑂) = ((𝑀 + 𝑁)C𝑀)

Theoremballotlemelo 29677* Elementhood in 𝑂. (Contributed by Thierry Arnoux, 17-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}       (𝐶𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (#‘𝐶) = 𝑀))

Theoremballotlem2 29678* The probability that the first vote picked in a count is a B. (Contributed by Thierry Arnoux, 23-Nov-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))       (𝑃‘{𝑐𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))

Theoremballotlemfval 29679* The value of F. (Contributed by Thierry Arnoux, 23-Nov-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   (𝜑𝐶𝑂)    &   (𝜑𝐽 ∈ ℤ)       (𝜑 → ((𝐹𝐶)‘𝐽) = ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))))

Theoremballotlemfelz 29680* (𝐹𝐶) has values in . (Contributed by Thierry Arnoux, 23-Nov-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   (𝜑𝐶𝑂)    &   (𝜑𝐽 ∈ ℤ)       (𝜑 → ((𝐹𝐶)‘𝐽) ∈ ℤ)

Theoremballotlemfp1 29681* If the 𝐽 th ballot is for A, (𝐹𝐶) goes up 1. If the 𝐽 th ballot is for B, (𝐹𝐶) goes down 1. (Contributed by Thierry Arnoux, 24-Nov-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   (𝜑𝐶𝑂)    &   (𝜑𝐽 ∈ ℕ)       (𝜑 → ((¬ 𝐽𝐶 → ((𝐹𝐶)‘𝐽) = (((𝐹𝐶)‘(𝐽 − 1)) − 1)) ∧ (𝐽𝐶 → ((𝐹𝐶)‘𝐽) = (((𝐹𝐶)‘(𝐽 − 1)) + 1))))

Theoremballotlemfc0 29682* 𝐹 takes value 0 between negative and positive values. (Contributed by Thierry Arnoux, 24-Nov-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   (𝜑𝐶𝑂)    &   (𝜑𝐽 ∈ ℕ)    &   (𝜑 → ∃𝑖 ∈ (1...𝐽)((𝐹𝐶)‘𝑖) ≤ 0)    &   (𝜑 → 0 < ((𝐹𝐶)‘𝐽))       (𝜑 → ∃𝑘 ∈ (1...𝐽)((𝐹𝐶)‘𝑘) = 0)

Theoremballotlemfcc 29683* 𝐹 takes value 0 between positive and negative values. (Contributed by Thierry Arnoux, 2-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   (𝜑𝐶𝑂)    &   (𝜑𝐽 ∈ ℕ)    &   (𝜑 → ∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹𝐶)‘𝑖))    &   (𝜑 → ((𝐹𝐶)‘𝐽) < 0)       (𝜑 → ∃𝑘 ∈ (1...𝐽)((𝐹𝐶)‘𝑘) = 0)

Theoremballotlemfmpn 29684* (𝐹𝐶) finishes counting at (𝑀𝑁). (Contributed by Thierry Arnoux, 25-Nov-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))       (𝐶𝑂 → ((𝐹𝐶)‘(𝑀 + 𝑁)) = (𝑀𝑁))

Theoremballotlemfval0 29685* (𝐹𝐶) always starts counting at 0 . (Contributed by Thierry Arnoux, 25-Nov-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))       (𝐶𝑂 → ((𝐹𝐶)‘0) = 0)

Theoremballotleme 29686* Elements of 𝐸. (Contributed by Thierry Arnoux, 14-Dec-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}       (𝐶𝐸 ↔ (𝐶𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝐶)‘𝑖)))

Theoremballotlemodife 29687* Elements of (𝑂𝐸). (Contributed by Thierry Arnoux, 7-Dec-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}       (𝐶 ∈ (𝑂𝐸) ↔ (𝐶𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑖) ≤ 0))

Theoremballotlem4 29688* If the first pick is a vote for B, A is not ahead throughout the count. (Contributed by Thierry Arnoux, 25-Nov-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}       (𝐶𝑂 → (¬ 1 ∈ 𝐶 → ¬ 𝐶𝐸))

Theoremballotlem5 29689* If A is not ahead throughout, there is a 𝑘 where votes are tied. (Contributed by Thierry Arnoux, 1-Dec-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀       (𝐶 ∈ (𝑂𝐸) → ∃𝑘 ∈ (1...(𝑀 + 𝑁))((𝐹𝐶)‘𝑘) = 0)

Theoremballotlemi 29690* Value of 𝐼 for a given counting 𝐶. (Contributed by Thierry Arnoux, 1-Dec-2016.) (Revised by AV, 6-Oct-2020.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))       (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}, ℝ, < ))

Theoremballotlemiex 29691* Properties of (𝐼𝐶). (Contributed by Thierry Arnoux, 12-Dec-2016.) (Revised by AV, 6-Oct-2020.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))       (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))

Theoremballotlemi1 29692* The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 12-Mar-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))       ((𝐶 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼𝐶) ≠ 1)

Theoremballotlemii 29693* The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 4-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))       ((𝐶 ∈ (𝑂𝐸) ∧ 1 ∈ 𝐶) → (𝐼𝐶) ≠ 1)

Theoremballotlemsup 29694* The set of zeroes of 𝐹 satisfies the conditions to have a supremum. (Contributed by Thierry Arnoux, 1-Dec-2016.) (Revised by AV, 6-Oct-2020.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))       (𝐶 ∈ (𝑂𝐸) → ∃𝑧 ∈ ℝ (∀𝑤 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0} ¬ 𝑤 < 𝑧 ∧ ∀𝑤 ∈ ℝ (𝑧 < 𝑤 → ∃𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}𝑦 < 𝑤)))

Theoremballotlemimin 29695* (𝐼𝐶) is the first tie. (Contributed by Thierry Arnoux, 1-Dec-2016.) (Revised by AV, 6-Oct-2020.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))       (𝐶 ∈ (𝑂𝐸) → ¬ ∃𝑘 ∈ (1...((𝐼𝐶) − 1))((𝐹𝐶)‘𝑘) = 0)

Theoremballotlemic 29696* If the first vote is for B, the vote on the first tie is for A. (Contributed by Thierry Arnoux, 1-Dec-2016.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))       ((𝐶 ∈ (𝑂𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼𝐶) ∈ 𝐶)

Theoremballotlem1c 29697* If the first vote is for A, the vote on the first tie is for B. (Contributed by Thierry Arnoux, 4-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))       ((𝐶 ∈ (𝑂𝐸) ∧ 1 ∈ 𝐶) → ¬ (𝐼𝐶) ∈ 𝐶)

Theoremballotlemsval 29698* Value of 𝑆. (Contributed by Thierry Arnoux, 12-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))       (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)))

Theoremballotlemsv 29699* Value of 𝑆 evaluated at 𝐽 for a given counting 𝐶. (Contributed by Thierry Arnoux, 12-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))       ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝐽) = if(𝐽 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝐽), 𝐽))

Theoremballotlemsgt1 29700* 𝑆 maps values less than (𝐼𝐶) to values greater than 1. (Contributed by Thierry Arnoux, 28-Apr-2017.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    &   𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((#‘𝑥) / (#‘𝑂)))    &   𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}    &   𝑁 < 𝑀    &   𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))    &   𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))       ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 1 < ((𝑆𝐶)‘𝐽))

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