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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ontgsucval 31601 | The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.) |
⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴) | ||
Theorem | onsuctop 31602 | A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.) |
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top) | ||
Theorem | onsuctopon 31603 | One of the topologies on an ordinal number is its successor. (Contributed by Chen-Pang He, 7-Nov-2015.) |
⊢ (𝐴 ∈ On → suc 𝐴 ∈ (TopOn‘𝐴)) | ||
Theorem | ordtoplem 31604 | Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.) |
⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆) ⇒ ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆)) | ||
Theorem | ordtop 31605 | An ordinal is a topology iff it is not its supremum (union), proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 1-Nov-2015.) |
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ≠ ∪ 𝐽)) | ||
Theorem | onsucconi 31606 | A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.) |
⊢ 𝐴 ∈ On ⇒ ⊢ suc 𝐴 ∈ Con | ||
Theorem | onsuccon 31607 | A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.) |
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Con) | ||
Theorem | ordtopcon 31608 | An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.) |
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Con)) | ||
Theorem | onintopsscon 31609 | An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.) |
⊢ (On ∩ Top) ⊆ Con | ||
Theorem | onsuct0 31610 | A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.) |
⊢ (𝐴 ∈ On → suc 𝐴 ∈ Kol2) | ||
Theorem | ordtopt0 31611 | An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.) |
⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2)) | ||
Theorem | onsucsuccmpi 31612 | The successor of a successor ordinal number is a compact topology, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 18-Oct-2015.) |
⊢ 𝐴 ∈ On ⇒ ⊢ suc suc 𝐴 ∈ Comp | ||
Theorem | onsucsuccmp 31613 | The successor of a successor ordinal number is a compact topology. (Contributed by Chen-Pang He, 18-Oct-2015.) |
⊢ (𝐴 ∈ On → suc suc 𝐴 ∈ Comp) | ||
Theorem | limsucncmpi 31614 | The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.) |
⊢ Lim 𝐴 ⇒ ⊢ ¬ suc 𝐴 ∈ Comp | ||
Theorem | limsucncmp 31615 | The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.) |
⊢ (Lim 𝐴 → ¬ suc 𝐴 ∈ Comp) | ||
Theorem | ordcmp 31616 | An ordinal topology is compact iff the underlying set is its supremum (union) only when the ordinal is 1𝑜. (Contributed by Chen-Pang He, 1-Nov-2015.) |
⊢ (Ord 𝐴 → (𝐴 ∈ Comp ↔ (∪ 𝐴 = ∪ ∪ 𝐴 → 𝐴 = 1𝑜))) | ||
Theorem | ssoninhaus 31617 | The ordinal topologies 1𝑜 and 2𝑜 are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.) |
⊢ {1𝑜, 2𝑜} ⊆ (On ∩ Haus) | ||
Theorem | onint1 31618 | The ordinal T1 spaces are 1𝑜 and 2𝑜, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 9-Nov-2015.) |
⊢ (On ∩ Fre) = {1𝑜, 2𝑜} | ||
Theorem | oninhaus 31619 | The ordinal Hausdorff spaces are 1𝑜 and 2𝑜. (Contributed by Chen-Pang He, 10-Nov-2015.) |
⊢ (On ∩ Haus) = {1𝑜, 2𝑜} | ||
Theorem | fveleq 31620 | Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.) |
⊢ (𝐴 = 𝐵 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹‘𝐵) ∈ 𝑃))) | ||
Theorem | findfvcl 31621* | Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.) |
⊢ (𝜑 → (𝐹‘∅) ∈ 𝑃) & ⊢ (𝑦 ∈ ω → (𝜑 → ((𝐹‘𝑦) ∈ 𝑃 → (𝐹‘suc 𝑦) ∈ 𝑃))) ⇒ ⊢ (𝐴 ∈ ω → (𝜑 → (𝐹‘𝐴) ∈ 𝑃)) | ||
Theorem | findreccl 31622* | Please add description here. (Contributed by Jeff Hoffman, 19-Feb-2008.) |
⊢ (𝑧 ∈ 𝑃 → (𝐺‘𝑧) ∈ 𝑃) ⇒ ⊢ (𝐶 ∈ ω → (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃)) | ||
Theorem | findabrcl 31623* | Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝑧 ∈ 𝑃 → (𝐺‘𝑧) ∈ 𝑃) ⇒ ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ 𝑃) → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) ∈ 𝑃) | ||
Theorem | nnssi2 31624 | Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
⊢ ℕ ⊆ 𝐷 & ⊢ (𝐵 ∈ ℕ → 𝜑) & ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑) → 𝜓) ⇒ ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝜓) | ||
Theorem | nnssi3 31625 | Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
⊢ ℕ ⊆ 𝐷 & ⊢ (𝐶 ∈ ℕ → 𝜑) & ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ∧ 𝜑) → 𝜓) ⇒ ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓) | ||
Theorem | nndivsub 31626 | Please add description here. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴 / 𝐶) ∈ ℕ ∧ 𝐴 < 𝐵)) → ((𝐵 / 𝐶) ∈ ℕ ↔ ((𝐵 − 𝐴) / 𝐶) ∈ ℕ)) | ||
Theorem | nndivlub 31627 | A factor of a positive integer cannot exceed it. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 / 𝐵) ∈ ℕ → 𝐵 ≤ 𝐴)) | ||
Syntax | cgcdOLD 31628 | Extend class notation to include the gdc function. (New usage is discouraged.) |
class gcdOLD (𝐴, 𝐵) | ||
Definition | df-gcdOLD 31629* | gcdOLD (𝐴, 𝐵) is the largest positive integer that evenly divides both 𝐴 and 𝐵. (Contributed by Jeff Hoffman, 17-Jun-2008.) (New usage is discouraged.) |
⊢ gcdOLD (𝐴, 𝐵) = sup({𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ (𝐵 / 𝑥) ∈ ℕ)}, ℕ, < ) | ||
Theorem | ee7.2aOLD 31630 | Lemma for Euclid's Elements, Book 7, proposition 2. The original mentions the smaller measure being 'continually subtracted' from the larger. Many authors interpret this phrase as 𝐴 mod 𝐵. Here, just one subtraction step is proved to preserve the gcdOLD. The rec function will be used in other proofs for iterated subtraction. (Contributed by Jeff Hoffman, 17-Jun-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 → gcdOLD (𝐴, 𝐵) = gcdOLD (𝐴, (𝐵 − 𝐴)))) | ||
Theorem | dnival 31631* | Value of the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) ⇒ ⊢ (𝐴 ∈ ℝ → (𝑇‘𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) | ||
Theorem | dnicld1 31632 | Closure theorem for the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ ℝ) | ||
Theorem | dnicld2 31633* | Closure theorem for the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑇‘𝐴) ∈ ℝ) | ||
Theorem | dnif 31634 | The "distance to nearest integer" function is a function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) ⇒ ⊢ 𝑇:ℝ⟶ℝ | ||
Theorem | dnizeq0 31635* | The distance to nearest integer is zero for integers. (Contributed by Asger C. Ipsen, 15-Jun-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑇‘𝐴) = 0) | ||
Theorem | dnizphlfeqhlf 31636* | The distance to nearest integer is a half for half-integers. (Contributed by Asger C. Ipsen, 15-Jun-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑇‘(𝐴 + (1 / 2))) = (1 / 2)) | ||
Theorem | rddif2 31637 | Variant of rddif 13928. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝐴 ∈ ℝ → 0 ≤ ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) | ||
Theorem | dnibndlem1 31638* | Lemma for dnibnd 31651. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ 𝑆 ↔ (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ 𝑆)) | ||
Theorem | dnibndlem2 31639* | Lemma for dnibnd 31651. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) = (⌊‘(𝐴 + (1 / 2)))) ⇒ ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) | ||
Theorem | dnibndlem3 31640 | Lemma for dnibnd 31651. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) = ((⌊‘(𝐴 + (1 / 2))) + 1)) ⇒ ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) = (abs‘((𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2))) + (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)))) | ||
Theorem | dnibndlem4 31641 | Lemma for dnibnd 31651. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝐵 ∈ ℝ → 0 ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2)))) | ||
Theorem | dnibndlem5 31642 | Lemma for dnibnd 31651. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝐴 ∈ ℝ → 0 < (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) | ||
Theorem | dnibndlem6 31643 | Lemma for dnibnd 31651. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (((1 / 2) − (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) + ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))))) | ||
Theorem | dnibndlem7 31644 | Lemma for dnibnd 31651. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2)))) | ||
Theorem | dnibndlem8 31645 | Lemma for dnibnd 31651. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ≤ (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) | ||
Theorem | dnibndlem9 31646* | Lemma for dnibnd 31651. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) = ((⌊‘(𝐴 + (1 / 2))) + 1)) ⇒ ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) | ||
Theorem | dnibndlem10 31647 | Lemma for dnibnd 31651. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2)))) ⇒ ⊢ (𝜑 → 1 ≤ (𝐵 − 𝐴)) | ||
Theorem | dnibndlem11 31648 | Lemma for dnibnd 31651. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (1 / 2)) | ||
Theorem | dnibndlem12 31649* | Lemma for dnibnd 31651. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2)))) ⇒ ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) | ||
Theorem | dnibndlem13 31650* | Lemma for dnibnd 31651. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2)))) ⇒ ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) | ||
Theorem | dnibnd 31651* | The "distance to nearest integer" function is Lipshitz continuous. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) | ||
Theorem | dnicn 31652 | The "distance to nearest integer" function is continuous. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) ⇒ ⊢ 𝑇 ∈ (ℝ–cn→ℝ) | ||
Theorem | knoppcnlem1 31653* | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴)‘𝑀) = ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴)))) | ||
Theorem | knoppcnlem2 31654* | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐶↑𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴))) ∈ ℝ) | ||
Theorem | knoppcnlem3 31655* | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴)‘𝑀) ∈ ℝ) | ||
Theorem | knoppcnlem4 31656* | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → (abs‘((𝐹‘𝐴)‘𝑀)) ≤ ((𝑚 ∈ ℕ0 ↦ ((abs‘𝐶)↑𝑚))‘𝑀)) | ||
Theorem | knoppcnlem5 31657* | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))):ℕ0⟶(ℂ ↑𝑚 ℝ)) | ||
Theorem | knoppcnlem6 31658* | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (abs‘𝐶) < 1) ⇒ ⊢ (𝜑 → seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)))) ∈ dom (⇝𝑢‘ℝ)) | ||
Theorem | knoppcnlem7 31659* | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → (seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))‘𝑀) = (𝑤 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑤))‘𝑀))) | ||
Theorem | knoppcnlem8 31660* | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)))):ℕ0⟶(ℂ ↑𝑚 ℝ)) | ||
Theorem | knoppcnlem9 31661* | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (abs‘𝐶) < 1) ⇒ ⊢ (𝜑 → seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))(⇝𝑢‘ℝ)𝑊) | ||
Theorem | knoppcnlem10 31662* | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑀)) ∈ ((topGen‘ran (,)) Cn (TopOpen‘ℂfld))) | ||
Theorem | knoppcnlem11 31663* | Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)))):ℕ0⟶(ℝ–cn→ℂ)) | ||
Theorem | knoppcn 31664* | The continuous nowhere differentiable function 𝑊 ( Knopp, K. (1918). Math. Z. 2, 1-26 ) is, in fact, continuous. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (abs‘𝐶) < 1) ⇒ ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℂ)) | ||
Theorem | knoppcld 31665* | Closure theorem for Knopp's function. (Contributed by Asger C. Ipsen, 26-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → (abs‘𝐶) < 1) ⇒ ⊢ (𝜑 → (𝑊‘𝐴) ∈ ℂ) | ||
Theorem | addgtge0d 31666 | Addition of positive and nonnegative numbers is positive. (Contributed by Asger C. Ipsen, 12-May-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → 0 < (𝐴 + 𝐵)) | ||
Theorem | unblimceq0lem 31667* | Lemma for unblimceq0 31668. (Contributed by Asger C. Ipsen, 12-May-2021.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝐹:𝑆⟶ℂ) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → ∀𝑏 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐹‘𝑥)))) ⇒ ⊢ (𝜑 → ∀𝑐 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑦 ∈ 𝑆 (𝑦 ≠ 𝐴 ∧ (abs‘(𝑦 − 𝐴)) < 𝑑 ∧ 𝑐 ≤ (abs‘(𝐹‘𝑦)))) | ||
Theorem | unblimceq0 31668* | If 𝐹 is unbounded near 𝐴 it has no limit at 𝐴. (Contributed by Asger C. Ipsen, 12-May-2021.) |
⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝐹:𝑆⟶ℂ) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → ∀𝑏 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑥 ∈ 𝑆 ((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐹‘𝑥)))) ⇒ ⊢ (𝜑 → (𝐹 limℂ 𝐴) = ∅) | ||
Theorem | unbdqndv1 31669* | If the difference quotient (((𝐹‘𝑧) − (𝐹‘𝐴)) / (𝑧 − 𝐴)) is unbounded near 𝐴 then 𝐹 is not differentiable at 𝐴. (Contributed by Asger C. Ipsen, 12-May-2021.) |
⊢ 𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐴)) / (𝑧 − 𝐴))) & ⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ (𝜑 → 𝑋 ⊆ 𝑆) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → ∀𝑏 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑥 ∈ (𝑋 ∖ {𝐴})((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐺‘𝑥)))) ⇒ ⊢ (𝜑 → ¬ 𝐴 ∈ dom (𝑆 D 𝐹)) | ||
Theorem | unbdqndv2lem1 31670 | Lemma for unbdqndv2 31672. (Contributed by Asger C. Ipsen, 12-May-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐷 ≠ 0) & ⊢ (𝜑 → (2 · 𝐸) ≤ (abs‘((𝐴 − 𝐵) / 𝐷))) ⇒ ⊢ (𝜑 → ((𝐸 · (abs‘𝐷)) ≤ (abs‘(𝐴 − 𝐶)) ∨ (𝐸 · (abs‘𝐷)) ≤ (abs‘(𝐵 − 𝐶)))) | ||
Theorem | unbdqndv2lem2 31671* | Lemma for unbdqndv2 31672. (Contributed by Asger C. Ipsen, 12-May-2021.) |
⊢ 𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐴)) / (𝑧 − 𝐴))) & ⊢ 𝑊 = if((𝐵 · (𝑉 − 𝑈)) ≤ (abs‘((𝐹‘𝑈) − (𝐹‘𝐴))), 𝑈, 𝑉) & ⊢ (𝜑 → 𝑋 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ (𝜑 → 𝑈 ∈ 𝑋) & ⊢ (𝜑 → 𝑉 ∈ 𝑋) & ⊢ (𝜑 → 𝑈 ≠ 𝑉) & ⊢ (𝜑 → 𝑈 ≤ 𝐴) & ⊢ (𝜑 → 𝐴 ≤ 𝑉) & ⊢ (𝜑 → (𝑉 − 𝑈) < 𝐷) & ⊢ (𝜑 → (2 · 𝐵) ≤ ((abs‘((𝐹‘𝑉) − (𝐹‘𝑈))) / (𝑉 − 𝑈))) ⇒ ⊢ (𝜑 → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊 − 𝐴)) < 𝐷 ∧ 𝐵 ≤ (abs‘(𝐺‘𝑊))))) | ||
Theorem | unbdqndv2 31672* | Variant of unbdqndv1 31669 with the hypothesis that (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) is unbounded where 𝑥 ≤ 𝐴 and 𝐴 ≤ 𝑦. (Contributed by Asger C. Ipsen, 12-May-2021.) |
⊢ (𝜑 → 𝑋 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) & ⊢ (𝜑 → ∀𝑏 ∈ ℝ+ ∀𝑑 ∈ ℝ+ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ((𝑥 ≤ 𝐴 ∧ 𝐴 ≤ 𝑦) ∧ ((𝑦 − 𝑥) < 𝑑 ∧ 𝑥 ≠ 𝑦) ∧ 𝑏 ≤ ((abs‘((𝐹‘𝑦) − (𝐹‘𝑥))) / (𝑦 − 𝑥)))) ⇒ ⊢ (𝜑 → ¬ 𝐴 ∈ dom (ℝ D 𝐹)) | ||
Theorem | knoppndvlem1 31673 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐽 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) ⇒ ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ) | ||
Theorem | knoppndvlem2 31674 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ ℤ) & ⊢ (𝜑 → 𝐽 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐽 < 𝐼) ⇒ ⊢ (𝜑 → (((2 · 𝑁)↑𝐼) · ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)) ∈ ℤ) | ||
Theorem | knoppndvlem3 31675 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) |
⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) ⇒ ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) | ||
Theorem | knoppndvlem4 31676* | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → seq0( + , (𝐹‘𝐴)) ⇝ (𝑊‘𝐴)) | ||
Theorem | knoppndvlem5 31677* | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) ∈ ℝ) | ||
Theorem | knoppndvlem6 31678* | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝑊‘𝐴) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖)) | ||
Theorem | knoppndvlem7 31679* | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = ((𝐶↑𝐽) · (𝑇‘(𝑀 / 2)))) | ||
Theorem | knoppndvlem8 31680* | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 2 ∥ 𝑀) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = 0) | ||
Theorem | knoppndvlem9 31681* | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → ¬ 2 ∥ 𝑀) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = ((𝐶↑𝐽) / 2)) | ||
Theorem | knoppndvlem10 31682* | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (((abs‘𝐶)↑𝐽) / 2)) | ||
Theorem | knoppndvlem11 31683* | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 28-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖))) ≤ ((abs‘(𝐵 − 𝐴)) · Σ𝑖 ∈ (0...(𝐽 − 1))(((2 · 𝑁) · (abs‘𝐶))↑𝑖))) | ||
Theorem | knoppndvlem12 31684 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 29-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))) | ||
Theorem | knoppndvlem13 31685 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 1-Jul-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → 𝐶 ≠ 0) | ||
Theorem | knoppndvlem14 31686* | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 1-Jul-2021.) (Revised by Asger C. Ipsen, 7-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖))) ≤ ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) | ||
Theorem | knoppndvlem15 31687* | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 6-Jul-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ (abs‘((𝑊‘𝐵) − (𝑊‘𝐴)))) | ||
Theorem | knoppndvlem16 31688 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 19-Jul-2021.) |
⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝐵 − 𝐴) = (((2 · 𝑁)↑-𝐽) / 2)) | ||
Theorem | knoppndvlem17 31689* | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 12-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) & ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → ((((2 · 𝑁) · (abs‘𝐶))↑𝐽) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ ((abs‘((𝑊‘𝐵) − (𝑊‘𝐴))) / (𝐵 − 𝐴))) | ||
Theorem | knoppndvlem18 31690* | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 14-Aug-2021.) |
⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐺 ∈ ℝ+) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ ℕ0 ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷 ∧ 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · 𝐺))) | ||
Theorem | knoppndvlem19 31691* | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 17-Aug-2021.) |
⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑚) & ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑚 + 1)) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝐻 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → ∃𝑚 ∈ ℤ (𝐴 ≤ 𝐻 ∧ 𝐻 ≤ 𝐵)) | ||
Theorem | knoppndvlem20 31692 | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 18-Aug-2021.) |
⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ+) | ||
Theorem | knoppndvlem21 31693* | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 18-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ 𝐺 = (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐻 ∈ ℝ) & ⊢ (𝜑 → 𝐽 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) & ⊢ (𝜑 → (((2 · 𝑁)↑-𝐽) / 2) < 𝐷) & ⊢ (𝜑 → 𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝐽) · 𝐺)) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎 ≤ 𝐻 ∧ 𝐻 ≤ 𝑏) ∧ ((𝑏 − 𝑎) < 𝐷 ∧ 𝑎 ≠ 𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊‘𝑏) − (𝑊‘𝑎))) / (𝑏 − 𝑎)))) | ||
Theorem | knoppndvlem22 31694* | Lemma for knoppndv 31695. (Contributed by Asger C. Ipsen, 19-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐻 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎 ≤ 𝐻 ∧ 𝐻 ≤ 𝑏) ∧ ((𝑏 − 𝑎) < 𝐷 ∧ 𝑎 ≠ 𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊‘𝑏) − (𝑊‘𝑎))) / (𝑏 − 𝑎)))) | ||
Theorem | knoppndv 31695* | The continuous nowhere differentiable function 𝑊 ( Knopp, K. (1918). Math. Z. 2, 1-26 ) is, in fact, nowhere differentiable. (Contributed by Asger C. Ipsen, 19-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶))) ⇒ ⊢ (𝜑 → dom (ℝ D 𝑊) = ∅) | ||
Theorem | knoppf 31696* | Knopp's function is a function. (Contributed by Asger C. Ipsen, 25-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝑊:ℝ⟶ℝ) | ||
Theorem | knoppcn2 31697* | Variant of knoppcn 31664 with different codomain. (Contributed by Asger C. Ipsen, 25-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) ⇒ ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℝ)) | ||
Theorem | cnndvlem1 31698* | Lemma for cnndv 31700. (Contributed by Asger C. Ipsen, 25-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ (((1 / 2)↑𝑛) · (𝑇‘(((2 · 3)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) ⇒ ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅) | ||
Theorem | cnndvlem2 31699* | Lemma for cnndv 31700. (Contributed by Asger C. Ipsen, 26-Aug-2021.) |
⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) & ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ (((1 / 2)↑𝑛) · (𝑇‘(((2 · 3)↑𝑛) · 𝑦))))) & ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) ⇒ ⊢ ∃𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅) | ||
Theorem | cnndv 31700 | There exists a continuous nowhere differentiable function. The result follows directly from knoppcn 31664 and knoppndv 31695. (Contributed by Asger C. Ipsen, 26-Aug-2021.) |
⊢ ∃𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅) |
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