P. 307 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 30601-30700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	domprobsiga 30601	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 30602	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 30603	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 30604	The probability of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.)
⊢ (𝑃 ∈ Prob → (𝑃‘∅) = 0)
Theorem	unveldomd 30605	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 30606	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 30607	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 30608*	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 30609	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 30610	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 30611	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 30612	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 30613	A probability measure is a measure. (Contributed by Thierry Arnoux, 2-Feb-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    ⇒   ⊢ (𝜑 → 𝑃 ∈ ∪ ran measures)
Theorem	probvalrnd 30614	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 30615	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 30616*	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 30617*	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	probfinmeasbOLD 30618*	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+) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /𝑒 (𝑀‘∪ 𝑆))) ∈ Prob)
Theorem	probfinmeasb 30619	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	probmeasb 30620*	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)
20.3.21.2  Conditional Probabilities
Syntax	ccprob 30621	Extends class notation with the conditional probability builder.
class cprob
Definition	df-cndprob 30622*	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 30623	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 30624	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 30625	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 30626	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 30627	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 30628*	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 30629	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)))
20.3.21.3  Real Valued Random Variables
Syntax	crrv 30630	Extend class notation with the class of real valued random variables.
class rRndVar
Definition	df-rrv 30631	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 30632	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 30633*	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 30634	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 30635	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 30636	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 30637	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 30638	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 30639	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 30640*	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 30641*	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 30642	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 30643	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 30644	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))
20.3.21.4  Preimage set mapping operator
Syntax	corvc 30645	Extend class notation to include the preimage set mapping operator.
class ∘RV/𝑐𝑅
Definition	df-orvc 30646*	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 5307- elementhood: {𝑋 ∈ 𝐴} as (𝑋∘RV/𝑐 E 𝐴) cf. epel 5061- 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 30647*	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 30648*	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 30649*	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 30650*	The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 30647. (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 30651*	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 30652*	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 30653*	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 30654*	The value of the preimage mapping operator can be restricted to preimages of subsets of RR. (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 30655*	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 30656*	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 30657	Preimage maps produced by the "greater than or equal" 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)
20.3.21.5  Distribution Functions
Theorem	orvcelval 30658	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 30659	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 30660*	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 30661*	The distribution of a random variable is a probability law. (TODO: could be shortened using dstrvval 30660) (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)
20.3.21.6  Cumulative Distribution Functions
Theorem	orvclteel 30662	Preimage maps produced by the "lower than or equal" 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 30663	Elementhood of preimage maps produced by the "lower than or equal" 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 30664*	The limit of all preimage maps by the "lower than or equal" 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 30665	Preimage maps produced by the "lower than or equal" 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))
Theorem	dstfrvinc 30666*	A cumulative distribution function is non-decreasing. (Contributed by Thierry Arnoux, 11-Feb-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ (𝑃‘(𝑋∘RV/𝑐 ≤ 𝑥))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) ≤ (𝐹‘𝐵))
Theorem	dstfrvclim1 30667*	The limit of the cumulative distribution function is one. (Contributed by Thierry Arnoux, 12-Feb-2017.) (Revised by Thierry Arnoux, 11-Jul-2017.)
⊢ (𝜑 → 𝑃 ∈ Prob)    &   ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ (𝑃‘(𝑋∘RV/𝑐 ≤ 𝑥))))    ⇒   ⊢ (𝜑 → 𝐹 ⇝ 1)
20.3.21.7  Probabilities - example
Theorem	coinfliplem 30668	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    &   ⊢ 𝐻 ≠ 𝑇    &   ⊢ 𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)    &   ⊢ 𝑋 = {⟨𝐻, 1⟩, ⟨𝑇, 0⟩}    ⇒   ⊢ 𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 /𝑒 2)
Theorem	coinflipprob 30669	The 𝑃 we defined for coin-flip is a probability law. (Contributed by Thierry Arnoux, 15-Jan-2017.)
⊢ 𝐻 ∈ V    &   ⊢ 𝑇 ∈ V    &   ⊢ 𝐻 ≠ 𝑇    &   ⊢ 𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)    &   ⊢ 𝑋 = {⟨𝐻, 1⟩, ⟨𝑇, 0⟩}    ⇒   ⊢ 𝑃 ∈ Prob
Theorem	coinflipspace 30670	The space of our coin-flip probability. (Contributed by Thierry Arnoux, 15-Jan-2017.)
⊢ 𝐻 ∈ V    &   ⊢ 𝑇 ∈ V    &   ⊢ 𝐻 ≠ 𝑇    &   ⊢ 𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)    &   ⊢ 𝑋 = {⟨𝐻, 1⟩, ⟨𝑇, 0⟩}    ⇒   ⊢ dom 𝑃 = 𝒫 {𝐻, 𝑇}
Theorem	coinflipuniv 30671	The universe of our coin-flip probability is {𝐻, 𝑇}. (Contributed by Thierry Arnoux, 15-Jan-2017.)
⊢ 𝐻 ∈ V    &   ⊢ 𝑇 ∈ V    &   ⊢ 𝐻 ≠ 𝑇    &   ⊢ 𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)    &   ⊢ 𝑋 = {⟨𝐻, 1⟩, ⟨𝑇, 0⟩}    ⇒   ⊢ ∪ dom 𝑃 = {𝐻, 𝑇}
Theorem	coinfliprv 30672	The 𝑋 we defined for coin-flip is a random variable. (Contributed by Thierry Arnoux, 12-Jan-2017.)
⊢ 𝐻 ∈ V    &   ⊢ 𝑇 ∈ V    &   ⊢ 𝐻 ≠ 𝑇    &   ⊢ 𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)    &   ⊢ 𝑋 = {⟨𝐻, 1⟩, ⟨𝑇, 0⟩}    ⇒   ⊢ 𝑋 ∈ (rRndVar‘𝑃)
Theorem	coinflippv 30673	The probability of heads is one-half. (Contributed by Thierry Arnoux, 15-Jan-2017.)
⊢ 𝐻 ∈ V    &   ⊢ 𝑇 ∈ V    &   ⊢ 𝐻 ≠ 𝑇    &   ⊢ 𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)    &   ⊢ 𝑋 = {⟨𝐻, 1⟩, ⟨𝑇, 0⟩}    ⇒   ⊢ (𝑃‘{𝐻}) = (1 / 2)
Theorem	coinflippvt 30674	The probability of tails is one-half. (Contributed by Thierry Arnoux, 5-Feb-2017.)
⊢ 𝐻 ∈ V    &   ⊢ 𝑇 ∈ V    &   ⊢ 𝐻 ≠ 𝑇    &   ⊢ 𝑃 = ((# ↾ 𝒫 {𝐻, 𝑇})∘𝑓/𝑐 / 2)    &   ⊢ 𝑋 = {⟨𝐻, 1⟩, ⟨𝑇, 0⟩}    ⇒   ⊢ (𝑃‘{𝑇}) = (1 / 2)
20.3.21.8  Bertrand's Ballot Problem
Theorem	ballotlemoex 30675*	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
Theorem	ballotlem1 30676*	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)
Theorem	ballotlemelo 30677*	Elementhood in 𝑂. (Contributed by Thierry Arnoux, 17-Apr-2017.)
⊢ 𝑀 ∈ ℕ    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}    ⇒   ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (#‘𝐶) = 𝑀))
Theorem	ballotlem2 30678*	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‘{𝑐 ∈ 𝑂 ∣ ¬ 1 ∈ 𝑐}) = (𝑁 / (𝑀 + 𝑁))
Theorem	ballotlemfval 30679*	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑂)    &   ⊢ (𝜑 → 𝐽 ∈ ℤ)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) = ((#‘((1...𝐽) ∩ 𝐶)) − (#‘((1...𝐽) ∖ 𝐶))))
Theorem	ballotlemfelz 30680*	(𝐹‘𝐶) has values in ℤ. (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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑂)    &   ⊢ (𝜑 → 𝐽 ∈ ℤ)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) ∈ ℤ)
Theorem	ballotlemfp1 30681*	If the 𝐽 th ballot is for A, (𝐹‘𝐶) goes up 1. If the 𝐽 th ballot is for B, (𝐹‘𝐶) 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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑂)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ)    ⇒   ⊢ (𝜑 → ((¬ 𝐽 ∈ 𝐶 → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) − 1)) ∧ (𝐽 ∈ 𝐶 → ((𝐹‘𝐶)‘𝐽) = (((𝐹‘𝐶)‘(𝐽 − 1)) + 1))))
Theorem	ballotlemfc0 30682*	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑂)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ)    &   ⊢ (𝜑 → ∃𝑖 ∈ (1...𝐽)((𝐹‘𝐶)‘𝑖) ≤ 0)    &   ⊢ (𝜑 → 0 < ((𝐹‘𝐶)‘𝐽))    ⇒   ⊢ (𝜑 → ∃𝑘 ∈ (1...𝐽)((𝐹‘𝐶)‘𝑘) = 0)
Theorem	ballotlemfcc 30683*	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ (𝜑 → 𝐶 ∈ 𝑂)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ)    &   ⊢ (𝜑 → ∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹‘𝐶)‘𝑖))    &   ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) < 0)    ⇒   ⊢ (𝜑 → ∃𝑘 ∈ (1...𝐽)((𝐹‘𝐶)‘𝑘) = 0)
Theorem	ballotlemfmpn 30684*	(𝐹‘𝐶) finishes counting at (𝑀 − 𝑁). (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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    ⇒   ⊢ (𝐶 ∈ 𝑂 → ((𝐹‘𝐶)‘(𝑀 + 𝑁)) = (𝑀 − 𝑁))
Theorem	ballotlemfval0 30685*	(𝐹‘𝐶) 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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    ⇒   ⊢ (𝐶 ∈ 𝑂 → ((𝐹‘𝐶)‘0) = 0)
Theorem	ballotleme 30686*	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    ⇒   ⊢ (𝐶 ∈ 𝐸 ↔ (𝐶 ∈ 𝑂 ∧ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝐶)‘𝑖)))
Theorem	ballotlemodife 30687*	Elements of (𝑂 ∖ 𝐸). (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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    ⇒   ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) ↔ (𝐶 ∈ 𝑂 ∧ ∃𝑖 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑖) ≤ 0))
Theorem	ballotlem4 30688*	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    ⇒   ⊢ (𝐶 ∈ 𝑂 → (¬ 1 ∈ 𝐶 → ¬ 𝐶 ∈ 𝐸))
Theorem	ballotlem5 30689*	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   ⊢ 𝑁 < 𝑀    ⇒   ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ∃𝑘 ∈ (1...(𝑀 + 𝑁))((𝐹‘𝐶)‘𝑘) = 0)
Theorem	ballotlemi 30690*	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   ⊢ 𝑁 < 𝑀    &   ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))    ⇒   ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))
Theorem	ballotlemiex 30691*	Properties of (𝐼‘𝐶). (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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   ⊢ 𝑁 < 𝑀    &   ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))    ⇒   ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
Theorem	ballotlemi1 30692*	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   ⊢ 𝑁 < 𝑀    &   ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))    ⇒   ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼‘𝐶) ≠ 1)
Theorem	ballotlemii 30693*	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   ⊢ 𝑁 < 𝑀    &   ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))    ⇒   ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → (𝐼‘𝐶) ≠ 1)
Theorem	ballotlemsup 30694*	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   ⊢ 𝑁 < 𝑀    &   ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))    ⇒   ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ∃𝑧 ∈ ℝ (∀𝑤 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ¬ 𝑤 < 𝑧 ∧ ∀𝑤 ∈ ℝ (𝑧 < 𝑤 → ∃𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}𝑦 < 𝑤)))
Theorem	ballotlemimin 30695*	(𝐼‘𝐶) 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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   ⊢ 𝑁 < 𝑀    &   ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))    ⇒   ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ¬ ∃𝑘 ∈ (1...((𝐼‘𝐶) − 1))((𝐹‘𝐶)‘𝑘) = 0)
Theorem	ballotlemic 30696*	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   ⊢ 𝑁 < 𝑀    &   ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))    ⇒   ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝐶) → (𝐼‘𝐶) ∈ 𝐶)
Theorem	ballotlem1c 30697*	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   ⊢ 𝑁 < 𝑀    &   ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))    ⇒   ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝐶) → ¬ (𝐼‘𝐶) ∈ 𝐶)
Theorem	ballotlemsval 30698*	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   ⊢ 𝑁 < 𝑀    &   ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))    &   ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))    ⇒   ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)))
Theorem	ballotlemsv 30699*	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   ⊢ 𝑁 < 𝑀    &   ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))    &   ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))    ⇒   ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) = if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽))
Theorem	ballotlemsgt1 30700*	𝑆 maps values less than (𝐼‘𝐶) 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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((#‘((1...𝑖) ∩ 𝑐)) − (#‘((1...𝑖) ∖ 𝑐)))))    &   ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}    &   ⊢ 𝑁 < 𝑀    &   ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))    &   ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))    ⇒   ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 < ((𝑆‘𝐶)‘𝐽))