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Theorem List for Metamath Proof Explorer - 33801-33900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremonint1 33801 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 33802 The ordinal Hausdorff spaces are 1o and 2o. (Contributed by Chen-Pang He, 10-Nov-2015.)
(On ∩ Haus) = {1o, 2o}
 
20.13  Mathbox for Jeff Hoffman
 
20.13.1  Inferences for finite induction on generic function values
 
Theoremfveleq 33803 Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
(𝐴 = 𝐵 → ((𝜑 → (𝐹𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹𝐵) ∈ 𝑃)))
 
Theoremfindfvcl 33804* 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 33805* Please add description here. (Contributed by Jeff Hoffman, 19-Feb-2008.)
(𝑧𝑃 → (𝐺𝑧) ∈ 𝑃)       (𝐶 ∈ ω → (𝐴𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))
 
Theoremfindabrcl 33806* 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)
 
20.13.2  gdc.mm
 
Theoremnnssi2 33807 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 33808 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 33809 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 33810 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 33811 Extend class notation to include the gdc function. (New usage is discouraged.)
class gcdOLD (𝐴, 𝐵)
 
Definitiondf-gcdOLD 33812* 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 33813 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))))
 
20.14  Mathbox for Asger C. Ipsen
 
20.14.1  Continuous nowhere differentiable functions
 
Theoremdnival 33814* 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 33815 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 33816* 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 33817 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 33818* 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 33819* 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 33820 Variant of rddif 14703. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐴 ∈ ℝ → 0 ≤ ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))))
 
Theoremdnibndlem1 33821* Lemma for dnibnd 33834. (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 33822* Lemma for dnibnd 33834. (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 33823 Lemma for dnibnd 33834. (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 33824 Lemma for dnibnd 33834. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐵 ∈ ℝ → 0 ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2))))
 
Theoremdnibndlem5 33825 Lemma for dnibnd 33834. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐴 ∈ ℝ → 0 < (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴))
 
Theoremdnibndlem6 33826 Lemma for dnibnd 33834. (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 33827 Lemma for dnibnd 33834. (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 33828 Lemma for dnibnd 33834. (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 33829* Lemma for dnibnd 33834. (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 33830 Lemma for dnibnd 33834. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2))))       (𝜑 → 1 ≤ (𝐵𝐴))
 
Theoremdnibndlem11 33831 Lemma for dnibnd 33834. (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 33832* Lemma for dnibnd 33834. (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 33833* Lemma for dnibnd 33834. (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 33834* 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 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (abs‘((𝑇𝐵) − (𝑇𝐴))) ≤ (abs‘(𝐵𝐴)))
 
Theoremdnicn 33835 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 33836* Lemma for knoppcn 33847. (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 33837* Lemma for knoppcn 33847. (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 33838* Lemma for knoppcn 33847. (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 33839* Lemma for knoppcn 33847. (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 33840* Lemma for knoppcn 33847. (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 33841* Lemma for knoppcn 33847. (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 33842* Lemma for knoppcn 33847. (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 33843* Lemma for knoppcn 33847. (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 33844* Lemma for knoppcn 33847. (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 33845* Lemma for knoppcn 33847. (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 33846* Lemma for knoppcn 33847. (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 33847* 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 33848* 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)
 
Theoremunblimceq0lem 33849* Lemma for unblimceq0 33850. (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 33850* 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 33851* 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 33852 Lemma for unbdqndv2 33854. (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 33853* Lemma for unbdqndv2 33854. (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 33854* Variant of unbdqndv1 33851 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 33855 Lemma for knoppndv 33877. (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 33856 Lemma for knoppndv 33877. (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 33857 Lemma for knoppndv 33877. (Contributed by Asger C. Ipsen, 15-Jun-2021.)
(𝜑𝐶 ∈ (-1(,)1))       (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
 
Theoremknoppndvlem4 33858* Lemma for knoppndv 33877. (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 33859* Lemma for knoppndv 33877. (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 33860* Lemma for knoppndv 33877. (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 33861* Lemma for knoppndv 33877. (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 33862* Lemma for knoppndv 33877. (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 33863* Lemma for knoppndv 33877. (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 33864* Lemma for knoppndv 33877. (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 33865* Lemma for knoppndv 33877. (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 33866 Lemma for knoppndv 33877. (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 33867 Lemma for knoppndv 33877. (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 33868* Lemma for knoppndv 33877. (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 33869* Lemma for knoppndv 33877. (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 33870 Lemma for knoppndv 33877. (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 33871* Lemma for knoppndv 33877. (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 33872* Lemma for knoppndv 33877. (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 33873* Lemma for knoppndv 33877. (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 33874 Lemma for knoppndv 33877. (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 33875* Lemma for knoppndv 33877. (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 33876* Lemma for knoppndv 33877. (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 33877* 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) = ∅)
 
Theoremknoppf 33878* 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 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)       (𝜑𝑊:ℝ⟶ℝ)
 
Theoremknoppcn2 33879* Variant of knoppcn 33847 with different codomain. (Contributed by Asger C. Ipsen, 25-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))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ (-1(,)1))       (𝜑𝑊 ∈ (ℝ–cn→ℝ))
 
Theoremcnndvlem1 33880* Lemma for cnndv 33882. (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 ((𝐹𝑤)‘𝑖))       (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅)
 
Theoremcnndvlem2 33881* Lemma for cnndv 33882. (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 ((𝐹𝑤)‘𝑖))       𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅)
 
Theoremcnndv 33882 There exists a continuous nowhere differentiable function. The result follows directly from knoppcn 33847 and knoppndv 33877. (Contributed by Asger C. Ipsen, 26-Aug-2021.)
𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅)
 
20.15  Mathbox for BJ

In this mathbox, we try to respect the ordering of the sections of the main part. There are strengthenings of theorems of the main part, as well as work on reducing axiom dependencies.

 
20.15.1  Propositional calculus

Miscellaneous utility theorems of propositional calculus.

 
20.15.1.1  Derived rules of inference

In this section, we prove a few rules of inference derived from modus ponens ax-mp 5, and which do not depend on any other axioms.

 
Theorembj-mp2c 33883 A double modus ponens inference. Inference associated with mpd 15. (Contributed by BJ, 24-Sep-2019.)
𝜑    &   (𝜑𝜓)    &   (𝜑 → (𝜓𝜒))       𝜒
 
Theorembj-mp2d 33884 A double modus ponens inference. Inference associated with mpcom 38. (Contributed by BJ, 24-Sep-2019.)
𝜑    &   (𝜑𝜓)    &   (𝜓 → (𝜑𝜒))       𝜒
 
20.15.1.2  A syntactic theorem

In this section, we prove a syntactic theorem (bj-0 33885) asserting that some formula is well-formed. Then, we use this syntactic theorem to shorten the proof of a "usual" theorem (bj-1 33886) and explain in the comment of that theorem why this phenomenon is unusual.

 
Theorembj-0 33885 A syntactic theorem. See the section comment and the comment of bj-1 33886. The full proof (that is, with the syntactic, non-essential steps) does not appear on this webpage. It has five steps and reads $= wph wps wi wch wi $. The only other syntactic theorems in the main part of set.mm are wel 2114 and weq 1963. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
wff ((𝜑𝜓) → 𝜒)
 
Theorembj-1 33886 In this proof, the use of the syntactic theorem bj-0 33885 allows to reduce the total length by one (non-essential) step. See also the section comment and the comment of bj-0 33885. Since bj-0 33885 is used in a non-essential step, this use does not appear on this webpage (but the present theorem appears on the webpage for bj-0 33885 as a theorem referencing it). The full proof reads $= wph wps wch bj-0 id $. (while, without using bj-0 33885, it would read $= wph wps wi wch wi id $.).

Now we explain why syntactic theorems are not useful in set.mm. Suppose that the syntactic theorem thm-0 proves that PHI is a well-formed formula, and that thm-0 is used to shorten the proof of thm-1. Assume that PHI does have proper non-atomic subformulas (which is not the case of the formula proved by weq 1963 or wel 2114). Then, the proof of thm-1 does not construct all the proper non-atomic subformulas of PHI (if it did, then using thm-0 would not shorten it). Therefore, thm-1 is a special instance of a more general theorem with essentially the same proof. In the present case, bj-1 33886 is a special instance of id 22. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

(((𝜑𝜓) → 𝜒) → ((𝜑𝜓) → 𝜒))
 
20.15.1.3  Minimal implicational calculus
 
Theorembj-a1k 33887 Weakening of ax-1 6. This shortens the proofs of dfwe2 7499 (937>925), ordunisuc2 7562 (789>777), r111 9207 (558>545), smo11 8004 (1176>1164). (Contributed by BJ, 11-Aug-2020.)
(𝜑 → (𝜓 → (𝜒𝜓)))
 
Theorembj-nnclav 33888 When is substituted for 𝜓, this formula is the Clavius law with a doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)
(((𝜑𝜓) → 𝜑) → ((𝜑𝜓) → 𝜓))
 
Theorembj-jarrii 33889 Inference associated with jarri 107. (Contributed by BJ, 29-Mar-2020.)
((𝜑𝜓) → 𝜒)    &   𝜓       𝜒
 
Theorembj-imim21 33890 The propositional function (𝜒 → (. → 𝜃)) is decreasing. (Contributed by BJ, 19-Jul-2019.)
((𝜑𝜓) → ((𝜒 → (𝜓𝜃)) → (𝜒 → (𝜑𝜃))))
 
Theorembj-imim21i 33891 Inference associated with bj-imim21 33890. Its associated inference is syl5 34. (Contributed by BJ, 19-Jul-2019.)
(𝜑𝜓)       ((𝜒 → (𝜓𝜃)) → (𝜒 → (𝜑𝜃)))
 
Theorembj-peircestab 33892 Over minimal implicational calculus, Peirce's law implies the double negation of the stability of any proposition (that is the interpretation when is substitued for 𝜓). (Contributed by BJ, 30-Nov-2023.) (Proof modification is discouraged.)
(((((𝜑𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)
 
Theorembj-stabpeirce 33893 Over minimal implicational calculus, Peirce's law is implied by the (classical refutation equivalent of) the double negation of the stability of any proposition. (Contributed by BJ, 30-Nov-2023.) (Proof modification is discouraged.)
((((((𝜑𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓) → (((𝜓𝜑) → 𝜓) → 𝜓))
 
20.15.1.4  Positive calculus

Positive calculus is understood to be intuitionistic.

 
Theorembj-syl66ib 33894 A mixed syllogism inference derived from syl6ib 253. In addition to bj-dvelimdv1 34180, it can also shorten alexsubALTlem4 22661 (4821>4812), supsrlem 10536 (2868>2863). (Contributed by BJ, 20-Oct-2021.)
(𝜑 → (𝜓𝜃))    &   (𝜃𝜏)    &   (𝜏𝜒)       (𝜑 → (𝜓𝜒))
 
Theorembj-orim2 33895 Proof of orim2 964 from the axiomatic definition of disjunction (olc 864, orc 863, jao 957) and minimal implicational calculus. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))
 
Theorembj-currypeirce 33896 Curry's axiom curryax 890 (a non-intuitionistic positive statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 204 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the elimination axiom jao 957 via its inference form jaoi 853; the introduction axioms olc 864 and orc 863 are not needed). Note that this theorem shows that actually, the standard instance of curryax 890 implies the standard instance of peirce 204, which is not the case for the converse bj-peircecurry 33897. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∨ (𝜑𝜓)) → (((𝜑𝜓) → 𝜑) → 𝜑))
 
Theorembj-peircecurry 33897 Peirce's axiom peirce 204 implies Curry's axiom curryax 890 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the introduction axioms olc 864 and orc 863; the elimination axiom jao 957 is not needed). See bj-currypeirce 33896 for the converse. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ∨ (𝜑𝜓))
 
Theorembj-animbi 33898 Conjunction in terms of implication and biconditional. Note that the proof is intuitionistic (use of ax-3 8 comes from the unusual definition of the biconditional in set.mm). (Contributed by BJ, 23-Sep-2023.)
((𝜑𝜓) ↔ (𝜑 ↔ (𝜑𝜓)))
 
Theorembj-currypara 33899 Curry's paradox. Note that the proof is intuitionistic (use ax-3 8 comes from the unusual definition of the biconditional in set.mm). The paradox comes from the case where 𝜑 is the self-referential sentence "If this sentence is true, then 𝜓", so that one can prove everything. Therefore, a consistent system cannot allow the formation of such self-referential sentences. This has lead to the study of logics rejecting contraction pm2.43 56, such as affine logic and linear logic. (Contributed by BJ, 23-Sep-2023.)
((𝜑 ↔ (𝜑𝜓)) → 𝜓)
 
20.15.1.5  Implication and negation
 
Theorembj-con2com 33900 A commuted form of the contrapositive, true in minimal calculus. (Contributed by BJ, 19-Mar-2020.)
(𝜑 → ((𝜓 → ¬ 𝜑) → ¬ 𝜓))
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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 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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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