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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | isrrvv 31601* | 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 31602 | 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 31603 | 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 31604 | 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 31605 | 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 31606 | 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 31607 | 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 31608* | 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 31609* | 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 31610 | 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 31611 | 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 31612 | 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 31613 | Extend class notation to include the preimage set mapping operator. |
class ∘RV/𝑐𝑅 | ||
Definition | df-orvc 31614* |
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 5717- elementhood: {𝑋 ∈ 𝐴} as (𝑋∘RV/𝑐 E 𝐴) cf. epel 5463- 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 31615* | 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 31616* | 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 31617* | 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 31618* | The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 31615. (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 31619* | 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 31620* | 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 31621* | 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 31622* | 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 31623* | 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 31624* | 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 31625 | 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 31626 | 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 31627 | 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 31628* | 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 31629* | The distribution of a random variable is a probability law. (TODO: could be shortened using dstrvval 31628) (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 31630 | 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 31631 | 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 31632* | 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 31633 | 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 31634* | 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 31635* | 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 31636 | 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 31637 | 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 31638 | 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 31639 | 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 31640 | 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 31641 | 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 31642 | 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 31643* | 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 31644* | 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 31645* | 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 31646* | 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 31647* | 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 31648* | (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 31649* | 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 31650* | 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 31651* | 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 31652* | (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 31653* | (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 31654* | 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 31655* | 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 31656* | 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 31657* | 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 31658* | 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 31659* | 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 31660* | 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 31661* | 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 31662* | 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 31663* | (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 31664* | 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 31665* | 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 31666* | 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 31667* | 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 31668* | 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)) | ||
Theorem | ballotlemsdom 31669* | Domain of 𝑆 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) ∈ (1...(𝑀 + 𝑁))) | ||
Theorem | ballotlemsel1i 31670* | The range (1...(𝐼‘𝐶)) is invariant under (𝑆‘𝐶). (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...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶))) | ||
Theorem | ballotlemsf1o 31671* | The defined 𝑆 is a bijection, and an involution. (Contributed by Thierry Arnoux, 14-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):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑆‘𝐶))) | ||
Theorem | ballotlemsi 31672* | The image by 𝑆 of the first tie pick is the first pick. (Contributed by Thierry Arnoux, 14-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‘𝐶)) = 1) | ||
Theorem | ballotlemsima 31673* | The image by 𝑆 of an interval before the first pick. (Contributed by Thierry Arnoux, 5-May-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...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ (1...𝐽)) = (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))) | ||
Theorem | ballotlemieq 31674* | If two countings share the same first tie, they also have the same swap function. (Contributed by Thierry Arnoux, 18-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) ∧ 𝐷 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) = (𝐼‘𝐷)) → (𝑆‘𝐶) = (𝑆‘𝐷)) | ||
Theorem | ballotlemrval 31675* | Value of 𝑅. (Contributed by Thierry Arnoux, 14-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) ⇒ ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) = ((𝑆‘𝐶) “ 𝐶)) | ||
Theorem | ballotlemscr 31676* | The image of (𝑅‘𝐶) by (𝑆‘𝐶). (Contributed by Thierry Arnoux, 21-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) ⇒ ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ (𝑅‘𝐶)) = 𝐶) | ||
Theorem | ballotlemrv 31677* | Value of 𝑅 evaluated at 𝐽. (Contributed by Thierry Arnoux, 17-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) ⇒ ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → (𝐽 ∈ (𝑅‘𝐶) ↔ if(𝐽 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝐽), 𝐽) ∈ 𝐶)) | ||
Theorem | ballotlemrv1 31678* | Value of 𝑅 before the tie. (Contributed by Thierry Arnoux, 11-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) ⇒ ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 ≤ (𝐼‘𝐶)) → (𝐽 ∈ (𝑅‘𝐶) ↔ (((𝐼‘𝐶) + 1) − 𝐽) ∈ 𝐶)) | ||
Theorem | ballotlemrv2 31679* | Value of 𝑅 after the tie. (Contributed by Thierry Arnoux, 11-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) ⇒ ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ (𝐼‘𝐶) < 𝐽) → (𝐽 ∈ (𝑅‘𝐶) ↔ 𝐽 ∈ 𝐶)) | ||
Theorem | ballotlemro 31680* | Range of 𝑅 is included in 𝑂. (Contributed by Thierry Arnoux, 17-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) ⇒ ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂) | ||
Theorem | ballotlemgval 31681* | Expand the value of ↑. (Contributed by Thierry Arnoux, 21-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) & ⊢ ↑ = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢)))) ⇒ ⊢ ((𝑈 ∈ Fin ∧ 𝑉 ∈ Fin) → (𝑈 ↑ 𝑉) = ((♯‘(𝑉 ∩ 𝑈)) − (♯‘(𝑉 ∖ 𝑈)))) | ||
Theorem | ballotlemgun 31682* | A property of the defined ↑ operator. (Contributed by Thierry Arnoux, 26-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) & ⊢ ↑ = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢)))) & ⊢ (𝜑 → 𝑈 ∈ Fin) & ⊢ (𝜑 → 𝑉 ∈ Fin) & ⊢ (𝜑 → 𝑊 ∈ Fin) & ⊢ (𝜑 → (𝑉 ∩ 𝑊) = ∅) ⇒ ⊢ (𝜑 → (𝑈 ↑ (𝑉 ∪ 𝑊)) = ((𝑈 ↑ 𝑉) + (𝑈 ↑ 𝑊))) | ||
Theorem | ballotlemfg 31683* | Express the value of (𝐹‘𝐶) in terms of ↑. (Contributed by Thierry Arnoux, 21-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) & ⊢ ↑ = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢)))) ⇒ ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘𝐽) = (𝐶 ↑ (1...𝐽))) | ||
Theorem | ballotlemfrc 31684* | Express the value of (𝐹‘(𝑅‘𝐶)) in terms of the newly defined ↑. (Contributed by Thierry Arnoux, 21-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) & ⊢ ↑ = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢)))) ⇒ ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)))) | ||
Theorem | ballotlemfrci 31685* | Reverse counting preserves a tie at the first tie. (Contributed by Thierry Arnoux, 21-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) & ⊢ ↑ = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢)))) ⇒ ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐹‘(𝑅‘𝐶))‘(𝐼‘𝐶)) = 0) | ||
Theorem | ballotlemfrceq 31686* | Value of 𝐹 for a reverse counting (𝑅‘𝐶). (Contributed by Thierry Arnoux, 27-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) & ⊢ ↑ = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢)))) ⇒ ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅‘𝐶))‘𝐽)) | ||
Theorem | ballotlemfrcn0 31687* | Value of 𝐹 for a reversed counting (𝑅‘𝐶), before the first tie, cannot be zero . (Contributed by Thierry Arnoux, 25-Apr-2017.) (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, < )) & ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) ⇒ ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ≠ 0) | ||
Theorem | ballotlemrc 31688* | Range of 𝑅. (Contributed by Thierry Arnoux, 19-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) ⇒ ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ (𝑂 ∖ 𝐸)) | ||
Theorem | ballotlemirc 31689* | Applying 𝑅 does not change first ties. (Contributed by Thierry Arnoux, 19-Apr-2017.) (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, < )) & ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) ⇒ ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘(𝑅‘𝐶)) = (𝐼‘𝐶)) | ||
Theorem | ballotlemrinv0 31690* | Lemma for ballotlemrinv 31691. (Contributed by Thierry Arnoux, 18-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) ⇒ ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐷 = ((𝑆‘𝐶) “ 𝐶)) → (𝐷 ∈ (𝑂 ∖ 𝐸) ∧ 𝐶 = ((𝑆‘𝐷) “ 𝐷))) | ||
Theorem | ballotlemrinv 31691* | 𝑅 is its own inverse : it is an involution. (Contributed by Thierry Arnoux, 10-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) ⇒ ⊢ ◡𝑅 = 𝑅 | ||
Theorem | ballotlem1ri 31692* | When the vote on the first tie is for A, the first vote is also for A on the reverse counting. (Contributed by Thierry Arnoux, 18-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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) ⇒ ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (1 ∈ (𝑅‘𝐶) ↔ (𝐼‘𝐶) ∈ 𝐶)) | ||
Theorem | ballotlem7 31693* | 𝑅 is a bijection between two subsets of (𝑂 ∖ 𝐸): one where a vote for A is picked first, and one where a vote for B is picked first. (Contributed by Thierry Arnoux, 12-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, < )) & ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) ⇒ ⊢ (𝑅 ↾ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}):{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}–1-1-onto→{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} | ||
Theorem | ballotlem8 31694* | There are as many countings with ties starting with a ballot for A as there are starting with a ballot for B. (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)} & ⊢ 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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) ⇒ ⊢ (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) = (♯‘{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) | ||
Theorem | ballotth 31695* | Bertrand's ballot problem : the probability that A is ahead throughout the counting. The proof formalized here is a proof "by reflection", as opposed to other known proofs "by induction" or "by permutation". This is Metamath 100 proof #30. (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)} & ⊢ 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))) & ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐)) ⇒ ⊢ (𝑃‘𝐸) = ((𝑀 − 𝑁) / (𝑀 + 𝑁)) | ||
Theorem | sgncl 31696 | Closure of the signum. (Contributed by Thierry Arnoux, 28-Sep-2018.) |
⊢ (𝐴 ∈ ℝ* → (sgn‘𝐴) ∈ {-1, 0, 1}) | ||
Theorem | sgnclre 31697 | Closure of the signum. (Contributed by Thierry Arnoux, 28-Sep-2018.) |
⊢ (𝐴 ∈ ℝ → (sgn‘𝐴) ∈ ℝ) | ||
Theorem | sgnneg 31698 | Negation of the signum. (Contributed by Thierry Arnoux, 1-Oct-2018.) |
⊢ (𝐴 ∈ ℝ → (sgn‘-𝐴) = -(sgn‘𝐴)) | ||
Theorem | sgn3da 31699 | A conditional containing a signum is true if it is true in all three possible cases. (Contributed by Thierry Arnoux, 1-Oct-2018.) |
⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ ((sgn‘𝐴) = 0 → (𝜓 ↔ 𝜒)) & ⊢ ((sgn‘𝐴) = 1 → (𝜓 ↔ 𝜃)) & ⊢ ((sgn‘𝐴) = -1 → (𝜓 ↔ 𝜏)) & ⊢ ((𝜑 ∧ 𝐴 = 0) → 𝜒) & ⊢ ((𝜑 ∧ 0 < 𝐴) → 𝜃) & ⊢ ((𝜑 ∧ 𝐴 < 0) → 𝜏) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | sgnmul 31700 | Signum of a product. (Contributed by Thierry Arnoux, 2-Oct-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (sgn‘(𝐴 · 𝐵)) = ((sgn‘𝐴) · (sgn‘𝐵))) |
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