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Theorem List for Metamath Proof Explorer - 32401-32500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsubsym1 32401 A symmetry with [𝑥 / 𝑦].

See negsym1 32391 for more information. (Contributed by Anthony Hart, 11-Sep-2011.)

([𝑥 / 𝑦][𝑥 / 𝑦]⊥ → [𝑥 / 𝑦]𝜑)
 
20.11  Mathbox for Chen-Pang He
 
20.11.1  Ordinal topology
 
Theoremontopbas 32402 An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)
(𝐵 ∈ On → 𝐵 ∈ TopBases)
 
Theoremonsstopbas 32403 The class of ordinal numbers is a subclass of the class of topological bases. (Contributed by Chen-Pang He, 8-Oct-2015.)
On ⊆ TopBases
 
Theoremonpsstopbas 32404 The class of ordinal numbers is a proper subclass of the class of topological bases. (Contributed by Chen-Pang He, 9-Oct-2015.)
On ⊊ TopBases
 
Theoremontgval 32405 The topology generated from an ordinal number 𝐵 is suc 𝐵. (Contributed by Chen-Pang He, 10-Oct-2015.)
(𝐵 ∈ On → (topGen‘𝐵) = suc 𝐵)
 
Theoremontgsucval 32406 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)
 
Theoremonsuctop 32407 A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.)
(𝐴 ∈ On → suc 𝐴 ∈ Top)
 
Theoremonsuctopon 32408 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))
 
Theoremordtoplem 32409 Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.)
( 𝐴 ∈ On → suc 𝐴𝑆)       (Ord 𝐴 → (𝐴 𝐴𝐴𝑆))
 
Theoremordtop 32410 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))
 
Theoremonsucconni 32411 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
𝐴 ∈ On       suc 𝐴 ∈ Conn
 
Theoremonsucconn 32412 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
(𝐴 ∈ On → suc 𝐴 ∈ Conn)
 
Theoremordtopconn 32413 An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.)
(Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Conn))
 
Theoremonintopssconn 32414 An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)
(On ∩ Top) ⊆ Conn
 
Theoremonsuct0 32415 A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)
(𝐴 ∈ On → suc 𝐴 ∈ Kol2)
 
Theoremordtopt0 32416 An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.)
(Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2))
 
Theoremonsucsuccmpi 32417 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
 
Theoremonsucsuccmp 32418 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)
 
Theoremlimsucncmpi 32419 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
Lim 𝐴        ¬ suc 𝐴 ∈ Comp
 
Theoremlimsucncmp 32420 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
(Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
 
Theoremordcmp 32421 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 = 𝐴𝐴 = 1𝑜)))
 
Theoremssoninhaus 32422 The ordinal topologies 1𝑜 and 2𝑜 are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)
{1𝑜, 2𝑜} ⊆ (On ∩ Haus)
 
Theoremonint1 32423 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}
 
Theoremoninhaus 32424 The ordinal Hausdorff spaces are 1𝑜 and 2𝑜. (Contributed by Chen-Pang He, 10-Nov-2015.)
(On ∩ Haus) = {1𝑜, 2𝑜}
 
20.12  Mathbox for Jeff Hoffman
 
20.12.1  Inferences for finite induction on generic function values
 
Theoremfveleq 32425 Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
(𝐴 = 𝐵 → ((𝜑 → (𝐹𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹𝐵) ∈ 𝑃)))
 
Theoremfindfvcl 32426* Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
(𝜑 → (𝐹‘∅) ∈ 𝑃)    &   (𝑦 ∈ ω → (𝜑 → ((𝐹𝑦) ∈ 𝑃 → (𝐹‘suc 𝑦) ∈ 𝑃)))       (𝐴 ∈ ω → (𝜑 → (𝐹𝐴) ∈ 𝑃))
 
Theoremfindreccl 32427* Please add description here. (Contributed by Jeff Hoffman, 19-Feb-2008.)
(𝑧𝑃 → (𝐺𝑧) ∈ 𝑃)       (𝐶 ∈ ω → (𝐴𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))
 
Theoremfindabrcl 32428* Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)
(𝑧𝑃 → (𝐺𝑧) ∈ 𝑃)       ((𝐶 ∈ ω ∧ 𝐴𝑃) → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) ∈ 𝑃)
 
20.12.2  gdc.mm
 
Theoremnnssi2 32429 Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
ℕ ⊆ 𝐷    &   (𝐵 ∈ ℕ → 𝜑)    &   ((𝐴𝐷𝐵𝐷𝜑) → 𝜓)       ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝜓)
 
Theoremnnssi3 32430 Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
ℕ ⊆ 𝐷    &   (𝐶 ∈ ℕ → 𝜑)    &   (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ 𝜑) → 𝜓)       ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓)
 
Theoremnndivsub 32431 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 ↔ ((𝐵𝐴) / 𝐶) ∈ ℕ))
 
Theoremnndivlub 32432 A factor of a positive integer cannot exceed it. (Contributed by Jeff Hoffman, 17-Jun-2008.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 / 𝐵) ∈ ℕ → 𝐵𝐴))
 
SyntaxcgcdOLD 32433 Extend class notation to include the gdc function. (New usage is discouraged.)
class gcdOLD (𝐴, 𝐵)
 
Definitiondf-gcdOLD 32434* 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, < )
 
Theoremee7.2aOLD 32435 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, (𝐵𝐴))))
 
20.13  Mathbox for Asger C. Ipsen
 
20.13.1  Continuous nowhere differentiable functions
 
Theoremdnival 32436* 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 → (𝑇𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))
 
Theoremdnicld1 32437 Closure theorem for the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ ℝ)
 
Theoremdnicld2 32438* 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)))    &   (𝜑𝐴 ∈ ℝ)       (𝜑 → (𝑇𝐴) ∈ ℝ)
 
Theoremdnif 32439 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
 
Theoremdnizeq0 32440* 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)))    &   (𝜑𝐴 ∈ ℤ)       (𝜑 → (𝑇𝐴) = 0)
 
Theoremdnizphlfeqhlf 32441* 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)))    &   (𝜑𝐴 ∈ ℤ)       (𝜑 → (𝑇‘(𝐴 + (1 / 2))) = (1 / 2))
 
Theoremrddif2 32442 Variant of rddif 14061. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐴 ∈ ℝ → 0 ≤ ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))))
 
Theoremdnibndlem1 32443* Lemma for dnibnd 32456. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → ((abs‘((𝑇𝐵) − (𝑇𝐴))) ≤ 𝑆 ↔ (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ 𝑆))
 
Theoremdnibndlem2 32444* Lemma for dnibnd 32456. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (⌊‘(𝐵 + (1 / 2))) = (⌊‘(𝐴 + (1 / 2))))       (𝜑 → (abs‘((𝑇𝐵) − (𝑇𝐴))) ≤ (abs‘(𝐵𝐴)))
 
Theoremdnibndlem3 32445 Lemma for dnibnd 32456. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (⌊‘(𝐵 + (1 / 2))) = ((⌊‘(𝐴 + (1 / 2))) + 1))       (𝜑 → (abs‘(𝐵𝐴)) = (abs‘((𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2))) + (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴))))
 
Theoremdnibndlem4 32446 Lemma for dnibnd 32456. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐵 ∈ ℝ → 0 ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2))))
 
Theoremdnibndlem5 32447 Lemma for dnibnd 32456. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐴 ∈ ℝ → 0 < (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴))
 
Theoremdnibndlem6 32448 Lemma for dnibnd 32456. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ 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)))))
 
Theoremdnibndlem7 32449 Lemma for dnibnd 32456. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝜑𝐵 ∈ ℝ)       (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2))))
 
Theoremdnibndlem8 32450 Lemma for dnibnd 32456. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ≤ (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴))
 
Theoremdnibndlem9 32451* Lemma for dnibnd 32456. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (⌊‘(𝐵 + (1 / 2))) = ((⌊‘(𝐴 + (1 / 2))) + 1))       (𝜑 → (abs‘((𝑇𝐵) − (𝑇𝐴))) ≤ (abs‘(𝐵𝐴)))
 
Theoremdnibndlem10 32452 Lemma for dnibnd 32456. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2))))       (𝜑 → 1 ≤ (𝐵𝐴))
 
Theoremdnibndlem11 32453 Lemma for dnibnd 32456. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (1 / 2))
 
Theoremdnibndlem12 32454* Lemma for dnibnd 32456. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2))))       (𝜑 → (abs‘((𝑇𝐵) − (𝑇𝐴))) ≤ (abs‘(𝐵𝐴)))
 
Theoremdnibndlem13 32455* Lemma for dnibnd 32456. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2))))       (𝜑 → (abs‘((𝑇𝐵) − (𝑇𝐴))) ≤ (abs‘(𝐵𝐴)))
 
Theoremdnibnd 32456* 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)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (abs‘((𝑇𝐵) − (𝑇𝐴))) ≤ (abs‘(𝐵𝐴)))
 
Theoremdnicn 32457 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)
 
Theoremknoppcnlem1 32458* Lemma for knoppcn 32469. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝑀 ∈ ℕ0)       (𝜑 → ((𝐹𝐴)‘𝑀) = ((𝐶𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴))))
 
Theoremknoppcnlem2 32459* Lemma for knoppcn 32469. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝑀 ∈ ℕ0)       (𝜑 → ((𝐶𝑀) · (𝑇‘(((2 · 𝑁)↑𝑀) · 𝐴))) ∈ ℝ)
 
Theoremknoppcnlem3 32460* Lemma for knoppcn 32469. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝑀 ∈ ℕ0)       (𝜑 → ((𝐹𝐴)‘𝑀) ∈ ℝ)
 
Theoremknoppcnlem4 32461* Lemma for knoppcn 32469. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝑀 ∈ ℕ0)       (𝜑 → (abs‘((𝐹𝐴)‘𝑀)) ≤ ((𝑚 ∈ ℕ0 ↦ ((abs‘𝐶)↑𝑚))‘𝑀))
 
Theoremknoppcnlem5 32462* Lemma for knoppcn 32469. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹𝑧)‘𝑚))):ℕ0⟶(ℂ ↑𝑚 ℝ))
 
Theoremknoppcnlem6 32463* Lemma for knoppcn 32469. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (abs‘𝐶) < 1)       (𝜑 → seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹𝑧)‘𝑚)))) ∈ dom (⇝𝑢‘ℝ))
 
Theoremknoppcnlem7 32464* Lemma for knoppcn 32469. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝑀 ∈ ℕ0)       (𝜑 → (seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹𝑧)‘𝑚))))‘𝑀) = (𝑤 ∈ ℝ ↦ (seq0( + , (𝐹𝑤))‘𝑀)))
 
Theoremknoppcnlem8 32465* Lemma for knoppcn 32469. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹𝑧)‘𝑚)))):ℕ0⟶(ℂ ↑𝑚 ℝ))
 
Theoremknoppcnlem9 32466* Lemma for knoppcn 32469. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (abs‘𝐶) < 1)       (𝜑 → seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹𝑧)‘𝑚))))(⇝𝑢‘ℝ)𝑊)
 
Theoremknoppcnlem10 32467* Lemma for knoppcn 32469. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝑀 ∈ ℕ0)       (𝜑 → (𝑧 ∈ ℝ ↦ ((𝐹𝑧)‘𝑀)) ∈ ((topGen‘ran (,)) Cn (TopOpen‘ℂfld)))
 
Theoremknoppcnlem11 32468* Lemma for knoppcn 32469. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹𝑧)‘𝑚)))):ℕ0⟶(ℝ–cn→ℂ))
 
Theoremknoppcn 32469* 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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (abs‘𝐶) < 1)       (𝜑𝑊 ∈ (ℝ–cn→ℂ))
 
Theoremknoppcld 32470* 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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (abs‘𝐶) < 1)       (𝜑 → (𝑊𝐴) ∈ ℂ)
 
Theoremaddgtge0d 32471 Addition of positive and nonnegative numbers is positive. (Contributed by Asger C. Ipsen, 12-May-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 0 ≤ 𝐵)       (𝜑 → 0 < (𝐴 + 𝐵))
 
Theoremunblimceq0lem 32472* Lemma for unblimceq0 32473. (Contributed by Asger C. Ipsen, 12-May-2021.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐹:𝑆⟶ℂ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑 → ∀𝑏 ∈ ℝ+𝑑 ∈ ℝ+𝑥𝑆 ((abs‘(𝑥𝐴)) < 𝑑𝑏 ≤ (abs‘(𝐹𝑥))))       (𝜑 → ∀𝑐 ∈ ℝ+𝑑 ∈ ℝ+𝑦𝑆 (𝑦𝐴 ∧ (abs‘(𝑦𝐴)) < 𝑑𝑐 ≤ (abs‘(𝐹𝑦))))
 
Theoremunblimceq0 32473* If 𝐹 is unbounded near 𝐴 it has no limit at 𝐴. (Contributed by Asger C. Ipsen, 12-May-2021.)
(𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐹:𝑆⟶ℂ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑 → ∀𝑏 ∈ ℝ+𝑑 ∈ ℝ+𝑥𝑆 ((abs‘(𝑥𝐴)) < 𝑑𝑏 ≤ (abs‘(𝐹𝑥))))       (𝜑 → (𝐹 lim 𝐴) = ∅)
 
Theoremunbdqndv1 32474* If the difference quotient (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)) is unbounded near 𝐴 then 𝐹 is not differentiable at 𝐴. (Contributed by Asger C. Ipsen, 12-May-2021.)
𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)))    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝑋𝑆)    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑 → ∀𝑏 ∈ ℝ+𝑑 ∈ ℝ+𝑥 ∈ (𝑋 ∖ {𝐴})((abs‘(𝑥𝐴)) < 𝑑𝑏 ≤ (abs‘(𝐺𝑥))))       (𝜑 → ¬ 𝐴 ∈ dom (𝑆 D 𝐹))
 
Theoremunbdqndv2lem1 32475 Lemma for unbdqndv2 32477. (Contributed by Asger C. Ipsen, 12-May-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐷 ≠ 0)    &   (𝜑 → (2 · 𝐸) ≤ (abs‘((𝐴𝐵) / 𝐷)))       (𝜑 → ((𝐸 · (abs‘𝐷)) ≤ (abs‘(𝐴𝐶)) ∨ (𝐸 · (abs‘𝐷)) ≤ (abs‘(𝐵𝐶))))
 
Theoremunbdqndv2lem2 32476* Lemma for unbdqndv2 32477. (Contributed by Asger C. Ipsen, 12-May-2021.)
𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹𝑧) − (𝐹𝐴)) / (𝑧𝐴)))    &   𝑊 = if((𝐵 · (𝑉𝑈)) ≤ (abs‘((𝐹𝑈) − (𝐹𝐴))), 𝑈, 𝑉)    &   (𝜑𝑋 ⊆ ℝ)    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝑈𝑋)    &   (𝜑𝑉𝑋)    &   (𝜑𝑈𝑉)    &   (𝜑𝑈𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑 → (𝑉𝑈) < 𝐷)    &   (𝜑 → (2 · 𝐵) ≤ ((abs‘((𝐹𝑉) − (𝐹𝑈))) / (𝑉𝑈)))       (𝜑 → (𝑊 ∈ (𝑋 ∖ {𝐴}) ∧ ((abs‘(𝑊𝐴)) < 𝐷𝐵 ≤ (abs‘(𝐺𝑊)))))
 
Theoremunbdqndv2 32477* Variant of unbdqndv1 32474 with the hypothesis that (((𝐹𝑦) − (𝐹𝑥)) / (𝑦𝑥)) is unbounded where 𝑥𝐴 and 𝐴𝑦. (Contributed by Asger C. Ipsen, 12-May-2021.)
(𝜑𝑋 ⊆ ℝ)    &   (𝜑𝐹:𝑋⟶ℂ)    &   (𝜑 → ∀𝑏 ∈ ℝ+𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐴𝐴𝑦) ∧ ((𝑦𝑥) < 𝑑𝑥𝑦) ∧ 𝑏 ≤ ((abs‘((𝐹𝑦) − (𝐹𝑥))) / (𝑦𝑥))))       (𝜑 → ¬ 𝐴 ∈ dom (ℝ D 𝐹))
 
Theoremknoppndvlem1 32478 Lemma for knoppndv 32500. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐽 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ)
 
Theoremknoppndvlem2 32479 Lemma for knoppndv 32500. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐼 ∈ ℤ)    &   (𝜑𝐽 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐽 < 𝐼)       (𝜑 → (((2 · 𝑁)↑𝐼) · ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)) ∈ ℤ)
 
Theoremknoppndvlem3 32480 Lemma for knoppndv 32500. (Contributed by Asger C. Ipsen, 15-Jun-2021.)
(𝜑𝐶 ∈ (-1(,)1))       (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
 
Theoremknoppndvlem4 32481* Lemma for knoppndv 32500. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → seq0( + , (𝐹𝐴)) ⇝ (𝑊𝐴))
 
Theoremknoppndvlem5 32482* Lemma for knoppndv 32500. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹𝐴)‘𝑖) ∈ ℝ)
 
Theoremknoppndvlem6 32483* Lemma for knoppndv 32500. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝑊𝐴) = Σ𝑖 ∈ (0...𝐽)((𝐹𝐴)‘𝑖))
 
Theoremknoppndvlem7 32484* Lemma for knoppndv 32500. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ((𝐹𝐴)‘𝐽) = ((𝐶𝐽) · (𝑇‘(𝑀 / 2))))
 
Theoremknoppndvlem8 32485* Lemma for knoppndv 32500. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 2 ∥ 𝑀)       (𝜑 → ((𝐹𝐴)‘𝐽) = 0)
 
Theoremknoppndvlem9 32486* Lemma for knoppndv 32500. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝑀)       (𝜑 → ((𝐹𝐴)‘𝐽) = ((𝐶𝐽) / 2))
 
Theoremknoppndvlem10 32487* Lemma for knoppndv 32500. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (abs‘(((𝐹𝐵)‘𝐽) − ((𝐹𝐴)‘𝐽))) = (((abs‘𝐶)↑𝐽) / 2))
 
Theoremknoppndvlem11 32488* Lemma for knoppndv 32500. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹𝐴)‘𝑖))) ≤ ((abs‘(𝐵𝐴)) · Σ𝑖 ∈ (0...(𝐽 − 1))(((2 · 𝑁) · (abs‘𝐶))↑𝑖)))
 
Theoremknoppndvlem12 32489 Lemma for knoppndv 32500. (Contributed by Asger C. Ipsen, 29-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
(𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1)))
 
Theoremknoppndvlem13 32490 Lemma for knoppndv 32500. (Contributed by Asger C. Ipsen, 1-Jul-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
(𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑𝐶 ≠ 0)
 
Theoremknoppndvlem14 32491* Lemma for knoppndv 32500. (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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹𝐴)‘𝑖))) ≤ ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))
 
Theoremknoppndvlem15 32492* Lemma for knoppndv 32500. (Contributed by Asger C. Ipsen, 6-Jul-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ (abs‘((𝑊𝐵) − (𝑊𝐴))))
 
Theoremknoppndvlem16 32493 Lemma for knoppndv 32500. (Contributed by Asger C. Ipsen, 19-Jul-2021.)
𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝐵𝐴) = (((2 · 𝑁)↑-𝐽) / 2))
 
Theoremknoppndvlem17 32494* Lemma for knoppndv 32500. (Contributed by Asger C. Ipsen, 12-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → ((((2 · 𝑁) · (abs‘𝐶))↑𝐽) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ ((abs‘((𝑊𝐵) − (𝑊𝐴))) / (𝐵𝐴)))
 
Theoremknoppndvlem18 32495* Lemma for knoppndv 32500. (Contributed by Asger C. Ipsen, 14-Aug-2021.)
(𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐺 ∈ ℝ+)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → ∃𝑗 ∈ ℕ0 ((((2 · 𝑁)↑-𝑗) / 2) < 𝐷𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝑗) · 𝐺)))
 
Theoremknoppndvlem19 32496* Lemma for knoppndv 32500. (Contributed by Asger C. Ipsen, 17-Aug-2021.)
𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑚)    &   𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑚 + 1))    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝐻 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ∃𝑚 ∈ ℤ (𝐴𝐻𝐻𝐵))
 
Theoremknoppndvlem20 32497 Lemma for knoppndv 32500. (Contributed by Asger C. Ipsen, 18-Aug-2021.)
(𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ+)
 
Theoremknoppndvlem21 32498* Lemma for knoppndv 32500. (Contributed by Asger C. Ipsen, 18-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   𝐺 = (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐻 ∈ ℝ)    &   (𝜑𝐽 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))    &   (𝜑 → (((2 · 𝑁)↑-𝐽) / 2) < 𝐷)    &   (𝜑𝐸 ≤ ((((2 · 𝑁) · (abs‘𝐶))↑𝐽) · 𝐺))       (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎𝐻𝐻𝑏) ∧ ((𝑏𝑎) < 𝐷𝑎𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊𝑏) − (𝑊𝑎))) / (𝑏𝑎))))
 
Theoremknoppndvlem22 32499* Lemma for knoppndv 32500. (Contributed by Asger C. Ipsen, 19-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐻 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎𝐻𝐻𝑏) ∧ ((𝑏𝑎) < 𝐷𝑎𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊𝑏) − (𝑊𝑎))) / (𝑏𝑎))))
 
Theoremknoppndv 32500* 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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → dom (ℝ D 𝑊) = ∅)
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