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Theorem List for Metamath Proof Explorer - 32501-32600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfilnetlem4 32501* Lemma for filnet 32502. (Contributed by Jeff Hankins, 15-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
𝐻 = 𝑛𝐹 ({𝑛} × 𝑛)    &   𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐻𝑦𝐻) ∧ (1st𝑦) ⊆ (1st𝑥))}       (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))

Theoremfilnet 32502* A filter has the same convergence and clustering properties as some net. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
(𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑𝑋𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))

20.10  Mathbox for Anthony Hart

20.10.1  Propositional Calculus

Theoremtb-ax1 32503 The first of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theoremtb-ax2 32504 The second of three axioms in the Tarski-Bernays axiom system. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))

Theoremtb-ax3 32505 The third of three axioms in the Tarski-Bernays axiom system.

This axiom, along with ax-mp 5, tb-ax1 32503, and tb-ax2 32504, can be used to derive any theorem or rule that uses only . (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(((𝜑𝜓) → 𝜑) → 𝜑)

Theoremtbsyl 32506 The weak syllogism from Tarski-Bernays'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theoremre1ax2lem 32507 Lemma for re1ax2 32508. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))

Theoremre1ax2 32508 ax-2 7 rederived from the Tarski-Bernays axiom system. Often tb-ax1 32503 is replaced with this theorem to make a "standard" system. This is because this theorem is easier to work with, despite it being longer. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Theoremnaim1 32509 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theoremnaim2 32510 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)
((𝜑𝜓) → ((𝜒𝜓) → (𝜒𝜑)))

Theoremnaim1i 32511 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theoremnaim2i 32512 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜒𝜓)       (𝜒𝜑)

Theoremnaim12i 32513 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜓𝜃)       (𝜑𝜒)

Theoremnabi1 32514 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)
((𝜑𝜓) → ((𝜑𝜒) ↔ (𝜓𝜒)))

Theoremnabi2 32515 Constructor theorem for . (Contributed by Anthony Hart, 1-Sep-2011.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))

Theoremnabi1i 32516 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑𝜒)

Theoremnabi2i 32517 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜒𝜓)       (𝜒𝜑)

Theoremnabi12i 32518 Constructor rule for . (Contributed by Anthony Hart, 2-Sep-2011.)
(𝜑𝜓)    &   (𝜒𝜃)    &   (𝜓𝜃)       (𝜑𝜒)

Syntaxw3nand 32519 The double nand.
wff (𝜑𝜓𝜒)

Definitiondf-3nand 32520 The double nand. This definition allows us to express the input of three variables only being false if all three are true. (Contributed by Anthony Hart, 2-Sep-2011.)
((𝜑𝜓𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒)))

Theoremdf3nandALT1 32521 The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011.)
((𝜑𝜓𝜒) ↔ (𝜑 ⊼ ((𝜓𝜒) ⊼ (𝜓𝜒))))

Theoremdf3nandALT2 32522 The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.)
((𝜑𝜓𝜒) ↔ ¬ (𝜑𝜓𝜒))

Theoremandnand1 32523 Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓𝜒) ⊼ (𝜑𝜓𝜒)))

Theoremimnand2 32524 An nand relation. (Contributed by Anthony Hart, 2-Sep-2011.)
((¬ 𝜑𝜓) ↔ ((𝜑𝜑) ⊼ (𝜓𝜓)))

20.10.2  Predicate Calculus

Theoremallt 32525 For all sets, is true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥

Theoremalnof 32526 For all sets, is not true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥 ¬ ⊥

Theoremnalf 32527 Not all sets hold as true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∀𝑥

Theoremextt 32528 There exists a set that holds as true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥

Theoremnextnt 32529 There does not exist a set, such that is not true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃𝑥 ¬ ⊤

Theoremnextf 32530 There does not exist a set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃𝑥

Theoremunnf 32531 There does not exist exactly one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃!𝑥

Theoremunnt 32532 There does not exist exactly one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃!𝑥

Theoremmont 32533 There does not exist at most one set, such that is true. (Contributed by Anthony Hart, 13-Sep-2011.)
¬ ∃*𝑥

Theoremmof 32534 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 32535 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜃𝜑) ∨ (𝜒 ∨ (𝜏𝜑))))

Theoremmeran2 32536 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜏𝜃) ∨ (𝜒 ∨ (𝜑𝜃))))

Theoremmeran3 32537 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(¬ (¬ (¬ 𝜑𝜓) ∨ (𝜒 ∨ (𝜃𝜏))) ∨ (¬ (¬ 𝜒𝜑) ∨ (𝜏 ∨ (𝜃𝜑))))

Theoremwaj-ax 32538 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 32539 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 32540 ? (Contributed by Anthony Hart, 14-Aug-2011.)
((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜑 ⊼ (𝜓𝜒)) ⊼ ((𝜃𝜒) ⊼ ((𝜒𝜃) ⊼ (𝜑𝜃)))))

20.10.4  Connective Symmetry

Theoremnegsym1 32541 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 32542 A symmetry with .

((𝜓 → (𝜓 → ⊥)) → (𝜓𝜑))

Theorembisym1 32543 A symmetry with .

((𝜓 ↔ (𝜓 ↔ ⊥)) → (𝜓𝜑))

Theoremconsym1 32544 A symmetry with .

((𝜓 ∧ (𝜓 ∧ ⊥)) → (𝜓𝜑))

Theoremdissym1 32545 A symmetry with .

((𝜓 ∨ (𝜓 ∨ ⊥)) → (𝜓𝜑))

Theoremnandsym1 32546 A symmetry with .

((𝜓 ⊼ (𝜓 ⊼ ⊥)) → (𝜓𝜑))

Theoremunisym1 32547 A symmetry with .

See negsym1 32541 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

(∀𝑥𝑥⊥ → ∀𝑥𝜑)

Theoremexisym1 32548 A symmetry with .

(∃𝑥𝑥⊥ → ∃𝑥𝜑)

Theoremunqsym1 32549 A symmetry with ∃!.

(∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑)

Theoremamosym1 32550 A symmetry with ∃*.

(∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)

Theoremsubsym1 32551 A symmetry with [𝑥 / 𝑦].

([𝑥 / 𝑦][𝑥 / 𝑦]⊥ → [𝑥 / 𝑦]𝜑)

20.11  Mathbox for Chen-Pang He

20.11.1  Ordinal topology

Theoremontopbas 32552 An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)
(𝐵 ∈ On → 𝐵 ∈ TopBases)

Theoremonsstopbas 32553 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 32554 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 32555 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 32556 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 32557 A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.)
(𝐴 ∈ On → suc 𝐴 ∈ Top)

Theoremonsuctopon 32558 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 32559 Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.)
( 𝐴 ∈ On → suc 𝐴𝑆)       (Ord 𝐴 → (𝐴 𝐴𝐴𝑆))

Theoremordtop 32560 An ordinal is a topology iff it is not its supremum (union), proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 1-Nov-2015.)
(Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 𝐽))

Theoremonsucconni 32561 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
𝐴 ∈ On       suc 𝐴 ∈ Conn

Theoremonsucconn 32562 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
(𝐴 ∈ On → suc 𝐴 ∈ Conn)

Theoremordtopconn 32563 An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.)
(Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Conn))

Theoremonintopssconn 32564 An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)
(On ∩ Top) ⊆ Conn

Theoremonsuct0 32565 A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)
(𝐴 ∈ On → suc 𝐴 ∈ Kol2)

Theoremordtopt0 32566 An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.)
(Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2))

Theoremonsucsuccmpi 32567 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 32568 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 32569 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
Lim 𝐴        ¬ suc 𝐴 ∈ Comp

Theoremlimsucncmp 32570 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
(Lim 𝐴 → ¬ suc 𝐴 ∈ Comp)

Theoremordcmp 32571 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 32572 The ordinal topologies 1𝑜 and 2𝑜 are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)
{1𝑜, 2𝑜} ⊆ (On ∩ Haus)

Theoremonint1 32573 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 32574 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

(𝐴 = 𝐵 → ((𝜑 → (𝐹𝐴) ∈ 𝑃) ↔ (𝜑 → (𝐹𝐵) ∈ 𝑃)))

(𝜑 → (𝐹‘∅) ∈ 𝑃)    &   (𝑦 ∈ ω → (𝜑 → ((𝐹𝑦) ∈ 𝑃 → (𝐹‘suc 𝑦) ∈ 𝑃)))       (𝐴 ∈ ω → (𝜑 → (𝐹𝐴) ∈ 𝑃))

(𝑧𝑃 → (𝐺𝑧) ∈ 𝑃)       (𝐶 ∈ ω → (𝐴𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))

Theoremfindabrcl 32578* 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 32579 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 32580 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)

(((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴 / 𝐶) ∈ ℕ ∧ 𝐴 < 𝐵)) → ((𝐵 / 𝐶) ∈ ℕ ↔ ((𝐵𝐴) / 𝐶) ∈ ℕ))

Theoremnndivlub 32582 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 32583 Extend class notation to include the gdc function. (New usage is discouraged.)
class gcdOLD (𝐴, 𝐵)

Definitiondf-gcdOLD 32584* 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 32585 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 32586* 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 32587 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 32588* 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 32589 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 32590* 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 32591* 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 32592 Variant of rddif 14124. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐴 ∈ ℝ → 0 ≤ ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))))

Theoremdnibndlem1 32593* Lemma for dnibnd 32606. (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 32594* Lemma for dnibnd 32606. (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 32595 Lemma for dnibnd 32606. (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 32596 Lemma for dnibnd 32606. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐵 ∈ ℝ → 0 ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2))))

Theoremdnibndlem5 32597 Lemma for dnibnd 32606. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐴 ∈ ℝ → 0 < (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴))

Theoremdnibndlem6 32598 Lemma for dnibnd 32606. (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 32599 Lemma for dnibnd 32606. (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 32600 Lemma for dnibnd 32606. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ≤ (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴))

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