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Theorem List for Metamath Proof Explorer - 35301-35400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdissym1 35301 A symmetry with ∨.

See negsym1 35297 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

((πœ“ ∨ (πœ“ ∨ βŠ₯)) β†’ (πœ“ ∨ πœ‘))
 
Theoremnandsym1 35302 A symmetry with ⊼.

See negsym1 35297 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

((πœ“ ⊼ (πœ“ ⊼ βŠ₯)) β†’ (πœ“ ⊼ πœ‘))
 
Theoremunisym1 35303 A symmetry with βˆ€.

See negsym1 35297 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

(βˆ€π‘₯βˆ€π‘₯βŠ₯ β†’ βˆ€π‘₯πœ‘)
 
Theoremexisym1 35304 A symmetry with βˆƒ.

See negsym1 35297 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

(βˆƒπ‘₯βˆƒπ‘₯βŠ₯ β†’ βˆƒπ‘₯πœ‘)
 
Theoremunqsym1 35305 A symmetry with βˆƒ!.

See negsym1 35297 for more information. (Contributed by Anthony Hart, 6-Sep-2011.)

(βˆƒ!π‘₯βˆƒ!π‘₯βŠ₯ β†’ βˆƒ!π‘₯πœ‘)
 
Theoremamosym1 35306 A symmetry with βˆƒ*.

See negsym1 35297 for more information. (Contributed by Anthony Hart, 13-Sep-2011.)

(βˆƒ*π‘₯βˆƒ*π‘₯βŠ₯ β†’ βˆƒ*π‘₯πœ‘)
 
Theoremsubsym1 35307 A symmetry with [π‘₯ / 𝑦].

See negsym1 35297 for more information. (Contributed by Anthony Hart, 11-Sep-2011.)

([𝑦 / π‘₯][𝑦 / π‘₯]βŠ₯ β†’ [𝑦 / π‘₯]πœ‘)
 
21.14  Mathbox for Chen-Pang He
 
21.14.1  Ordinal topology
 
Theoremontopbas 35308 An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)
(𝐡 ∈ On β†’ 𝐡 ∈ TopBases)
 
Theoremonsstopbas 35309 The class of ordinal numbers is a subclass of the class of topological bases. (Contributed by Chen-Pang He, 8-Oct-2015.)
On βŠ† TopBases
 
Theoremonpsstopbas 35310 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 35311 The topology generated from an ordinal number 𝐡 is suc βˆͺ 𝐡. (Contributed by Chen-Pang He, 10-Oct-2015.)
(𝐡 ∈ On β†’ (topGenβ€˜π΅) = suc βˆͺ 𝐡)
 
Theoremontgsucval 35312 The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.)
(𝐴 ∈ On β†’ (topGenβ€˜suc 𝐴) = suc 𝐴)
 
Theoremonsuctop 35313 A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.)
(𝐴 ∈ On β†’ suc 𝐴 ∈ Top)
 
Theoremonsuctopon 35314 One of the topologies on an ordinal number is its successor. (Contributed by Chen-Pang He, 7-Nov-2015.)
(𝐴 ∈ On β†’ suc 𝐴 ∈ (TopOnβ€˜π΄))
 
Theoremordtoplem 35315 Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.)
(βˆͺ 𝐴 ∈ On β†’ suc βˆͺ 𝐴 ∈ 𝑆)    β‡’   (Ord 𝐴 β†’ (𝐴 β‰  βˆͺ 𝐴 β†’ 𝐴 ∈ 𝑆))
 
Theoremordtop 35316 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 β†’ (𝐽 ∈ Top ↔ 𝐽 β‰  βˆͺ 𝐽))
 
Theoremonsucconni 35317 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
𝐴 ∈ On    β‡’   suc 𝐴 ∈ Conn
 
Theoremonsucconn 35318 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
(𝐴 ∈ On β†’ suc 𝐴 ∈ Conn)
 
Theoremordtopconn 35319 An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.)
(Ord 𝐽 β†’ (𝐽 ∈ Top ↔ 𝐽 ∈ Conn))
 
Theoremonintopssconn 35320 An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)
(On ∩ Top) βŠ† Conn
 
Theoremonsuct0 35321 A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)
(𝐴 ∈ On β†’ suc 𝐴 ∈ Kol2)
 
Theoremordtopt0 35322 An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.)
(Ord 𝐽 β†’ (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2))
 
Theoremonsucsuccmpi 35323 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    β‡’   suc suc 𝐴 ∈ Comp
 
Theoremonsucsuccmp 35324 The successor of a successor ordinal number is a compact topology. (Contributed by Chen-Pang He, 18-Oct-2015.)
(𝐴 ∈ On β†’ suc suc 𝐴 ∈ Comp)
 
Theoremlimsucncmpi 35325 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
Lim 𝐴    β‡’    Β¬ suc 𝐴 ∈ Comp
 
Theoremlimsucncmp 35326 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
(Lim 𝐴 β†’ Β¬ suc 𝐴 ∈ Comp)
 
Theoremordcmp 35327 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 β†’ (𝐴 ∈ Comp ↔ (βˆͺ 𝐴 = βˆͺ βˆͺ 𝐴 β†’ 𝐴 = 1o)))
 
Theoremssoninhaus 35328 The ordinal topologies 1o and 2o are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)
{1o, 2o} βŠ† (On ∩ Haus)
 
Theoremonint1 35329 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 35330 The ordinal Hausdorff spaces are 1o and 2o. (Contributed by Chen-Pang He, 10-Nov-2015.)
(On ∩ Haus) = {1o, 2o}
 
21.15  Mathbox for Jeff Hoffman
 
21.15.1  Inferences for finite induction on generic function values
 
Theoremfveleq 35331 Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
(𝐴 = 𝐡 β†’ ((πœ‘ β†’ (πΉβ€˜π΄) ∈ 𝑃) ↔ (πœ‘ β†’ (πΉβ€˜π΅) ∈ 𝑃)))
 
Theoremfindfvcl 35332* Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
(πœ‘ β†’ (πΉβ€˜βˆ…) ∈ 𝑃)    &   (𝑦 ∈ Ο‰ β†’ (πœ‘ β†’ ((πΉβ€˜π‘¦) ∈ 𝑃 β†’ (πΉβ€˜suc 𝑦) ∈ 𝑃)))    β‡’   (𝐴 ∈ Ο‰ β†’ (πœ‘ β†’ (πΉβ€˜π΄) ∈ 𝑃))
 
Theoremfindreccl 35333* Please add description here. (Contributed by Jeff Hoffman, 19-Feb-2008.)
(𝑧 ∈ 𝑃 β†’ (πΊβ€˜π‘§) ∈ 𝑃)    β‡’   (𝐢 ∈ Ο‰ β†’ (𝐴 ∈ 𝑃 β†’ (rec(𝐺, 𝐴)β€˜πΆ) ∈ 𝑃))
 
Theoremfindabrcl 35334* Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)
(𝑧 ∈ 𝑃 β†’ (πΊβ€˜π‘§) ∈ 𝑃)    β‡’   ((𝐢 ∈ Ο‰ ∧ 𝐴 ∈ 𝑃) β†’ ((π‘₯ ∈ V ↦ (rec(𝐺, 𝐴)β€˜π‘₯))β€˜πΆ) ∈ 𝑃)
 
21.15.2  gdc.mm
 
Theoremnnssi2 35335 Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
β„• βŠ† 𝐷    &   (𝐡 ∈ β„• β†’ πœ‘)    &   ((𝐴 ∈ 𝐷 ∧ 𝐡 ∈ 𝐷 ∧ πœ‘) β†’ πœ“)    β‡’   ((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) β†’ πœ“)
 
Theoremnnssi3 35336 Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
β„• βŠ† 𝐷    &   (𝐢 ∈ β„• β†’ πœ‘)    &   (((𝐴 ∈ 𝐷 ∧ 𝐡 ∈ 𝐷 ∧ 𝐢 ∈ 𝐷) ∧ πœ‘) β†’ πœ“)    β‡’   ((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ 𝐢 ∈ β„•) β†’ πœ“)
 
Theoremnndivsub 35337 Please add description here. (Contributed by Jeff Hoffman, 17-Jun-2008.)
(((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ 𝐢 ∈ β„•) ∧ ((𝐴 / 𝐢) ∈ β„• ∧ 𝐴 < 𝐡)) β†’ ((𝐡 / 𝐢) ∈ β„• ↔ ((𝐡 βˆ’ 𝐴) / 𝐢) ∈ β„•))
 
Theoremnndivlub 35338 A factor of a positive integer cannot exceed it. (Contributed by Jeff Hoffman, 17-Jun-2008.)
((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) β†’ ((𝐴 / 𝐡) ∈ β„• β†’ 𝐡 ≀ 𝐴))
 
SyntaxcgcdOLD 35339 Extend class notation to include the gdc function. (New usage is discouraged.)
class gcdOLD (𝐴, 𝐡)
 
Definitiondf-gcdOLD 35340* gcdOLD (𝐴, 𝐡) is the largest positive integer that evenly divides both 𝐴 and 𝐡. (Contributed by Jeff Hoffman, 17-Jun-2008.) (New usage is discouraged.)
gcdOLD (𝐴, 𝐡) = sup({π‘₯ ∈ β„• ∣ ((𝐴 / π‘₯) ∈ β„• ∧ (𝐡 / π‘₯) ∈ β„•)}, β„•, < )
 
Theoremee7.2aOLD 35341 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 𝐡. 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 ∈ β„• ∧ 𝐡 ∈ β„•) β†’ (𝐴 < 𝐡 β†’ gcdOLD (𝐴, 𝐡) = gcdOLD (𝐴, (𝐡 βˆ’ 𝐴))))
 
21.16  Mathbox for Asger C. Ipsen
 
21.16.1  Continuous nowhere differentiable functions
 
Theoremdnival 35342* Value of the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    β‡’   (𝐴 ∈ ℝ β†’ (π‘‡β€˜π΄) = (absβ€˜((βŒŠβ€˜(𝐴 + (1 / 2))) βˆ’ 𝐴)))
 
Theoremdnicld1 35343 Closure theorem for the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ (absβ€˜((βŒŠβ€˜(𝐴 + (1 / 2))) βˆ’ 𝐴)) ∈ ℝ)
 
Theoremdnicld2 35344* Closure theorem for the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ (π‘‡β€˜π΄) ∈ ℝ)
 
Theoremdnif 35345 The "distance to nearest integer" function is a function. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    β‡’   π‘‡:β„βŸΆβ„
 
Theoremdnizeq0 35346* The distance to nearest integer is zero for integers. (Contributed by Asger C. Ipsen, 15-Jun-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   (πœ‘ β†’ 𝐴 ∈ β„€)    β‡’   (πœ‘ β†’ (π‘‡β€˜π΄) = 0)
 
Theoremdnizphlfeqhlf 35347* The distance to nearest integer is a half for half-integers. (Contributed by Asger C. Ipsen, 15-Jun-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   (πœ‘ β†’ 𝐴 ∈ β„€)    β‡’   (πœ‘ β†’ (π‘‡β€˜(𝐴 + (1 / 2))) = (1 / 2))
 
Theoremrddif2 35348 Variant of rddif 15286. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐴 ∈ ℝ β†’ 0 ≀ ((1 / 2) βˆ’ (absβ€˜((βŒŠβ€˜(𝐴 + (1 / 2))) βˆ’ 𝐴))))
 
Theoremdnibndlem1 35349* Lemma for dnibnd 35362. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ ((absβ€˜((π‘‡β€˜π΅) βˆ’ (π‘‡β€˜π΄))) ≀ 𝑆 ↔ (absβ€˜((absβ€˜((βŒŠβ€˜(𝐡 + (1 / 2))) βˆ’ 𝐡)) βˆ’ (absβ€˜((βŒŠβ€˜(𝐴 + (1 / 2))) βˆ’ 𝐴)))) ≀ 𝑆))
 
Theoremdnibndlem2 35350* Lemma for dnibnd 35362. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (βŒŠβ€˜(𝐡 + (1 / 2))) = (βŒŠβ€˜(𝐴 + (1 / 2))))    β‡’   (πœ‘ β†’ (absβ€˜((π‘‡β€˜π΅) βˆ’ (π‘‡β€˜π΄))) ≀ (absβ€˜(𝐡 βˆ’ 𝐴)))
 
Theoremdnibndlem3 35351 Lemma for dnibnd 35362. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (βŒŠβ€˜(𝐡 + (1 / 2))) = ((βŒŠβ€˜(𝐴 + (1 / 2))) + 1))    β‡’   (πœ‘ β†’ (absβ€˜(𝐡 βˆ’ 𝐴)) = (absβ€˜((𝐡 βˆ’ ((βŒŠβ€˜(𝐡 + (1 / 2))) βˆ’ (1 / 2))) + (((βŒŠβ€˜(𝐴 + (1 / 2))) + (1 / 2)) βˆ’ 𝐴))))
 
Theoremdnibndlem4 35352 Lemma for dnibnd 35362. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐡 ∈ ℝ β†’ 0 ≀ (𝐡 βˆ’ ((βŒŠβ€˜(𝐡 + (1 / 2))) βˆ’ (1 / 2))))
 
Theoremdnibndlem5 35353 Lemma for dnibnd 35362. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐴 ∈ ℝ β†’ 0 < (((βŒŠβ€˜(𝐴 + (1 / 2))) + (1 / 2)) βˆ’ 𝐴))
 
Theoremdnibndlem6 35354 Lemma for dnibnd 35362. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (absβ€˜((absβ€˜((βŒŠβ€˜(𝐡 + (1 / 2))) βˆ’ 𝐡)) βˆ’ (absβ€˜((βŒŠβ€˜(𝐴 + (1 / 2))) βˆ’ 𝐴)))) ≀ (((1 / 2) βˆ’ (absβ€˜((βŒŠβ€˜(𝐡 + (1 / 2))) βˆ’ 𝐡))) + ((1 / 2) βˆ’ (absβ€˜((βŒŠβ€˜(𝐴 + (1 / 2))) βˆ’ 𝐴)))))
 
Theoremdnibndlem7 35355 Lemma for dnibnd 35362. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ ((1 / 2) βˆ’ (absβ€˜((βŒŠβ€˜(𝐡 + (1 / 2))) βˆ’ 𝐡))) ≀ (𝐡 βˆ’ ((βŒŠβ€˜(𝐡 + (1 / 2))) βˆ’ (1 / 2))))
 
Theoremdnibndlem8 35356 Lemma for dnibnd 35362. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ ((1 / 2) βˆ’ (absβ€˜((βŒŠβ€˜(𝐴 + (1 / 2))) βˆ’ 𝐴))) ≀ (((βŒŠβ€˜(𝐴 + (1 / 2))) + (1 / 2)) βˆ’ 𝐴))
 
Theoremdnibndlem9 35357* Lemma for dnibnd 35362. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (βŒŠβ€˜(𝐡 + (1 / 2))) = ((βŒŠβ€˜(𝐴 + (1 / 2))) + 1))    β‡’   (πœ‘ β†’ (absβ€˜((π‘‡β€˜π΅) βˆ’ (π‘‡β€˜π΄))) ≀ (absβ€˜(𝐡 βˆ’ 𝐴)))
 
Theoremdnibndlem10 35358 Lemma for dnibnd 35362. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ ((βŒŠβ€˜(𝐴 + (1 / 2))) + 2) ≀ (βŒŠβ€˜(𝐡 + (1 / 2))))    β‡’   (πœ‘ β†’ 1 ≀ (𝐡 βˆ’ 𝐴))
 
Theoremdnibndlem11 35359 Lemma for dnibnd 35362. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (absβ€˜((absβ€˜((βŒŠβ€˜(𝐡 + (1 / 2))) βˆ’ 𝐡)) βˆ’ (absβ€˜((βŒŠβ€˜(𝐴 + (1 / 2))) βˆ’ 𝐴)))) ≀ (1 / 2))
 
Theoremdnibndlem12 35360* Lemma for dnibnd 35362. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ ((βŒŠβ€˜(𝐴 + (1 / 2))) + 2) ≀ (βŒŠβ€˜(𝐡 + (1 / 2))))    β‡’   (πœ‘ β†’ (absβ€˜((π‘‡β€˜π΅) βˆ’ (π‘‡β€˜π΄))) ≀ (absβ€˜(𝐡 βˆ’ 𝐴)))
 
Theoremdnibndlem13 35361* Lemma for dnibnd 35362. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (βŒŠβ€˜(𝐴 + (1 / 2))) ≀ (βŒŠβ€˜(𝐡 + (1 / 2))))    β‡’   (πœ‘ β†’ (absβ€˜((π‘‡β€˜π΅) βˆ’ (π‘‡β€˜π΄))) ≀ (absβ€˜(𝐡 βˆ’ 𝐴)))
 
Theoremdnibnd 35362* The "distance to nearest integer" function is 1-Lipschitz continuous, i.e., is a short map. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (absβ€˜((π‘‡β€˜π΅) βˆ’ (π‘‡β€˜π΄))) ≀ (absβ€˜(𝐡 βˆ’ 𝐴)))
 
Theoremdnicn 35363 The "distance to nearest integer" function is continuous. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    β‡’   π‘‡ ∈ (ℝ–cn→ℝ)
 
Theoremknoppcnlem1 35364* Lemma for knoppcn 35375. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝑀 ∈ β„•0)    β‡’   (πœ‘ β†’ ((πΉβ€˜π΄)β€˜π‘€) = ((𝐢↑𝑀) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑀) Β· 𝐴))))
 
Theoremknoppcnlem2 35365* Lemma for knoppcn 35375. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝑀 ∈ β„•0)    β‡’   (πœ‘ β†’ ((𝐢↑𝑀) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑀) Β· 𝐴))) ∈ ℝ)
 
Theoremknoppcnlem3 35366* Lemma for knoppcn 35375. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝑀 ∈ β„•0)    β‡’   (πœ‘ β†’ ((πΉβ€˜π΄)β€˜π‘€) ∈ ℝ)
 
Theoremknoppcnlem4 35367* Lemma for knoppcn 35375. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝑀 ∈ β„•0)    β‡’   (πœ‘ β†’ (absβ€˜((πΉβ€˜π΄)β€˜π‘€)) ≀ ((π‘š ∈ β„•0 ↦ ((absβ€˜πΆ)β†‘π‘š))β€˜π‘€))
 
Theoremknoppcnlem5 35368* Lemma for knoppcn 35375. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ (π‘š ∈ β„•0 ↦ (𝑧 ∈ ℝ ↦ ((πΉβ€˜π‘§)β€˜π‘š))):β„•0⟢(β„‚ ↑m ℝ))
 
Theoremknoppcnlem6 35369* Lemma for knoppcn 35375. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ (absβ€˜πΆ) < 1)    β‡’   (πœ‘ β†’ seq0( ∘f + , (π‘š ∈ β„•0 ↦ (𝑧 ∈ ℝ ↦ ((πΉβ€˜π‘§)β€˜π‘š)))) ∈ dom (β‡π‘’β€˜β„))
 
Theoremknoppcnlem7 35370* Lemma for knoppcn 35375. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝑀 ∈ β„•0)    β‡’   (πœ‘ β†’ (seq0( ∘f + , (π‘š ∈ β„•0 ↦ (𝑧 ∈ ℝ ↦ ((πΉβ€˜π‘§)β€˜π‘š))))β€˜π‘€) = (𝑀 ∈ ℝ ↦ (seq0( + , (πΉβ€˜π‘€))β€˜π‘€)))
 
Theoremknoppcnlem8 35371* Lemma for knoppcn 35375. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ seq0( ∘f + , (π‘š ∈ β„•0 ↦ (𝑧 ∈ ℝ ↦ ((πΉβ€˜π‘§)β€˜π‘š)))):β„•0⟢(β„‚ ↑m ℝ))
 
Theoremknoppcnlem9 35372* Lemma for knoppcn 35375. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   π‘Š = (𝑀 ∈ ℝ ↦ Σ𝑖 ∈ β„•0 ((πΉβ€˜π‘€)β€˜π‘–))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ (absβ€˜πΆ) < 1)    β‡’   (πœ‘ β†’ seq0( ∘f + , (π‘š ∈ β„•0 ↦ (𝑧 ∈ ℝ ↦ ((πΉβ€˜π‘§)β€˜π‘š))))(β‡π‘’β€˜β„)π‘Š)
 
Theoremknoppcnlem10 35373* Lemma for knoppcn 35375. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝑀 ∈ β„•0)    β‡’   (πœ‘ β†’ (𝑧 ∈ ℝ ↦ ((πΉβ€˜π‘§)β€˜π‘€)) ∈ ((topGenβ€˜ran (,)) Cn (TopOpenβ€˜β„‚fld)))
 
Theoremknoppcnlem11 35374* Lemma for knoppcn 35375. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ seq0( ∘f + , (π‘š ∈ β„•0 ↦ (𝑧 ∈ ℝ ↦ ((πΉβ€˜π‘§)β€˜π‘š)))):β„•0⟢(ℝ–cnβ†’β„‚))
 
Theoremknoppcn 35375* The continuous nowhere differentiable function π‘Š ( 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 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   π‘Š = (𝑀 ∈ ℝ ↦ Σ𝑖 ∈ β„•0 ((πΉβ€˜π‘€)β€˜π‘–))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ (absβ€˜πΆ) < 1)    β‡’   (πœ‘ β†’ π‘Š ∈ (ℝ–cnβ†’β„‚))
 
Theoremknoppcld 35376* Closure theorem for Knopp's function. (Contributed by Asger C. Ipsen, 26-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   π‘Š = (𝑀 ∈ ℝ ↦ Σ𝑖 ∈ β„•0 ((πΉβ€˜π‘€)β€˜π‘–))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ (absβ€˜πΆ) < 1)    β‡’   (πœ‘ β†’ (π‘Šβ€˜π΄) ∈ β„‚)
 
Theoremunblimceq0lem 35377* Lemma for unblimceq0 35378. (Contributed by Asger C. Ipsen, 12-May-2021.)
(πœ‘ β†’ 𝑆 βŠ† β„‚)    &   (πœ‘ β†’ 𝐹:π‘†βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ βˆ€π‘ ∈ ℝ+ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘₯ ∈ 𝑆 ((absβ€˜(π‘₯ βˆ’ 𝐴)) < 𝑑 ∧ 𝑏 ≀ (absβ€˜(πΉβ€˜π‘₯))))    β‡’   (πœ‘ β†’ βˆ€π‘ ∈ ℝ+ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘¦ ∈ 𝑆 (𝑦 β‰  𝐴 ∧ (absβ€˜(𝑦 βˆ’ 𝐴)) < 𝑑 ∧ 𝑐 ≀ (absβ€˜(πΉβ€˜π‘¦))))
 
Theoremunblimceq0 35378* If 𝐹 is unbounded near 𝐴 it has no limit at 𝐴. (Contributed by Asger C. Ipsen, 12-May-2021.)
(πœ‘ β†’ 𝑆 βŠ† β„‚)    &   (πœ‘ β†’ 𝐹:π‘†βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ βˆ€π‘ ∈ ℝ+ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘₯ ∈ 𝑆 ((absβ€˜(π‘₯ βˆ’ 𝐴)) < 𝑑 ∧ 𝑏 ≀ (absβ€˜(πΉβ€˜π‘₯))))    β‡’   (πœ‘ β†’ (𝐹 limβ„‚ 𝐴) = βˆ…)
 
Theoremunbdqndv1 35379* If the difference quotient (((πΉβ€˜π‘§) βˆ’ (πΉβ€˜π΄)) / (𝑧 βˆ’ 𝐴)) is unbounded near 𝐴 then 𝐹 is not differentiable at 𝐴. (Contributed by Asger C. Ipsen, 12-May-2021.)
𝐺 = (𝑧 ∈ (𝑋 βˆ– {𝐴}) ↦ (((πΉβ€˜π‘§) βˆ’ (πΉβ€˜π΄)) / (𝑧 βˆ’ 𝐴)))    &   (πœ‘ β†’ 𝑆 βŠ† β„‚)    &   (πœ‘ β†’ 𝑋 βŠ† 𝑆)    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ βˆ€π‘ ∈ ℝ+ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘₯ ∈ (𝑋 βˆ– {𝐴})((absβ€˜(π‘₯ βˆ’ 𝐴)) < 𝑑 ∧ 𝑏 ≀ (absβ€˜(πΊβ€˜π‘₯))))    β‡’   (πœ‘ β†’ Β¬ 𝐴 ∈ dom (𝑆 D 𝐹))
 
Theoremunbdqndv2lem1 35380 Lemma for unbdqndv2 35382. (Contributed by Asger C. Ipsen, 12-May-2021.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐷 ∈ β„‚)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝐷 β‰  0)    &   (πœ‘ β†’ (2 Β· 𝐸) ≀ (absβ€˜((𝐴 βˆ’ 𝐡) / 𝐷)))    β‡’   (πœ‘ β†’ ((𝐸 Β· (absβ€˜π·)) ≀ (absβ€˜(𝐴 βˆ’ 𝐢)) ∨ (𝐸 Β· (absβ€˜π·)) ≀ (absβ€˜(𝐡 βˆ’ 𝐢))))
 
Theoremunbdqndv2lem2 35381* Lemma for unbdqndv2 35382. (Contributed by Asger C. Ipsen, 12-May-2021.)
𝐺 = (𝑧 ∈ (𝑋 βˆ– {𝐴}) ↦ (((πΉβ€˜π‘§) βˆ’ (πΉβ€˜π΄)) / (𝑧 βˆ’ 𝐴)))    &   π‘Š = if((𝐡 Β· (𝑉 βˆ’ π‘ˆ)) ≀ (absβ€˜((πΉβ€˜π‘ˆ) βˆ’ (πΉβ€˜π΄))), π‘ˆ, 𝑉)    &   (πœ‘ β†’ 𝑋 βŠ† ℝ)    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ 𝐴 ∈ 𝑋)    &   (πœ‘ β†’ 𝐡 ∈ ℝ+)    &   (πœ‘ β†’ 𝐷 ∈ ℝ+)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑋)    &   (πœ‘ β†’ 𝑉 ∈ 𝑋)    &   (πœ‘ β†’ π‘ˆ β‰  𝑉)    &   (πœ‘ β†’ π‘ˆ ≀ 𝐴)    &   (πœ‘ β†’ 𝐴 ≀ 𝑉)    &   (πœ‘ β†’ (𝑉 βˆ’ π‘ˆ) < 𝐷)    &   (πœ‘ β†’ (2 Β· 𝐡) ≀ ((absβ€˜((πΉβ€˜π‘‰) βˆ’ (πΉβ€˜π‘ˆ))) / (𝑉 βˆ’ π‘ˆ)))    β‡’   (πœ‘ β†’ (π‘Š ∈ (𝑋 βˆ– {𝐴}) ∧ ((absβ€˜(π‘Š βˆ’ 𝐴)) < 𝐷 ∧ 𝐡 ≀ (absβ€˜(πΊβ€˜π‘Š)))))
 
Theoremunbdqndv2 35382* Variant of unbdqndv1 35379 with the hypothesis that (((πΉβ€˜π‘¦) βˆ’ (πΉβ€˜π‘₯)) / (𝑦 βˆ’ π‘₯)) is unbounded where π‘₯ ≀ 𝐴 and 𝐴 ≀ 𝑦. (Contributed by Asger C. Ipsen, 12-May-2021.)
(πœ‘ β†’ 𝑋 βŠ† ℝ)    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„‚)    &   (πœ‘ β†’ βˆ€π‘ ∈ ℝ+ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘₯ ∈ 𝑋 βˆƒπ‘¦ ∈ 𝑋 ((π‘₯ ≀ 𝐴 ∧ 𝐴 ≀ 𝑦) ∧ ((𝑦 βˆ’ π‘₯) < 𝑑 ∧ π‘₯ β‰  𝑦) ∧ 𝑏 ≀ ((absβ€˜((πΉβ€˜π‘¦) βˆ’ (πΉβ€˜π‘₯))) / (𝑦 βˆ’ π‘₯))))    β‡’   (πœ‘ β†’ Β¬ 𝐴 ∈ dom (ℝ D 𝐹))
 
Theoremknoppndvlem1 35383 Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐽 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    β‡’   (πœ‘ β†’ ((((2 Β· 𝑁)↑-𝐽) / 2) Β· 𝑀) ∈ ℝ)
 
Theoremknoppndvlem2 35384 Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐼 ∈ β„€)    &   (πœ‘ β†’ 𝐽 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐽 < 𝐼)    β‡’   (πœ‘ β†’ (((2 Β· 𝑁)↑𝐼) Β· ((((2 Β· 𝑁)↑-𝐽) / 2) Β· 𝑀)) ∈ β„€)
 
Theoremknoppndvlem3 35385 Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 15-Jun-2021.)
(πœ‘ β†’ 𝐢 ∈ (-1(,)1))    β‡’   (πœ‘ β†’ (𝐢 ∈ ℝ ∧ (absβ€˜πΆ) < 1))
 
Theoremknoppndvlem4 35386* Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   π‘Š = (𝑀 ∈ ℝ ↦ Σ𝑖 ∈ β„•0 ((πΉβ€˜π‘€)β€˜π‘–))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ (-1(,)1))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ seq0( + , (πΉβ€˜π΄)) ⇝ (π‘Šβ€˜π΄))
 
Theoremknoppndvlem5 35387* Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ Σ𝑖 ∈ (0...𝐽)((πΉβ€˜π΄)β€˜π‘–) ∈ ℝ)
 
Theoremknoppndvlem6 35388* Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   π‘Š = (𝑀 ∈ ℝ ↦ Σ𝑖 ∈ β„•0 ((πΉβ€˜π‘€)β€˜π‘–))    &   π΄ = ((((2 Β· 𝑁)↑-𝐽) / 2) Β· 𝑀)    &   (πœ‘ β†’ 𝐢 ∈ (-1(,)1))    &   (πœ‘ β†’ 𝐽 ∈ β„•0)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (π‘Šβ€˜π΄) = Σ𝑖 ∈ (0...𝐽)((πΉβ€˜π΄)β€˜π‘–))
 
Theoremknoppndvlem7 35389* Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   π΄ = ((((2 Β· 𝑁)↑-𝐽) / 2) Β· 𝑀)    &   (πœ‘ β†’ 𝐽 ∈ β„•0)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ ((πΉβ€˜π΄)β€˜π½) = ((𝐢↑𝐽) Β· (π‘‡β€˜(𝑀 / 2))))
 
Theoremknoppndvlem8 35390* Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   π΄ = ((((2 Β· 𝑁)↑-𝐽) / 2) Β· 𝑀)    &   (πœ‘ β†’ 𝐢 ∈ (-1(,)1))    &   (πœ‘ β†’ 𝐽 ∈ β„•0)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 2 βˆ₯ 𝑀)    β‡’   (πœ‘ β†’ ((πΉβ€˜π΄)β€˜π½) = 0)
 
Theoremknoppndvlem9 35391* Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   π΄ = ((((2 Β· 𝑁)↑-𝐽) / 2) Β· 𝑀)    &   (πœ‘ β†’ 𝐢 ∈ (-1(,)1))    &   (πœ‘ β†’ 𝐽 ∈ β„•0)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ Β¬ 2 βˆ₯ 𝑀)    β‡’   (πœ‘ β†’ ((πΉβ€˜π΄)β€˜π½) = ((𝐢↑𝐽) / 2))
 
Theoremknoppndvlem10 35392* Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   π΄ = ((((2 Β· 𝑁)↑-𝐽) / 2) Β· 𝑀)    &   π΅ = ((((2 Β· 𝑁)↑-𝐽) / 2) Β· (𝑀 + 1))    &   (πœ‘ β†’ 𝐢 ∈ (-1(,)1))    &   (πœ‘ β†’ 𝐽 ∈ β„•0)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (absβ€˜(((πΉβ€˜π΅)β€˜π½) βˆ’ ((πΉβ€˜π΄)β€˜π½))) = (((absβ€˜πΆ)↑𝐽) / 2))
 
Theoremknoppndvlem11 35393* Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 28-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ (-1(,)1))    &   (πœ‘ β†’ 𝐽 ∈ β„•0)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (absβ€˜(Σ𝑖 ∈ (0...(𝐽 βˆ’ 1))((πΉβ€˜π΅)β€˜π‘–) βˆ’ Σ𝑖 ∈ (0...(𝐽 βˆ’ 1))((πΉβ€˜π΄)β€˜π‘–))) ≀ ((absβ€˜(𝐡 βˆ’ 𝐴)) Β· Σ𝑖 ∈ (0...(𝐽 βˆ’ 1))(((2 Β· 𝑁) Β· (absβ€˜πΆ))↑𝑖)))
 
Theoremknoppndvlem12 35394 Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 29-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
(πœ‘ β†’ 𝐢 ∈ (-1(,)1))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 1 < (𝑁 Β· (absβ€˜πΆ)))    β‡’   (πœ‘ β†’ (((2 Β· 𝑁) Β· (absβ€˜πΆ)) β‰  1 ∧ 1 < (((2 Β· 𝑁) Β· (absβ€˜πΆ)) βˆ’ 1)))
 
Theoremknoppndvlem13 35395 Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 1-Jul-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)
(πœ‘ β†’ 𝐢 ∈ (-1(,)1))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 1 < (𝑁 Β· (absβ€˜πΆ)))    β‡’   (πœ‘ β†’ 𝐢 β‰  0)
 
Theoremknoppndvlem14 35396* Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 1-Jul-2021.) (Revised by Asger C. Ipsen, 7-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   π΄ = ((((2 Β· 𝑁)↑-𝐽) / 2) Β· 𝑀)    &   π΅ = ((((2 Β· 𝑁)↑-𝐽) / 2) Β· (𝑀 + 1))    &   (πœ‘ β†’ 𝐢 ∈ (-1(,)1))    &   (πœ‘ β†’ 𝐽 ∈ β„•0)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 1 < (𝑁 Β· (absβ€˜πΆ)))    β‡’   (πœ‘ β†’ (absβ€˜(Σ𝑖 ∈ (0...(𝐽 βˆ’ 1))((πΉβ€˜π΅)β€˜π‘–) βˆ’ Σ𝑖 ∈ (0...(𝐽 βˆ’ 1))((πΉβ€˜π΄)β€˜π‘–))) ≀ ((((absβ€˜πΆ)↑𝐽) / 2) Β· (1 / (((2 Β· 𝑁) Β· (absβ€˜πΆ)) βˆ’ 1))))
 
Theoremknoppndvlem15 35397* Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 6-Jul-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   π‘Š = (𝑀 ∈ ℝ ↦ Σ𝑖 ∈ β„•0 ((πΉβ€˜π‘€)β€˜π‘–))    &   π΄ = ((((2 Β· 𝑁)↑-𝐽) / 2) Β· 𝑀)    &   π΅ = ((((2 Β· 𝑁)↑-𝐽) / 2) Β· (𝑀 + 1))    &   (πœ‘ β†’ 𝐢 ∈ (-1(,)1))    &   (πœ‘ β†’ 𝐽 ∈ β„•0)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 1 < (𝑁 Β· (absβ€˜πΆ)))    β‡’   (πœ‘ β†’ ((((absβ€˜πΆ)↑𝐽) / 2) Β· (1 βˆ’ (1 / (((2 Β· 𝑁) Β· (absβ€˜πΆ)) βˆ’ 1)))) ≀ (absβ€˜((π‘Šβ€˜π΅) βˆ’ (π‘Šβ€˜π΄))))
 
Theoremknoppndvlem16 35398 Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 19-Jul-2021.)
𝐴 = ((((2 Β· 𝑁)↑-𝐽) / 2) Β· 𝑀)    &   π΅ = ((((2 Β· 𝑁)↑-𝐽) / 2) Β· (𝑀 + 1))    &   (πœ‘ β†’ 𝐽 ∈ β„•0)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (𝐡 βˆ’ 𝐴) = (((2 Β· 𝑁)↑-𝐽) / 2))
 
Theoremknoppndvlem17 35399* Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 12-Aug-2021.)
𝑇 = (π‘₯ ∈ ℝ ↦ (absβ€˜((βŒŠβ€˜(π‘₯ + (1 / 2))) βˆ’ π‘₯)))    &   πΉ = (𝑦 ∈ ℝ ↦ (𝑛 ∈ β„•0 ↦ ((𝐢↑𝑛) Β· (π‘‡β€˜(((2 Β· 𝑁)↑𝑛) Β· 𝑦)))))    &   π‘Š = (𝑀 ∈ ℝ ↦ Σ𝑖 ∈ β„•0 ((πΉβ€˜π‘€)β€˜π‘–))    &   π΄ = ((((2 Β· 𝑁)↑-𝐽) / 2) Β· 𝑀)    &   π΅ = ((((2 Β· 𝑁)↑-𝐽) / 2) Β· (𝑀 + 1))    &   (πœ‘ β†’ 𝐢 ∈ (-1(,)1))    &   (πœ‘ β†’ 𝐽 ∈ β„•0)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 1 < (𝑁 Β· (absβ€˜πΆ)))    β‡’   (πœ‘ β†’ ((((2 Β· 𝑁) Β· (absβ€˜πΆ))↑𝐽) Β· (1 βˆ’ (1 / (((2 Β· 𝑁) Β· (absβ€˜πΆ)) βˆ’ 1)))) ≀ ((absβ€˜((π‘Šβ€˜π΅) βˆ’ (π‘Šβ€˜π΄))) / (𝐡 βˆ’ 𝐴)))
 
Theoremknoppndvlem18 35400* Lemma for knoppndv 35405. (Contributed by Asger C. Ipsen, 14-Aug-2021.)
(πœ‘ β†’ 𝐢 ∈ (-1(,)1))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐷 ∈ ℝ+)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝐺 ∈ ℝ+)    &   (πœ‘ β†’ 1 < (𝑁 Β· (absβ€˜πΆ)))    β‡’   (πœ‘ β†’ βˆƒπ‘— ∈ β„•0 ((((2 Β· 𝑁)↑-𝑗) / 2) < 𝐷 ∧ 𝐸 ≀ ((((2 Β· 𝑁) Β· (absβ€˜πΆ))↑𝑗) Β· 𝐺)))
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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