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Theorem List for Metamath Proof Explorer - 35801-35900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfilnetlem4 35801* Lemma for filnet 35802. (Contributed by Jeff Hankins, 15-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
𝐻 = βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛)    &   π· = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st β€˜π‘¦) βŠ† (1st β€˜π‘₯))}    β‡’   (𝐹 ∈ (Filβ€˜π‘‹) β†’ βˆƒπ‘‘ ∈ DirRel βˆƒπ‘“(𝑓:dom π‘‘βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π‘‘))))
 
Theoremfilnet 35802* 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 ∈ (Filβ€˜π‘‹) β†’ βˆƒπ‘‘ ∈ DirRel βˆƒπ‘“(𝑓:dom π‘‘βŸΆπ‘‹ ∧ 𝐹 = ((𝑋 FilMap 𝑓)β€˜ran (tailβ€˜π‘‘))))
 
21.13  Mathbox for Anthony Hart
 
21.13.1  Propositional Calculus
 
Theoremtb-ax1 35803 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.)
((πœ‘ β†’ πœ“) β†’ ((πœ“ β†’ πœ’) β†’ (πœ‘ β†’ πœ’)))
 
Theoremtb-ax2 35804 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.)
(πœ‘ β†’ (πœ“ β†’ πœ‘))
 
Theoremtb-ax3 35805 The third of three axioms in the Tarski-Bernays axiom system.

This axiom, along with ax-mp 5, tb-ax1 35803, and tb-ax2 35804, can be used to derive any theorem or rule that uses only β†’. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

(((πœ‘ β†’ πœ“) β†’ πœ‘) β†’ πœ‘)
 
Theoremtbsyl 35806 The weak syllogism from Tarski-Bernays'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ πœ“)    &   (πœ“ β†’ πœ’)    β‡’   (πœ‘ β†’ πœ’)
 
Theoremre1ax2lem 35807 Lemma for re1ax2 35808. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((πœ‘ β†’ (πœ“ β†’ πœ’)) β†’ (πœ“ β†’ (πœ‘ β†’ πœ’)))
 
Theoremre1ax2 35808 ax-2 7 rederived from the Tarski-Bernays axiom system. Often tb-ax1 35803 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.)
((πœ‘ β†’ (πœ“ β†’ πœ’)) β†’ ((πœ‘ β†’ πœ“) β†’ (πœ‘ β†’ πœ’)))
 
Theoremnaim1 35809 Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.)
((πœ‘ β†’ πœ“) β†’ ((πœ“ ⊼ πœ’) β†’ (πœ‘ ⊼ πœ’)))
 
Theoremnaim2 35810 Constructor theorem for ⊼. (Contributed by Anthony Hart, 1-Sep-2011.)
((πœ‘ β†’ πœ“) β†’ ((πœ’ ⊼ πœ“) β†’ (πœ’ ⊼ πœ‘)))
 
Theoremnaim1i 35811 Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
(πœ‘ β†’ πœ“)    &   (πœ“ ⊼ πœ’)    β‡’   (πœ‘ ⊼ πœ’)
 
Theoremnaim2i 35812 Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
(πœ‘ β†’ πœ“)    &   (πœ’ ⊼ πœ“)    β‡’   (πœ’ ⊼ πœ‘)
 
Theoremnaim12i 35813 Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
(πœ‘ β†’ πœ“)    &   (πœ’ β†’ πœƒ)    &   (πœ“ ⊼ πœƒ)    β‡’   (πœ‘ ⊼ πœ’)
 
Theoremnabi1i 35814 Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
(πœ‘ ↔ πœ“)    &   (πœ“ ⊼ πœ’)    β‡’   (πœ‘ ⊼ πœ’)
 
Theoremnabi2i 35815 Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
(πœ‘ ↔ πœ“)    &   (πœ’ ⊼ πœ“)    β‡’   (πœ’ ⊼ πœ‘)
 
Theoremnabi12i 35816 Constructor rule for ⊼. (Contributed by Anthony Hart, 2-Sep-2011.)
(πœ‘ ↔ πœ“)    &   (πœ’ ↔ πœƒ)    &   (πœ“ ⊼ πœƒ)    β‡’   (πœ‘ ⊼ πœ’)
 
Syntaxw3nand 35817 The double nand.
wff (πœ‘ ⊼ πœ“ ⊼ πœ’)
 
Definitiondf-3nand 35818 The double nand. This definition allows to express the input of three variables only being false if all three are true. (Contributed by Anthony Hart, 2-Sep-2011.)
((πœ‘ ⊼ πœ“ ⊼ πœ’) ↔ (πœ‘ β†’ (πœ“ β†’ Β¬ πœ’)))
 
Theoremdf3nandALT1 35819 The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011.)
((πœ‘ ⊼ πœ“ ⊼ πœ’) ↔ (πœ‘ ⊼ ((πœ“ ⊼ πœ’) ⊼ (πœ“ ⊼ πœ’))))
 
Theoremdf3nandALT2 35820 The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.)
((πœ‘ ⊼ πœ“ ⊼ πœ’) ↔ Β¬ (πœ‘ ∧ πœ“ ∧ πœ’))
 
Theoremandnand1 35821 Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.)
((πœ‘ ∧ πœ“ ∧ πœ’) ↔ ((πœ‘ ⊼ πœ“ ⊼ πœ’) ⊼ (πœ‘ ⊼ πœ“ ⊼ πœ’)))
 
Theoremimnand2 35822 An β†’ nand relation. (Contributed by Anthony Hart, 2-Sep-2011.)
((Β¬ πœ‘ β†’ πœ“) ↔ ((πœ‘ ⊼ πœ‘) ⊼ (πœ“ ⊼ πœ“)))
 
21.13.2  Predicate Calculus
 
Theoremnalfal 35823 Not all sets hold βŠ₯ as true. (Contributed by Anthony Hart, 13-Sep-2011.)
Β¬ βˆ€π‘₯βŠ₯
 
Theoremnexntru 35824 There does not exist a set such that ⊀ is not true. (Contributed by Anthony Hart, 13-Sep-2011.)
Β¬ βˆƒπ‘₯ Β¬ ⊀
 
Theoremnexfal 35825 There does not exist a set such that βŠ₯ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
Β¬ βˆƒπ‘₯βŠ₯
 
Theoremneufal 35826 There does not exist exactly one set such that βŠ₯ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
Β¬ βˆƒ!π‘₯βŠ₯
 
Theoremneutru 35827 There does not exist exactly one set such that ⊀ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
Β¬ βˆƒ!π‘₯⊀
 
Theoremnmotru 35828 There does not exist at most one set such that ⊀ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
Β¬ βˆƒ*π‘₯⊀
 
Theoremmofal 35829 There exist at most one set such that βŠ₯ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
βˆƒ*π‘₯βŠ₯
 
Theoremnrmo 35830 "At most one" restricted existential quantifier for a statement which is never true. (Contributed by Thierry Arnoux, 27-Nov-2023.)
(π‘₯ ∈ 𝐴 β†’ Β¬ πœ‘)    β‡’   βˆƒ*π‘₯ ∈ 𝐴 πœ‘
 
21.13.3  Miscellaneous single axioms
 
Theoremmeran1 35831 A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(Β¬ (Β¬ (Β¬ πœ‘ ∨ πœ“) ∨ (πœ’ ∨ (πœƒ ∨ 𝜏))) ∨ (Β¬ (Β¬ πœƒ ∨ πœ‘) ∨ (πœ’ ∨ (𝜏 ∨ πœ‘))))
 
Theoremmeran2 35832 A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(Β¬ (Β¬ (Β¬ πœ‘ ∨ πœ“) ∨ (πœ’ ∨ (πœƒ ∨ 𝜏))) ∨ (Β¬ (Β¬ 𝜏 ∨ πœƒ) ∨ (πœ’ ∨ (πœ‘ ∨ πœƒ))))
 
Theoremmeran3 35833 A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
(Β¬ (Β¬ (Β¬ πœ‘ ∨ πœ“) ∨ (πœ’ ∨ (πœƒ ∨ 𝜏))) ∨ (Β¬ (Β¬ πœ’ ∨ πœ‘) ∨ (𝜏 ∨ (πœƒ ∨ πœ‘))))
 
Theoremwaj-ax 35834 A single axiom for propositional calculus discovered by Mordchaj Wajsberg (Logical Works, Polish Academy of Sciences, 1977). See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom W on slide 8). (Contributed by Anthony Hart, 13-Aug-2011.)
((πœ‘ ⊼ (πœ“ ⊼ πœ’)) ⊼ (((πœƒ ⊼ πœ’) ⊼ ((πœ‘ ⊼ πœƒ) ⊼ (πœ‘ ⊼ πœƒ))) ⊼ (πœ‘ ⊼ (πœ‘ ⊼ πœ“))))
 
Theoremlukshef-ax2 35835 A single axiom for propositional calculus discovered by Jan Lukasiewicz. See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom L2 on slide 8). (Contributed by Anthony Hart, 14-Aug-2011.)
((πœ‘ ⊼ (πœ“ ⊼ πœ’)) ⊼ ((πœ‘ ⊼ (πœ’ ⊼ πœ‘)) ⊼ ((πœƒ ⊼ πœ“) ⊼ ((πœ‘ ⊼ πœƒ) ⊼ (πœ‘ ⊼ πœƒ)))))
 
Theoremarg-ax 35836 A single axiom for propositional calculus discovered by Ken Harris and Branden Fitelson. See: Fitelson, Some recent results in algebra and logical calculi obtained using automated reasoning, 2003 (axiom HF1 on slide 8). (Contributed by Anthony Hart, 14-Aug-2011.)
((πœ‘ ⊼ (πœ“ ⊼ πœ’)) ⊼ ((πœ‘ ⊼ (πœ“ ⊼ πœ’)) ⊼ ((πœƒ ⊼ πœ’) ⊼ ((πœ’ ⊼ πœƒ) ⊼ (πœ‘ ⊼ πœƒ)))))
 
21.13.4  Connective Symmetry
 
Theoremnegsym1 35837 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 πœ‘ " means that "something is true of πœ‘". The expression "delta πœ‘ " can be substituted with Β¬ πœ‘, πœ“ ∧ πœ‘, βˆ€π‘₯πœ‘, etc.

Later on, Meredith discovered a single axiom, in the form of ( delta delta βŠ₯ β†’ delta πœ‘ ). 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 Β¬. (Contributed by Anthony Hart, 4-Sep-2011.)

(Β¬ Β¬ βŠ₯ β†’ Β¬ πœ‘)
 
Theoremimsym1 35838 A symmetry with β†’.

See negsym1 35837 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

((πœ“ β†’ (πœ“ β†’ βŠ₯)) β†’ (πœ“ β†’ πœ‘))
 
Theorembisym1 35839 A symmetry with ↔.

See negsym1 35837 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

((πœ“ ↔ (πœ“ ↔ βŠ₯)) β†’ (πœ“ ↔ πœ‘))
 
Theoremconsym1 35840 A symmetry with ∧.

See negsym1 35837 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

((πœ“ ∧ (πœ“ ∧ βŠ₯)) β†’ (πœ“ ∧ πœ‘))
 
Theoremdissym1 35841 A symmetry with ∨.

See negsym1 35837 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

((πœ“ ∨ (πœ“ ∨ βŠ₯)) β†’ (πœ“ ∨ πœ‘))
 
Theoremnandsym1 35842 A symmetry with ⊼.

See negsym1 35837 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

((πœ“ ⊼ (πœ“ ⊼ βŠ₯)) β†’ (πœ“ ⊼ πœ‘))
 
Theoremunisym1 35843 A symmetry with βˆ€.

See negsym1 35837 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 35844 A symmetry with βˆƒ.

See negsym1 35837 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

(βˆƒπ‘₯βˆƒπ‘₯βŠ₯ β†’ βˆƒπ‘₯πœ‘)
 
Theoremunqsym1 35845 A symmetry with βˆƒ!.

See negsym1 35837 for more information. (Contributed by Anthony Hart, 6-Sep-2011.)

(βˆƒ!π‘₯βˆƒ!π‘₯βŠ₯ β†’ βˆƒ!π‘₯πœ‘)
 
Theoremamosym1 35846 A symmetry with βˆƒ*.

See negsym1 35837 for more information. (Contributed by Anthony Hart, 13-Sep-2011.)

(βˆƒ*π‘₯βˆƒ*π‘₯βŠ₯ β†’ βˆƒ*π‘₯πœ‘)
 
Theoremsubsym1 35847 A symmetry with [π‘₯ / 𝑦].

See negsym1 35837 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 35848 An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)
(𝐡 ∈ On β†’ 𝐡 ∈ TopBases)
 
Theoremonsstopbas 35849 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 35850 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 35851 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 35852 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 35853 A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.)
(𝐴 ∈ On β†’ suc 𝐴 ∈ Top)
 
Theoremonsuctopon 35854 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 35855 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 35856 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 35857 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
𝐴 ∈ On    β‡’   suc 𝐴 ∈ Conn
 
Theoremonsucconn 35858 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
(𝐴 ∈ On β†’ suc 𝐴 ∈ Conn)
 
Theoremordtopconn 35859 An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.)
(Ord 𝐽 β†’ (𝐽 ∈ Top ↔ 𝐽 ∈ Conn))
 
Theoremonintopssconn 35860 An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)
(On ∩ Top) βŠ† Conn
 
Theoremonsuct0 35861 A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)
(𝐴 ∈ On β†’ suc 𝐴 ∈ Kol2)
 
Theoremordtopt0 35862 An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.)
(Ord 𝐽 β†’ (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2))
 
Theoremonsucsuccmpi 35863 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 35864 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 35865 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 35866 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 35867 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 35868 The ordinal topologies 1o and 2o are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)
{1o, 2o} βŠ† (On ∩ Haus)
 
Theoremonint1 35869 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 35870 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 35871 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 35872* 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 35873* 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 35874* 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 35875 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 35876 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 35877 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 35878 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 35879 Extend class notation to include the gdc function. (New usage is discouraged.)
class gcdOLD (𝐴, 𝐡)
 
Definitiondf-gcdOLD 35880* 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 35881 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 35882* 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 35883 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 35884* 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 35885 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 35886* 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 35887* 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 35888 Variant of rddif 15311. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐴 ∈ ℝ β†’ 0 ≀ ((1 / 2) βˆ’ (absβ€˜((βŒŠβ€˜(𝐴 + (1 / 2))) βˆ’ 𝐴))))
 
Theoremdnibndlem1 35889* Lemma for dnibnd 35902. (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 35890* Lemma for dnibnd 35902. (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 35891 Lemma for dnibnd 35902. (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 35892 Lemma for dnibnd 35902. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐡 ∈ ℝ β†’ 0 ≀ (𝐡 βˆ’ ((βŒŠβ€˜(𝐡 + (1 / 2))) βˆ’ (1 / 2))))
 
Theoremdnibndlem5 35893 Lemma for dnibnd 35902. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
(𝐴 ∈ ℝ β†’ 0 < (((βŒŠβ€˜(𝐴 + (1 / 2))) + (1 / 2)) βˆ’ 𝐴))
 
Theoremdnibndlem6 35894 Lemma for dnibnd 35902. (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 35895 Lemma for dnibnd 35902. (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 35896 Lemma for dnibnd 35902. (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 35897* Lemma for dnibnd 35902. (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 35898 Lemma for dnibnd 35902. (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 35899 Lemma for dnibnd 35902. (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 35900* Lemma for dnibnd 35902. (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)))
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