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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sitgclbn 31501 | Closure of the Bochner integral on a simple function. This version is specific to Banach spaces, with additional conditions on its scalar field. (Contributed by Thierry Arnoux, 24-Feb-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) & ⊢ (𝜑 → 𝑊 ∈ Ban) & ⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt ) ⇒ ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵) | ||
Theorem | sitgclcn 31502 | Closure of the Bochner integral on a simple function. This version is specific to Banach spaces on the complex numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) & ⊢ (𝜑 → 𝑊 ∈ Ban) & ⊢ (𝜑 → (Scalar‘𝑊) = ℂfld) ⇒ ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵) | ||
Theorem | sitgclre 31503 | Closure of the Bochner integral on a simple function. This version is specific to Banach spaces on the real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) & ⊢ (𝜑 → 𝑊 ∈ Ban) & ⊢ (𝜑 → (Scalar‘𝑊) = ℝfld) ⇒ ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵) | ||
Theorem | sitg0 31504 | The integral of the constant zero function is zero. (Contributed by Thierry Arnoux, 13-Mar-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝑊 ∈ TopSp) & ⊢ (𝜑 → 𝑊 ∈ Mnd) ⇒ ⊢ (𝜑 → ((𝑊sitg𝑀)‘(∪ dom 𝑀 × { 0 })) = 0 ) | ||
Theorem | sitgf 31505* | The integral for simple functions is itself a function. (Contributed by Thierry Arnoux, 13-Feb-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ ((𝜑 ∧ 𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵) | ||
Theorem | sitgaddlemb 31506 | Lemma for * sitgadd . (Contributed by Thierry Arnoux, 10-Mar-2019.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝑆 = (sigaGen‘𝐽) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝑊 ∈ TopSp) & ⊢ (𝜑 → (𝑊 ↾v (𝐻 “ (0[,)+∞))) ∈ SLMod) & ⊢ (𝜑 → 𝐽 ∈ Fre) & ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) & ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀)) & ⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt ) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝜑 ∧ 𝑝 ∈ ((ran 𝐹 × ran 𝐺) ∖ {〈 0 , 0 〉})) → ((𝐻‘(𝑀‘((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))) · (2nd ‘𝑝)) ∈ 𝐵) | ||
Theorem | sitmval 31507* | Value of the simple function integral metric for a given space 𝑊 and measure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
⊢ 𝐷 = (dist‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) ⇒ ⊢ (𝜑 → (𝑊sitm𝑀) = (𝑓 ∈ dom (𝑊sitg𝑀), 𝑔 ∈ dom (𝑊sitg𝑀) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝑓 ∘f 𝐷𝑔)))) | ||
Theorem | sitmfval 31508 | Value of the integral distance between two simple functions. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
⊢ 𝐷 = (dist‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) & ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀)) ⇒ ⊢ (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) = (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑀)‘(𝐹 ∘f 𝐷𝐺))) | ||
Theorem | sitmcl 31509 | Closure of the integral distance between two simple functions, for an extended metric space. (Contributed by Thierry Arnoux, 13-Feb-2018.) |
⊢ (𝜑 → 𝑊 ∈ Mnd) & ⊢ (𝜑 → 𝑊 ∈ ∞MetSp) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) & ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀)) & ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀)) ⇒ ⊢ (𝜑 → (𝐹(𝑊sitm𝑀)𝐺) ∈ (0[,]+∞)) | ||
Theorem | sitmf 31510 | The integral metric as a function. (Contributed by Thierry Arnoux, 13-Mar-2018.) |
⊢ (𝜑 → 𝑊 ∈ Mnd) & ⊢ (𝜑 → 𝑊 ∈ ∞MetSp) & ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) ⇒ ⊢ (𝜑 → (𝑊sitm𝑀):(dom (𝑊sitg𝑀) × dom (𝑊sitg𝑀))⟶(0[,]+∞)) | ||
Definition | df-itgm 31511* |
Define the Bochner integral as the extension by continuity of the
Bochnel integral for simple functions.
Bogachev first defines 'fundamental in the mean' sequences, in definition 2.3.1 of [Bogachev] p. 116, and notes that those are actually Cauchy sequences for the pseudometric (𝑤sitm𝑚). He then defines the Bochner integral in chapter 2.4.4 in [Bogachev] p. 118. The definition of the Lebesgue integral, df-itg 24153. (Contributed by Thierry Arnoux, 13-Feb-2018.) |
⊢ itgm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (((metUnif‘(𝑤sitm𝑚))CnExt(UnifSt‘𝑤))‘(𝑤sitg𝑚))) | ||
Theorem | oddpwdc 31512* | Lemma for eulerpart 31540. The function 𝐹 that decomposes a number into its "odd" and "even" parts, which is to say the largest power of two and largest odd divisor of a number, is a bijection from pairs of a nonnegative integer and an odd number to positive integers. (Contributed by Thierry Arnoux, 15-Aug-2017.) |
⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) ⇒ ⊢ 𝐹:(𝐽 × ℕ0)–1-1-onto→ℕ | ||
Theorem | oddpwdcv 31513* | Lemma for eulerpart 31540: value of the 𝐹 function. (Contributed by Thierry Arnoux, 9-Sep-2017.) |
⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) ⇒ ⊢ (𝑊 ∈ (𝐽 × ℕ0) → (𝐹‘𝑊) = ((2↑(2nd ‘𝑊)) · (1st ‘𝑊))) | ||
Theorem | eulerpartlemsv1 31514* | Lemma for eulerpart 31540. Value of the sum of a partition 𝐴. (Contributed by Thierry Arnoux, 26-Aug-2018.) |
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) ⇒ ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘)) | ||
Theorem | eulerpartlemelr 31515* | Lemma for eulerpart 31540. (Contributed by Thierry Arnoux, 8-Aug-2018.) |
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) ⇒ ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin)) | ||
Theorem | eulerpartlemsv2 31516* | Lemma for eulerpart 31540. Value of the sum of a finite partition 𝐴 (Contributed by Thierry Arnoux, 19-Aug-2018.) |
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) ⇒ ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑘) · 𝑘)) | ||
Theorem | eulerpartlemsf 31517* | Lemma for eulerpart 31540. (Contributed by Thierry Arnoux, 8-Aug-2018.) |
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) ⇒ ⊢ 𝑆:((ℕ0 ↑m ℕ) ∩ 𝑅)⟶ℕ0 | ||
Theorem | eulerpartlems 31518* | Lemma for eulerpart 31540. (Contributed by Thierry Arnoux, 6-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) ⇒ ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))) → (𝐴‘𝑡) = 0) | ||
Theorem | eulerpartlemsv3 31519* | Lemma for eulerpart 31540. Value of the sum of a finite partition 𝐴 (Contributed by Thierry Arnoux, 19-Aug-2018.) |
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) ⇒ ⊢ (𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘)) | ||
Theorem | eulerpartlemgc 31520* | Lemma for eulerpart 31540. (Contributed by Thierry Arnoux, 9-Aug-2018.) |
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) ⇒ ⊢ ((𝐴 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ≤ (𝑆‘𝐴)) | ||
Theorem | eulerpartleme 31521* | Lemma for eulerpart 31540. (Contributed by Mario Carneiro, 26-Jan-2015.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} ⇒ ⊢ (𝐴 ∈ 𝑃 ↔ (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝐴‘𝑘) · 𝑘) = 𝑁)) | ||
Theorem | eulerpartlemv 31522* | Lemma for eulerpart 31540. (Contributed by Thierry Arnoux, 19-Aug-2018.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} ⇒ ⊢ (𝐴 ∈ 𝑃 ↔ (𝐴:ℕ⟶ℕ0 ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ (◡𝐴 “ ℕ)((𝐴‘𝑘) · 𝑘) = 𝑁)) | ||
Theorem | eulerpartlemo 31523* | Lemma for eulerpart 31540: 𝑂 is the set of odd partitions of 𝑁. (Contributed by Thierry Arnoux, 10-Aug-2017.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} ⇒ ⊢ (𝐴 ∈ 𝑂 ↔ (𝐴 ∈ 𝑃 ∧ ∀𝑛 ∈ (◡𝐴 “ ℕ) ¬ 2 ∥ 𝑛)) | ||
Theorem | eulerpartlemd 31524* | Lemma for eulerpart 31540: 𝐷 is the set of distinct part. of 𝑁. (Contributed by Thierry Arnoux, 11-Aug-2017.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} ⇒ ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ 𝑃 ∧ (𝐴 “ ℕ) ⊆ {0, 1})) | ||
Theorem | eulerpartlem1 31525* | Lemma for eulerpart 31540. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} & ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) & ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} & ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ⇒ ⊢ 𝑀:𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) | ||
Theorem | eulerpartlemb 31526* | Lemma for eulerpart 31540. The set of all partitions of 𝑁 is finite. (Contributed by Mario Carneiro, 26-Jan-2015.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} & ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) & ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} & ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ⇒ ⊢ 𝑃 ∈ Fin | ||
Theorem | eulerpartlemt0 31527* | Lemma for eulerpart 31540. (Contributed by Thierry Arnoux, 19-Sep-2017.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} & ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) & ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} & ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) & ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} ⇒ ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) ↔ (𝐴 ∈ (ℕ0 ↑m ℕ) ∧ (◡𝐴 “ ℕ) ∈ Fin ∧ (◡𝐴 “ ℕ) ⊆ 𝐽)) | ||
Theorem | eulerpartlemf 31528* | Lemma for eulerpart 31540: Odd partitions are zero for even numbers. (Contributed by Thierry Arnoux, 9-Sep-2017.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} & ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) & ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} & ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) & ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} ⇒ ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ 𝐽)) → (𝐴‘𝑡) = 0) | ||
Theorem | eulerpartlemt 31529* | Lemma for eulerpart 31540. (Contributed by Thierry Arnoux, 19-Sep-2017.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} & ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) & ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} & ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) & ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} ⇒ ⊢ ((ℕ0 ↑m 𝐽) ∩ 𝑅) = ran (𝑚 ∈ (𝑇 ∩ 𝑅) ↦ (𝑚 ↾ 𝐽)) | ||
Theorem | eulerpartgbij 31530* | Lemma for eulerpart 31540: The 𝐺 function is a bijection. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} & ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) & ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} & ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) & ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} & ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) ⇒ ⊢ 𝐺:(𝑇 ∩ 𝑅)–1-1-onto→(({0, 1} ↑m ℕ) ∩ 𝑅) | ||
Theorem | eulerpartlemgv 31531* | Lemma for eulerpart 31540: value of the function 𝐺. (Contributed by Thierry Arnoux, 13-Nov-2017.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} & ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) & ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} & ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) & ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} & ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) ⇒ ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐺‘𝐴) = ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝐴 ↾ 𝐽)))))) | ||
Theorem | eulerpartlemr 31532* | Lemma for eulerpart 31540. (Contributed by Thierry Arnoux, 13-Nov-2017.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} & ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) & ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} & ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) & ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} & ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) ⇒ ⊢ 𝑂 = ((𝑇 ∩ 𝑅) ∩ 𝑃) | ||
Theorem | eulerpartlemmf 31533* | Lemma for eulerpart 31540. (Contributed by Thierry Arnoux, 30-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} & ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) & ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} & ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) & ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} & ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) ⇒ ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (bits ∘ (𝐴 ↾ 𝐽)) ∈ 𝐻) | ||
Theorem | eulerpartlemgvv 31534* | Lemma for eulerpart 31540: value of the function 𝐺 evaluated. (Contributed by Thierry Arnoux, 10-Aug-2018.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} & ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) & ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} & ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) & ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} & ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) ⇒ ⊢ ((𝐴 ∈ (𝑇 ∩ 𝑅) ∧ 𝐵 ∈ ℕ) → ((𝐺‘𝐴)‘𝐵) = if(∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝐵, 1, 0)) | ||
Theorem | eulerpartlemgu 31535* | Lemma for eulerpart 31540: Rewriting the 𝑈 set for an odd partition Note that interestingly, this proof reuses marypha2lem2 8889. (Contributed by Thierry Arnoux, 10-Aug-2018.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} & ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) & ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} & ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) & ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} & ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) & ⊢ 𝑈 = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⇒ ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → 𝑈 = {〈𝑡, 𝑛〉 ∣ (𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽) ∧ 𝑛 ∈ ((bits ∘ 𝐴)‘𝑡))}) | ||
Theorem | eulerpartlemgh 31536* | Lemma for eulerpart 31540: The 𝐹 function is a bijection on the 𝑈 subsets. (Contributed by Thierry Arnoux, 15-Aug-2018.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} & ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) & ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} & ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) & ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} & ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) & ⊢ 𝑈 = ∪ 𝑡 ∈ ((◡𝐴 “ ℕ) ∩ 𝐽)({𝑡} × (bits‘(𝐴‘𝑡))) ⇒ ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝐹 ↾ 𝑈):𝑈–1-1-onto→{𝑚 ∈ ℕ ∣ ∃𝑡 ∈ ℕ ∃𝑛 ∈ (bits‘(𝐴‘𝑡))((2↑𝑛) · 𝑡) = 𝑚}) | ||
Theorem | eulerpartlemgf 31537* | Lemma for eulerpart 31540: Images under 𝐺 have finite support. (Contributed by Thierry Arnoux, 29-Aug-2018.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} & ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) & ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} & ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) & ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} & ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) ⇒ ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (◡(𝐺‘𝐴) “ ℕ) ∈ Fin) | ||
Theorem | eulerpartlemgs2 31538* | Lemma for eulerpart 31540: The 𝐺 function also preserves partition sums. (Contributed by Thierry Arnoux, 10-Sep-2017.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} & ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) & ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} & ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) & ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} & ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) & ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) ⇒ ⊢ (𝐴 ∈ (𝑇 ∩ 𝑅) → (𝑆‘(𝐺‘𝐴)) = (𝑆‘𝐴)) | ||
Theorem | eulerpartlemn 31539* | Lemma for eulerpart 31540. (Contributed by Thierry Arnoux, 30-Aug-2018.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} & ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) & ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} & ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) & ⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈ Fin} & ⊢ 𝑇 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ (◡𝑓 “ ℕ) ⊆ 𝐽} & ⊢ 𝐺 = (𝑜 ∈ (𝑇 ∩ 𝑅) ↦ ((𝟭‘ℕ)‘(𝐹 “ (𝑀‘(bits ∘ (𝑜 ↾ 𝐽)))))) & ⊢ 𝑆 = (𝑓 ∈ ((ℕ0 ↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) ⇒ ⊢ (𝐺 ↾ 𝑂):𝑂–1-1-onto→𝐷 | ||
Theorem | eulerpart 31540* | Euler's theorem on partitions, also known as a special case of Glaisher's theorem. Let 𝑃 be the set of all partitions of 𝑁, represented as multisets of positive integers, which is to say functions from ℕ to ℕ0 where the value of the function represents the number of repetitions of an individual element, and the sum of all the elements with repetition equals 𝑁. Then the set 𝑂 of all partitions that only consist of odd numbers and the set 𝐷 of all partitions which have no repeated elements have the same cardinality. This is Metamath 100 proof #45. (Contributed by Thierry Arnoux, 14-Aug-2018.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑m ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} & ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} & ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} ⇒ ⊢ (♯‘𝑂) = (♯‘𝐷) | ||
Syntax | csseq 31541 | Sequences defined by strong recursion. |
class seqstr | ||
Definition | df-sseq 31542* | Define a builder for sequences by strong recursion, i.e., by computing the value of the n-th element of the sequence from all preceding elements and not just the previous one. (Contributed by Thierry Arnoux, 21-Apr-2019.) |
⊢ seqstr = (𝑚 ∈ V, 𝑓 ∈ V ↦ (𝑚 ∪ (lastS ∘ seq(♯‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝑓‘𝑥)”〉)), (ℕ0 × {(𝑚 ++ 〈“(𝑓‘𝑚)”〉)}))))) | ||
Theorem | subiwrd 31543 | Lemma for sseqp1 31553. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (𝜑 → 𝑆 ∈ V) & ⊢ (𝜑 → 𝐹:ℕ0⟶𝑆) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐹 ↾ (0..^𝑁)) ∈ Word 𝑆) | ||
Theorem | subiwrdlen 31544 | Length of a subword of an infinite word. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (𝜑 → 𝑆 ∈ V) & ⊢ (𝜑 → 𝐹:ℕ0⟶𝑆) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (♯‘(𝐹 ↾ (0..^𝑁))) = 𝑁) | ||
Theorem | iwrdsplit 31545 | Lemma for sseqp1 31553. (Contributed by Thierry Arnoux, 25-Apr-2019.) (Proof shortened by AV, 14-Oct-2022.) |
⊢ (𝜑 → 𝑆 ∈ V) & ⊢ (𝜑 → 𝐹:ℕ0⟶𝑆) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐹 ↾ (0..^(𝑁 + 1))) = ((𝐹 ↾ (0..^𝑁)) ++ 〈“(𝐹‘𝑁)”〉)) | ||
Theorem | sseqval 31546* | Value of the strong sequence builder function. The set 𝑊 represents here the words of length greater than or equal to the lenght of the initial sequence 𝑀. (Contributed by Thierry Arnoux, 21-Apr-2019.) |
⊢ (𝜑 → 𝑆 ∈ V) & ⊢ (𝜑 → 𝑀 ∈ Word 𝑆) & ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀)))) & ⊢ (𝜑 → 𝐹:𝑊⟶𝑆) ⇒ ⊢ (𝜑 → (𝑀seqstr𝐹) = (𝑀 ∪ (lastS ∘ seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))))) | ||
Theorem | sseqfv1 31547 | Value of the strong sequence builder function at one of its initial values. (Contributed by Thierry Arnoux, 21-Apr-2019.) |
⊢ (𝜑 → 𝑆 ∈ V) & ⊢ (𝜑 → 𝑀 ∈ Word 𝑆) & ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀)))) & ⊢ (𝜑 → 𝐹:𝑊⟶𝑆) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝑀))) ⇒ ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (𝑀‘𝑁)) | ||
Theorem | sseqfn 31548 | A strong recursive sequence is a function over the nonnegative integers. (Contributed by Thierry Arnoux, 23-Apr-2019.) |
⊢ (𝜑 → 𝑆 ∈ V) & ⊢ (𝜑 → 𝑀 ∈ Word 𝑆) & ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀)))) & ⊢ (𝜑 → 𝐹:𝑊⟶𝑆) ⇒ ⊢ (𝜑 → (𝑀seqstr𝐹) Fn ℕ0) | ||
Theorem | sseqmw 31549 | Lemma for sseqf 31550 amd sseqp1 31553. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (𝜑 → 𝑆 ∈ V) & ⊢ (𝜑 → 𝑀 ∈ Word 𝑆) & ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀)))) & ⊢ (𝜑 → 𝐹:𝑊⟶𝑆) ⇒ ⊢ (𝜑 → 𝑀 ∈ 𝑊) | ||
Theorem | sseqf 31550 | A strong recursive sequence is a function over the nonnegative integers. (Contributed by Thierry Arnoux, 23-Apr-2019.) (Proof shortened by AV, 7-Mar-2022.) |
⊢ (𝜑 → 𝑆 ∈ V) & ⊢ (𝜑 → 𝑀 ∈ Word 𝑆) & ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀)))) & ⊢ (𝜑 → 𝐹:𝑊⟶𝑆) ⇒ ⊢ (𝜑 → (𝑀seqstr𝐹):ℕ0⟶𝑆) | ||
Theorem | sseqfres 31551 | The first elements in the strong recursive sequence are the sequence initializer. (Contributed by Thierry Arnoux, 23-Apr-2019.) |
⊢ (𝜑 → 𝑆 ∈ V) & ⊢ (𝜑 → 𝑀 ∈ Word 𝑆) & ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀)))) & ⊢ (𝜑 → 𝐹:𝑊⟶𝑆) ⇒ ⊢ (𝜑 → ((𝑀seqstr𝐹) ↾ (0..^(♯‘𝑀))) = 𝑀) | ||
Theorem | sseqfv2 31552* | Value of the strong sequence builder function. (Contributed by Thierry Arnoux, 21-Apr-2019.) |
⊢ (𝜑 → 𝑆 ∈ V) & ⊢ (𝜑 → 𝑀 ∈ Word 𝑆) & ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀)))) & ⊢ (𝜑 → 𝐹:𝑊⟶𝑆) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(♯‘𝑀))) ⇒ ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (lastS‘(seq(♯‘𝑀)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝐹‘𝑥)”〉)), (ℕ0 × {(𝑀 ++ 〈“(𝐹‘𝑀)”〉)}))‘𝑁))) | ||
Theorem | sseqp1 31553 | Value of the strong sequence builder function at a successor. (Contributed by Thierry Arnoux, 24-Apr-2019.) |
⊢ (𝜑 → 𝑆 ∈ V) & ⊢ (𝜑 → 𝑀 ∈ Word 𝑆) & ⊢ 𝑊 = (Word 𝑆 ∩ (◡♯ “ (ℤ≥‘(♯‘𝑀)))) & ⊢ (𝜑 → 𝐹:𝑊⟶𝑆) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(♯‘𝑀))) ⇒ ⊢ (𝜑 → ((𝑀seqstr𝐹)‘𝑁) = (𝐹‘((𝑀seqstr𝐹) ↾ (0..^𝑁)))) | ||
Syntax | cfib 31554 | The Fibonacci sequence. |
class Fibci | ||
Definition | df-fib 31555 | Define the Fibonacci sequence, where that each element is the sum of the two preceding ones, starting from 0 and 1. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ Fibci = (〈“01”〉seqstr(𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1))))) | ||
Theorem | fiblem 31556 | Lemma for fib0 31557, fib1 31558 and fibp1 31559. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))):(Word ℕ0 ∩ (◡♯ “ (ℤ≥‘(♯‘〈“01”〉))))⟶ℕ0 | ||
Theorem | fib0 31557 | Value of the Fibonacci sequence at index 0. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (Fibci‘0) = 0 | ||
Theorem | fib1 31558 | Value of the Fibonacci sequence at index 1. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (Fibci‘1) = 1 | ||
Theorem | fibp1 31559 | 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 31560 | Value of the Fibonacci sequence at index 2. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (Fibci‘2) = 1 | ||
Theorem | fib3 31561 | Value of the Fibonacci sequence at index 3. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (Fibci‘3) = 2 | ||
Theorem | fib4 31562 | Value of the Fibonacci sequence at index 4. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (Fibci‘4) = 3 | ||
Theorem | fib5 31563 | Value of the Fibonacci sequence at index 5. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (Fibci‘5) = 5 | ||
Theorem | fib6 31564 | Value of the Fibonacci sequence at index 6. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
⊢ (Fibci‘6) = 8 | ||
Syntax | cprb 31565 | Extend class notation to include the class of probability measures. |
class Prob | ||
Definition | df-prob 31566 | 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 31567 | 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 31568 | A probability measure is a measure on its domain. (Contributed by Thierry Arnoux, 23-Dec-2016.) |
⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃)) | ||
Theorem | domprobsiga 31569 | 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 31570 | 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 31571 | 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 31572 | The probability of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
⊢ (𝑃 ∈ Prob → (𝑃‘∅) = 0) | ||
Theorem | unveldomd 31573 | 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 31574 | 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 31575 | 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 31576* | 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 31577 | 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 31578 | 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 31579 | 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 31580 | 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 31581 | A probability measure is a measure. (Contributed by Thierry Arnoux, 2-Feb-2017.) |
⊢ (𝜑 → 𝑃 ∈ Prob) ⇒ ⊢ (𝜑 → 𝑃 ∈ ∪ ran measures) | ||
Theorem | probvalrnd 31582 | 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 31583 | 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 31584* | 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 31585* | 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 31586 | 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 31587* | Alternate version of probfinmeasb 31586. (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 31588* | 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 31589 | Extends class notation with the conditional probability builder. |
class cprob | ||
Definition | df-cndprob 31590* | 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 31591 | 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 31592 | 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 31593 | 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 31594 | 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 31595 | 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 31596* | 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 31597 | 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 31598 | Extend class notation with the class of real-valued random variables. |
class rRndVar | ||
Definition | df-rrv 31599 | 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 31600 | 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))) |
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