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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fibp1 33001 | 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 33002 | Value of the Fibonacci sequence at index 2. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (Fibci‘2) = 1 | ||
Theorem | fib3 33003 | Value of the Fibonacci sequence at index 3. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (Fibci‘3) = 2 | ||
Theorem | fib4 33004 | Value of the Fibonacci sequence at index 4. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (Fibci‘4) = 3 | ||
Theorem | fib5 33005 | Value of the Fibonacci sequence at index 5. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (Fibci‘5) = 5 | ||
Theorem | fib6 33006 | Value of the Fibonacci sequence at index 6. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (Fibci‘6) = 8 | ||
Syntax | cprb 33007 | Extend class notation to include the class of probability measures. |
class Prob | ||
Definition | df-prob 33008 | 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 33009 | 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 33010 | A probability measure is a measure on its domain. (Contributed by Thierry Arnoux, 23-Dec-2016.) |
⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃)) | ||
Theorem | domprobsiga 33011 | 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 33012 | 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 33013 | 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 33014 | The probability of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
⊢ (𝑃 ∈ Prob → (𝑃‘∅) = 0) | ||
Theorem | unveldomd 33015 | 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 33016 | 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 33017 | 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 33018* | 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 33019 | 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 33020 | 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 33021 | 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 33022 | 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 33023 | A probability measure is a measure. (Contributed by Thierry Arnoux, 2-Feb-2017.) |
⊢ (𝜑 → 𝑃 ∈ Prob) ⇒ ⊢ (𝜑 → 𝑃 ∈ ∪ ran measures) | ||
Theorem | probvalrnd 33024 | 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 33025 | 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 33026* | 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 33027* | 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 33028 | 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 33029* | Alternate version of probfinmeasb 33028. (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 33030* | 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) | ||
Syntax | ccprob 33031 | Extends class notation with the conditional probability builder. |
class cprob | ||
Definition | df-cndprob 33032* | 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 33033 | 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 33034 | 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 33035 | 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 33036 | 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 33037 | 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 33038* | 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 33039 | 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))) | ||
Syntax | crrv 33040 | Extend class notation with the class of real-valued random variables. |
class rRndVar | ||
Definition | df-rrv 33041 | 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 33042 | 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 33043* | 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 33044 | 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 33045 | 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 33046 | 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 33047 | 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 33048 | 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 33049 | 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 33050* | 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 33051* | 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 33052 | 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 33053 | 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 33054 | 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)) | ||
Syntax | corvc 33055 | Extend class notation to include the preimage set mapping operator. |
class ∘RV/𝑐𝑅 | ||
Definition | df-orvc 33056* |
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 5808- elementhood: {𝑋 ∈ 𝐴} as (𝑋∘RV/𝑐 E 𝐴) cf. epel 5540- 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 33057* | 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 33058* | 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 33059* | 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 33060* | The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 33057. (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 33061* | 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 33062* | 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 33063* | 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 33064* | 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 33065* | 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 33066* | 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 33067 | 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) | ||
Theorem | orvcelval 33068 | 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 33069 | 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 33070* | 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 33071* | The distribution of a random variable is a probability law. (TODO: could be shortened using dstrvval 33070). (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) | ||
Theorem | orvclteel 33072 | 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 33073 | 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 33074* | 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 33075 | 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)) | ||
Theorem | dstfrvinc 33076* | A cumulative distribution function is nondecreasing. (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 33077* | 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) | ||
Theorem | coinfliplem 33078 | 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 33079 | 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 33080 | 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 33081 | 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 33082 | 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 33083 | 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 33084 | 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) | ||
Theorem | ballotlemoex 33085* | 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 33086* | 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 33087* | 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 33088* | 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 33089* | 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 33090* | (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 33091* | 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 33092* | 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 33093* | 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 33094* | (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 33095* | (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 33096* | 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 33097* | 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 33098* | 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 33099* | 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 33100* | 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, < )) |
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