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Theorem List for Metamath Proof Explorer - 31401-31500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdf3nandALT1 31401 The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011.)
((𝜑𝜓𝜒) ↔ (𝜑 ⊼ ((𝜓𝜒) ⊼ (𝜓𝜒))))
 
Theoremdf3nandALT2 31402 The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.)
((𝜑𝜓𝜒) ↔ ¬ (𝜑𝜓𝜒))
 
Theoremandnand1 31403 Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓𝜒) ⊼ (𝜑𝜓𝜒)))
 
Theoremimnand2 31404 An nand relation. (Contributed by Anthony Hart, 2-Sep-2011.)
((¬ 𝜑𝜓) ↔ ((𝜑𝜑) ⊼ (𝜓𝜓)))
 
20.10.2  Predicate Calculus
 
Theoremallt 31405 For all sets, is true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥
 
Theoremalnof 31406 For all sets, is not true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥 ¬ ⊥
 
Theoremnalf 31407 Not all sets hold as true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∀𝑥
 
Theoremextt 31408 There exists a set that holds as true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥
 
Theoremnextnt 31409 There does not exist a set, such that is not true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃𝑥 ¬ ⊤
 
Theoremnextf 31410 There does not exist a set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃𝑥
 
Theoremunnf 31411 There does not exist exactly one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃!𝑥
 
Theoremunnt 31412 There does not exist exactly one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃!𝑥
 
Theoremmont 31413 There does not exist at most one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃*𝑥
 
Theoremmof 31414 There exist at most one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
∃*𝑥
 
20.10.3  Misc. Single Axiom Systems
 
Theoremmeran1 31415 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜃𝜑) ∨ (𝜒 ∨ (𝜏𝜑))))
 
Theoremmeran2 31416 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜏𝜃) ∨ (𝜒 ∨ (𝜑𝜃))))
 
Theoremmeran3 31417 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜒𝜑) ∨ (𝜏 ∨ (𝜃𝜑))))
 
Theoremwaj-ax 31418 A single axiom for propositional calculus offered by Wajsberg. (Contributed by Anthony Hart, 13-Aug-2011.)
((𝜑 ⊼ (𝜓𝜒)) ⊼ (((𝜃𝜒) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃))) ⊼ (𝜑 ⊼ (𝜑𝜓))))
 
Theoremlukshef-ax2 31419 A single axiom for propositional calculus offered by Lukasiewicz. (Contributed by Anthony Hart, 14-Aug-2011.)
((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜑 ⊼ (𝜒𝜑)) ⊼ ((𝜃𝜓) ⊼ ((𝜑𝜃) ⊼ (𝜑𝜃)))))
 
Theoremarg-ax 31420 ? (Contributed by Anthony Hart, 14-Aug-2011.)
((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜃𝜒) ⊼ ((𝜒𝜃) ⊼ (𝜑𝜃)))))
 
20.10.4  Connective Symmetry
 
Theoremnegsym1 31421 In the paper "On Variable Functors of Propositional Arguments", Lukasiewicz introduced a system that can handle variable connectives. This was done by introducing a variable, marked with a lowercase delta, which takes a wff as input. In the system, "delta 𝜑 " means that "something is true of 𝜑." "delta 𝜑 " can be substituted with ¬ 𝜑, 𝜓𝜑, 𝑥𝜑, etc.

Later on, Meredith discovered a single axiom, in the form of ( delta delta ⊥ → delta 𝜑 ). This represents the shortest theorem in the extended propositional calculus that cannot be derived as an instance of a theorem in propositional calculus.

A symmetry with ¬. (Contributed by Anthony Hart, 4-Sep-2011.)

(¬ ¬ ⊥ → ¬ 𝜑)
 
Theoremimsym1 31422 A symmetry with .

See negsym1 31421 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

((𝜓 → (𝜓 → ⊥)) → (𝜓𝜑))
 
Theorembisym1 31423 A symmetry with .

See negsym1 31421 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

((𝜓 ↔ (𝜓 ↔ ⊥)) → (𝜓𝜑))
 
Theoremconsym1 31424 A symmetry with .

See negsym1 31421 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓𝜑))
 
Theoremdissym1 31425 A symmetry with .

See negsym1 31421 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

((𝜓 ∨ (𝜓 ∨ ⊥)) → (𝜓𝜑))
 
Theoremnandsym1 31426 A symmetry with .

See negsym1 31421 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

((𝜓 ⊼ (𝜓 ⊼ ⊥)) → (𝜓𝜑))
 
Theoremunisym1 31427 A symmetry with .

See negsym1 31421 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

(∀𝑥𝑥⊥ → ∀𝑥𝜑)
 
Theoremexisym1 31428 A symmetry with .

See negsym1 31421 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

(∃𝑥𝑥⊥ → ∃𝑥𝜑)
 
Theoremunqsym1 31429 A symmetry with ∃!.

See negsym1 31421 for more information. (Contributed by Anthony Hart, 6-Sep-2011.)

(∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑)
 
Theoremamosym1 31430 A symmetry with ∃*.

See negsym1 31421 for more information. (Contributed by Anthony Hart, 13-Sep-2011.)

(∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
 
Theoremsubsym1 31431 A symmetry with [𝑥 / 𝑦].

See negsym1 31421 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 31432 An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)
(𝐵 ∈ On → 𝐵 ∈ TopBases)
 
Theoremonsstopbas 31433 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 31434 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 31435 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 31436 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 31437 A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.)
(𝐴 ∈ On → suc 𝐴 ∈ Top)
 
Theoremonsuctopon 31438 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 31439 Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.)
( 𝐴 ∈ On → suc 𝐴𝑆)       (Ord 𝐴 → (𝐴 𝐴𝐴𝑆))
 
Theoremordtop 31440 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))
 
Theoremonsucconi 31441 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
𝐴 ∈ On       suc 𝐴 ∈ Con
 
Theoremonsuccon 31442 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
(𝐴 ∈ On → suc 𝐴 ∈ Con)
 
Theoremordtopcon 31443 An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.)
(Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Con))
 
Theoremonintopsscon 31444 An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)
(On ∩ Top) ⊆ Con
 
Theoremonsuct0 31445 A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)
(𝐴 ∈ On → suc 𝐴 ∈ Kol2)
 
Theoremordtopt0 31446 An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.)
(Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2))
 
Theoremonsucsuccmpi 31447 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 31448 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 31449 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
Lim 𝐴        ¬ suc 𝐴 ∈ Comp
 
Theoremlimsucncmp 31450 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
(Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)
 
Theoremordcmp 31451 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 31452 The ordinal topologies 1𝑜 and 2𝑜 are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)
{1𝑜, 2𝑜} ⊆ (On ∩ Haus)
 
Theoremonint1 31453 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 31454 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 31455 Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
(𝐴 = 𝐵 → ((𝜑 → (𝐹𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹𝐵) ∈ 𝑃)))
 
Theoremfindfvcl 31456* 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 31457* Please add description here. (Contributed by Jeff Hoffman, 19-Feb-2008.)
(𝑧𝑃 → (𝐺𝑧) ∈ 𝑃)       (𝐶 ∈ ω → (𝐴𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))
 
Theoremfindabrcl 31458* 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 31459 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 31460 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 31461 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 31462 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 31463 Extend class notation to include the gdc function. (New usage is discouraged.)
class gcdOLD (𝐴, 𝐵)
 
Definitiondf-gcdOLD 31464* 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 31465 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 31466* 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 31467 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 31468* 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 31469 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 31470* 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 31471* 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 31472 Variant of rddif 13785. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐴 ∈ ℝ → 0 ≤ ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))))
 
Theoremdnibndlem1 31473* Lemma for dnibnd 31486. (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 31474* Lemma for dnibnd 31486. (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 31475 Lemma for dnibnd 31486. (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 31476 Lemma for dnibnd 31486. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐵 ∈ ℝ → 0 ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2))))
 
Theoremdnibndlem5 31477 Lemma for dnibnd 31486. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐴 ∈ ℝ → 0 < (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴))
 
Theoremdnibndlem6 31478 Lemma for dnibnd 31486. (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 31479 Lemma for dnibnd 31486. (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 31480 Lemma for dnibnd 31486. (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 31481* Lemma for dnibnd 31486. (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 31482 Lemma for dnibnd 31486. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2))))       (𝜑 → 1 ≤ (𝐵𝐴))
 
Theoremdnibndlem11 31483 Lemma for dnibnd 31486. (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 31484* Lemma for dnibnd 31486. (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 31485* Lemma for dnibnd 31486. (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 31486* 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 31487 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 31488* Lemma for knoppcn 31499. (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 31489* Lemma for knoppcn 31499. (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 31490* Lemma for knoppcn 31499. (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 31491* Lemma for knoppcn 31499. (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 31492* Lemma for knoppcn 31499. (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 31493* Lemma for knoppcn 31499. (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 31494* Lemma for knoppcn 31499. (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 31495* Lemma for knoppcn 31499. (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 31496* Lemma for knoppcn 31499. (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 31497* Lemma for knoppcn 31499. (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 31498* Lemma for knoppcn 31499. (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 31499* 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 31500* 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)
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