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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | omssubaddlem 34301* | For any small margin 𝐸, we can find a covering approaching the outer measure of a set 𝐴 by that margin. (Contributed by Thierry Arnoux, 18-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
| ⊢ 𝑀 = (toOMeas‘𝑅) & ⊢ (𝜑 → 𝑄 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑄) & ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑅 ∣ (𝐴 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)}Σ*𝑤 ∈ 𝑥(𝑅‘𝑤) < ((𝑀‘𝐴) + 𝐸)) | ||
| Theorem | omssubadd 34302* | A constructed outer measure is countably sub-additive. Lemma 1.5.4 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 21-Sep-2019.) (Revised by AV, 4-Oct-2020.) |
| ⊢ 𝑀 = (toOMeas‘𝑅) & ⊢ (𝜑 → 𝑄 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞)) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝐴 ⊆ ∪ 𝑄) & ⊢ (𝜑 → 𝑋 ≼ ω) ⇒ ⊢ (𝜑 → (𝑀‘∪ 𝑦 ∈ 𝑋 𝐴) ≤ Σ*𝑦 ∈ 𝑋(𝑀‘𝐴)) | ||
| Syntax | ccarsg 34303 | Class declaration for the Caratheodory sigma-Algebra construction. |
| class toCaraSiga | ||
| Definition | df-carsg 34304* | Define a function constructing Caratheodory measurable sets for a given outer measure. See carsgval 34305 for its value. Definition 1.11.2 of [Bogachev] p. 41. (Contributed by Thierry Arnoux, 17-May-2020.) |
| ⊢ toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 ∪ dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 ∪ dom 𝑚((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)}) | ||
| Theorem | carsgval 34305* | Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) ⇒ ⊢ (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝑎)) +𝑒 (𝑀‘(𝑒 ∖ 𝑎))) = (𝑀‘𝑒)}) | ||
| Theorem | carsgcl 34306 | Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) ⇒ ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂) | ||
| Theorem | elcarsg 34307* | Property of being a Caratheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) ⇒ ⊢ (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ (𝐴 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ 𝐴)) +𝑒 (𝑀‘(𝑒 ∖ 𝐴))) = (𝑀‘𝑒)))) | ||
| Theorem | baselcarsg 34308 | The universe set, 𝑂, is Caratheodory measurable. (Contributed by Thierry Arnoux, 17-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) ⇒ ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀)) | ||
| Theorem | 0elcarsg 34309 | The empty set is Caratheodory measurable. (Contributed by Thierry Arnoux, 30-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) ⇒ ⊢ (𝜑 → ∅ ∈ (toCaraSiga‘𝑀)) | ||
| Theorem | carsguni 34310 | The union of all Caratheodory measurable sets is the universe. (Contributed by Thierry Arnoux, 22-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) ⇒ ⊢ (𝜑 → ∪ (toCaraSiga‘𝑀) = 𝑂) | ||
| Theorem | elcarsgss 34311 | Caratheodory measurable sets are subsets of the universe. (Contributed by Thierry Arnoux, 21-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝑂) | ||
| Theorem | difelcarsg 34312 | The Caratheodory measurable sets are closed under complement. (Contributed by Thierry Arnoux, 17-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → (𝑂 ∖ 𝐴) ∈ (toCaraSiga‘𝑀)) | ||
| Theorem | inelcarsg 34313* | The Caratheodory measurable sets are closed under intersection. (Contributed by Thierry Arnoux, 18-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏))) & ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ (toCaraSiga‘𝑀)) | ||
| Theorem | unelcarsg 34314* | The Caratheodory-measurable sets are closed under pairwise unions. (Contributed by Thierry Arnoux, 21-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏))) & ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ (toCaraSiga‘𝑀)) | ||
| Theorem | difelcarsg2 34315* | The Caratheodory-measurable sets are closed under class difference. (Contributed by Thierry Arnoux, 30-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ (toCaraSiga‘𝑀)) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑂 ∧ 𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎 ∪ 𝑏)) ≤ ((𝑀‘𝑎) +𝑒 (𝑀‘𝑏))) & ⊢ (𝜑 → 𝐵 ∈ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ (toCaraSiga‘𝑀)) | ||
| Theorem | carsgmon 34316* | Utility lemma: Apply monotony. (Contributed by Thierry Arnoux, 29-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝑂) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) | ||
| Theorem | carsgsigalem 34317* | Lemma for the following theorems. (Contributed by Thierry Arnoux, 23-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) ⇒ ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∪ 𝑓)) ≤ ((𝑀‘𝑒) +𝑒 (𝑀‘𝑓))) | ||
| Theorem | fiunelcarsg 34318* | The Caratheodory measurable sets are closed under finite union. (Contributed by Thierry Arnoux, 23-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀)) | ||
| Theorem | carsgclctunlem1 34319* | Lemma for carsgclctun 34323. (Contributed by Thierry Arnoux, 23-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀)) & ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂) ⇒ ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝐴)) = Σ*𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦))) | ||
| Theorem | carsggect 34320* | The outer measure is countably superadditive on Caratheodory measurable sets. (Contributed by Thierry Arnoux, 31-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ (𝜑 → ¬ ∅ ∈ 𝐴) & ⊢ (𝜑 → 𝐴 ≼ ω) & ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀)) & ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) ⇒ ⊢ (𝜑 → Σ*𝑧 ∈ 𝐴(𝑀‘𝑧) ≤ (𝑀‘∪ 𝐴)) | ||
| Theorem | carsgclctunlem2 34321* | Lemma for carsgclctun 34323. (Contributed by Thierry Arnoux, 25-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) & ⊢ (𝜑 → Disj 𝑘 ∈ ℕ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀)) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂) & ⊢ (𝜑 → (𝑀‘𝐸) ≠ +∞) ⇒ ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ≤ (𝑀‘𝐸)) | ||
| Theorem | carsgclctunlem3 34322* | Lemma for carsgclctun 34323. (Contributed by Thierry Arnoux, 24-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) & ⊢ (𝜑 → 𝐴 ≼ ω) & ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀)) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂) ⇒ ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝐴))) ≤ (𝑀‘𝐸)) | ||
| Theorem | carsgclctun 34323* | The Caratheodory measurable sets are closed under countable union. (Contributed by Thierry Arnoux, 21-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) & ⊢ (𝜑 → 𝐴 ≼ ω) & ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀)) ⇒ ⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀)) | ||
| Theorem | carsgsiga 34324* | The Caratheodory measurable sets constructed from outer measures form a Sigma-algebra. Statement (iii) of Theorem 1.11.4 of [Bogachev] p. 42. (Contributed by Thierry Arnoux, 17-May-2020.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑀‘∅) = 0) & ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) ⇒ ⊢ (𝜑 → (toCaraSiga‘𝑀) ∈ (sigAlgebra‘𝑂)) | ||
| Theorem | omsmeas 34325 | The restriction of a constructed outer measure to Caratheodory measurable sets is a measure. This theorem allows to construct measures from pre-measures with the required characteristics, as for the Lebesgue measure. (Contributed by Thierry Arnoux, 17-May-2020.) |
| ⊢ 𝑀 = (toOMeas‘𝑅) & ⊢ 𝑆 = (toCaraSiga‘𝑀) & ⊢ (𝜑 → 𝑄 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝑄⟶(0[,]+∞)) & ⊢ (𝜑 → ∅ ∈ dom 𝑅) & ⊢ (𝜑 → (𝑅‘∅) = 0) ⇒ ⊢ (𝜑 → (𝑀 ↾ 𝑆) ∈ (measures‘𝑆)) | ||
| Theorem | pmeasmono 34326* | This theorem's hypotheses define a pre-measure. A pre-measure is monotone. (Contributed by Thierry Arnoux, 19-Jul-2020.) |
| ⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑃‘∅) = 0) & ⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑃‘𝑦)) & ⊢ (𝜑 → 𝐴 ∈ 𝑅) & ⊢ (𝜑 → 𝐵 ∈ 𝑅) & ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑅) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑃‘𝐴) ≤ (𝑃‘𝐵)) | ||
| Theorem | pmeasadd 34327* | A premeasure on a ring of sets is additive on disjoint countable collections. This is called sigma-additivity. (Contributed by Thierry Arnoux, 19-Jul-2020.) |
| ⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞)) & ⊢ (𝜑 → (𝑃‘∅) = 0) & ⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑃‘𝑦)) & ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))} & ⊢ (𝜑 → 𝑅 ∈ 𝑄) & ⊢ (𝜑 → 𝐴 ≼ ω) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑅) & ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵) ⇒ ⊢ (𝜑 → (𝑃‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ*𝑘 ∈ 𝐴(𝑃‘𝐵)) | ||
| Theorem | itgeq12dv 34328* | Equality theorem for an integral. (Contributed by Thierry Arnoux, 14-Feb-2017.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥) | ||
| Syntax | citgm 34329 | Extend class notation with the (measure) Bochner integral. |
| class itgm | ||
| Syntax | csitm 34330 | Extend class notation with the integral metric for simple functions. |
| class sitm | ||
| Syntax | csitg 34331 | Extend class notation with the integral of simple functions. |
| class sitg | ||
| Definition | df-sitg 34332* |
Define the integral of simple functions from a measurable space
dom 𝑚 to a generic space 𝑤
equipped with the right scalar
product. 𝑤 will later be required to be a Banach
space.
These simple functions are required to take finitely many different values: this is expressed by ran 𝑔 ∈ Fin in the definition. Moreover, for each 𝑥, the pre-image (◡𝑔 “ {𝑥}) is requested to be measurable, of finite measure. In this definition, (sigaGen‘(TopOpen‘𝑤)) is the Borel sigma-algebra on 𝑤, and the functions 𝑔 range over the measurable functions over that Borel algebra. Definition 2.4.1 of [Bogachev] p. 118. (Contributed by Thierry Arnoux, 21-Oct-2017.) |
| ⊢ sitg = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ {𝑔 ∈ (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ {(0g‘𝑤)})(𝑚‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑤 Σg (𝑥 ∈ (ran 𝑓 ∖ {(0g‘𝑤)}) ↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(◡𝑓 “ {𝑥})))( ·𝑠 ‘𝑤)𝑥))))) | ||
| Definition | df-sitm 34333* | Define the integral metric for simple functions, as the integral of the distances between the function values. Since distances take nonnegative values in ℝ*, the range structure for this integral is (ℝ*𝑠 ↾s (0[,]+∞)). See definition 2.3.1 of [Bogachev] p. 116. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
| ⊢ sitm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (𝑓 ∈ dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦ (((ℝ*𝑠 ↾s (0[,]+∞))sitg𝑚)‘(𝑓 ∘f (dist‘𝑤)𝑔)))) | ||
| Theorem | sitgval 34334* | Value of the simple function integral builder for a given space 𝑊 and measure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-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 → (𝑊sitg𝑀) = (𝑓 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) · 𝑥))))) | ||
| Theorem | issibf 34335* | The predicate "𝐹 is a simple function" relative to the Bochner integral. (Contributed by Thierry Arnoux, 19-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) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))) | ||
| Theorem | sibf0 34336 | The constant zero function is a simple function. (Contributed by Thierry Arnoux, 4-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 → (∪ dom 𝑀 × { 0 }) ∈ dom (𝑊sitg𝑀)) | ||
| Theorem | sibfmbl 34337 | A simple function is measurable. (Contributed by Thierry Arnoux, 19-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 → 𝐹 ∈ (dom 𝑀MblFnM𝑆)) | ||
| Theorem | sibff 34338 | A simple function is a function. (Contributed by Thierry Arnoux, 19-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 → 𝐹:∪ dom 𝑀⟶∪ 𝐽) | ||
| Theorem | sibfrn 34339 | A simple function has finite range. (Contributed by Thierry Arnoux, 19-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 → ran 𝐹 ∈ Fin) | ||
| Theorem | sibfima 34340 | Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-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 ∧ 𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞)) | ||
| Theorem | sibfinima 34341 | The measure of the intersection of any two preimages by simple functions is a real number. (Contributed by Thierry Arnoux, 21-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 → 𝐹 ∈ dom (𝑊sitg𝑀)) & ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀)) & ⊢ (𝜑 → 𝑊 ∈ TopSp) & ⊢ (𝜑 → 𝐽 ∈ Fre) ⇒ ⊢ (((𝜑 ∧ 𝑋 ∈ ran 𝐹 ∧ 𝑌 ∈ ran 𝐺) ∧ (𝑋 ≠ 0 ∨ 𝑌 ≠ 0 )) → (𝑀‘((◡𝐹 “ {𝑋}) ∩ (◡𝐺 “ {𝑌}))) ∈ (0[,)+∞)) | ||
| Theorem | sibfof 34342 | Applying function operations on simple functions results in simple functions with regard to the destination space, provided the operation fulfills a simple condition. (Contributed by Thierry Arnoux, 12-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 → 𝐹 ∈ dom (𝑊sitg𝑀)) & ⊢ 𝐶 = (Base‘𝐾) & ⊢ (𝜑 → 𝑊 ∈ TopSp) & ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐶) & ⊢ (𝜑 → 𝐺 ∈ dom (𝑊sitg𝑀)) & ⊢ (𝜑 → 𝐾 ∈ TopSp) & ⊢ (𝜑 → 𝐽 ∈ Fre) & ⊢ (𝜑 → ( 0 + 0 ) = (0g‘𝐾)) ⇒ ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom (𝐾sitg𝑀)) | ||
| Theorem | sitgfval 34343* | Value of the Bochner integral for a simple function 𝐹. (Contributed by Thierry Arnoux, 30-Jan-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 → ((𝑊sitg𝑀)‘𝐹) = (𝑊 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝐹 “ {𝑥}))) · 𝑥)))) | ||
| Theorem | sitgclg 34344* | Closure of the Bochner integral on simple functions, generic version. See sitgclbn 34345 for the version for Banach spaces. (Contributed by Thierry Arnoux, 24-Feb-2018.) (Proof shortened by AV, 12-Dec-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 → 𝐹 ∈ dom (𝑊sitg𝑀)) & ⊢ 𝐺 = (Scalar‘𝑊) & ⊢ 𝐷 = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) & ⊢ (𝜑 → 𝑊 ∈ TopSp) & ⊢ (𝜑 → 𝑊 ∈ CMnd) & ⊢ (𝜑 → (Scalar‘𝑊) ∈ ℝExt ) & ⊢ ((𝜑 ∧ 𝑚 ∈ (𝐻 “ (0[,)+∞)) ∧ 𝑥 ∈ 𝐵) → (𝑚 · 𝑥) ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑊sitg𝑀)‘𝐹) ∈ 𝐵) | ||
| Theorem | sitgclbn 34345 | 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 34346 | 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 34347 | 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 34348 | 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 34349* | 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 34350 | 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 34351* | 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 34352 | 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 34353 | 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 34354 | 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 34355* |
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 25658. (Contributed by Thierry Arnoux, 13-Feb-2018.) |
| ⊢ itgm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran measures ↦ (((metUnif‘(𝑤sitm𝑚))CnExt(UnifSt‘𝑤))‘(𝑤sitg𝑚))) | ||
| Theorem | oddpwdc 34356* | Lemma for eulerpart 34384. 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 34357* | Lemma for eulerpart 34384: 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 34358* | Lemma for eulerpart 34384. 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 34359* | Lemma for eulerpart 34384. (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 34360* | Lemma for eulerpart 34384. 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 34361* | Lemma for eulerpart 34384. (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 34362* | Lemma for eulerpart 34384. (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 34363* | Lemma for eulerpart 34384. 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 34364* | Lemma for eulerpart 34384. (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 34365* | Lemma for eulerpart 34384. (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 34366* | Lemma for eulerpart 34384. (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 34367* | Lemma for eulerpart 34384: 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 34368* | Lemma for eulerpart 34384: 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 34369* | Lemma for eulerpart 34384. (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 34370* | Lemma for eulerpart 34384. 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 34371* | Lemma for eulerpart 34384. (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 34372* | Lemma for eulerpart 34384: 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 34373* | Lemma for eulerpart 34384. (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 34374* | Lemma for eulerpart 34384: 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 34375* | Lemma for eulerpart 34384: 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 34376* | Lemma for eulerpart 34384. (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 34377* | Lemma for eulerpart 34384. (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 34378* | Lemma for eulerpart 34384: 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 34379* | Lemma for eulerpart 34384: Rewriting the 𝑈 set for an odd partition Note that interestingly, this proof reuses marypha2lem2 9476. (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 34380* | Lemma for eulerpart 34384: 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 34381* | Lemma for eulerpart 34384: 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 34382* | Lemma for eulerpart 34384: 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 34383* | Lemma for eulerpart 34384. (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 34384* | 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 34385 | Sequences defined by strong recursion. |
| class seqstr | ||
| Definition | df-sseq 34386* | 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 34387 | Lemma for sseqp1 34397. (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 34388 | 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 34389 | Lemma for sseqp1 34397. (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 34390* | 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 34391 | 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 34392 | 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 34393 | Lemma for sseqf 34394 amd sseqp1 34397. (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 34394 | 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 34395 | 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 34396* | 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 34397 | 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 34398 | The Fibonacci sequence. |
| class Fibci | ||
| Definition | df-fib 34399 | 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 34400 | Lemma for fib0 34401, fib1 34402 and fibp1 34403. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
| ⊢ (𝑤 ∈ (Word ℕ0 ∩ (◡♯ “ (ℤ≥‘2))) ↦ ((𝑤‘((♯‘𝑤) − 2)) + (𝑤‘((♯‘𝑤) − 1)))):(Word ℕ0 ∩ (◡♯ “ (ℤ≥‘(♯‘〈“01”〉))))⟶ℕ0 | ||
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