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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | filnetlem4 35801* | Lemma for filnet 35802. (Contributed by Jeff Hankins, 15-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.) |
β’ π» = βͺ π β πΉ ({π} Γ π) & β’ π· = {β¨π₯, π¦β© β£ ((π₯ β π» β§ π¦ β π») β§ (1st βπ¦) β (1st βπ₯))} β β’ (πΉ β (Filβπ) β βπ β DirRel βπ(π:dom πβΆπ β§ πΉ = ((π FilMap π)βran (tailβπ)))) | ||
Theorem | filnet 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.) |
β’ (πΉ β (Filβπ) β βπ β DirRel βπ(π:dom πβΆπ β§ πΉ = ((π FilMap π)βran (tailβπ)))) | ||
Theorem | tb-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.) |
β’ ((π β π) β ((π β π) β (π β π))) | ||
Theorem | tb-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.) |
β’ (π β (π β π)) | ||
Theorem | tb-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.) |
β’ (((π β π) β π) β π) | ||
Theorem | tbsyl 35806 | The weak syllogism from Tarski-Bernays'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β π) & β’ (π β π) β β’ (π β π) | ||
Theorem | re1ax2lem 35807 | Lemma for re1ax2 35808. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((π β (π β π)) β (π β (π β π))) | ||
Theorem | re1ax2 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.) |
β’ ((π β (π β π)) β ((π β π) β (π β π))) | ||
Theorem | naim1 35809 | Constructor theorem for βΌ. (Contributed by Anthony Hart, 1-Sep-2011.) |
β’ ((π β π) β ((π βΌ π) β (π βΌ π))) | ||
Theorem | naim2 35810 | Constructor theorem for βΌ. (Contributed by Anthony Hart, 1-Sep-2011.) |
β’ ((π β π) β ((π βΌ π) β (π βΌ π))) | ||
Theorem | naim1i 35811 | Constructor rule for βΌ. (Contributed by Anthony Hart, 2-Sep-2011.) |
β’ (π β π) & β’ (π βΌ π) β β’ (π βΌ π) | ||
Theorem | naim2i 35812 | Constructor rule for βΌ. (Contributed by Anthony Hart, 2-Sep-2011.) |
β’ (π β π) & β’ (π βΌ π) β β’ (π βΌ π) | ||
Theorem | naim12i 35813 | Constructor rule for βΌ. (Contributed by Anthony Hart, 2-Sep-2011.) |
β’ (π β π) & β’ (π β π) & β’ (π βΌ π) β β’ (π βΌ π) | ||
Theorem | nabi1i 35814 | Constructor rule for βΌ. (Contributed by Anthony Hart, 2-Sep-2011.) |
β’ (π β π) & β’ (π βΌ π) β β’ (π βΌ π) | ||
Theorem | nabi2i 35815 | Constructor rule for βΌ. (Contributed by Anthony Hart, 2-Sep-2011.) |
β’ (π β π) & β’ (π βΌ π) β β’ (π βΌ π) | ||
Theorem | nabi12i 35816 | Constructor rule for βΌ. (Contributed by Anthony Hart, 2-Sep-2011.) |
β’ (π β π) & β’ (π β π) & β’ (π βΌ π) β β’ (π βΌ π) | ||
Syntax | w3nand 35817 | The double nand. |
wff (π βΌ π βΌ π) | ||
Definition | df-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.) |
β’ ((π βΌ π βΌ π) β (π β (π β Β¬ π))) | ||
Theorem | df3nandALT1 35819 | The double nand expressed in terms of pure nand. (Contributed by Anthony Hart, 2-Sep-2011.) |
β’ ((π βΌ π βΌ π) β (π βΌ ((π βΌ π) βΌ (π βΌ π)))) | ||
Theorem | df3nandALT2 35820 | The double nand expressed in terms of negation and and not. (Contributed by Anthony Hart, 13-Sep-2011.) |
β’ ((π βΌ π βΌ π) β Β¬ (π β§ π β§ π)) | ||
Theorem | andnand1 35821 | Double and in terms of double nand. (Contributed by Anthony Hart, 2-Sep-2011.) |
β’ ((π β§ π β§ π) β ((π βΌ π βΌ π) βΌ (π βΌ π βΌ π))) | ||
Theorem | imnand2 35822 | An β nand relation. (Contributed by Anthony Hart, 2-Sep-2011.) |
β’ ((Β¬ π β π) β ((π βΌ π) βΌ (π βΌ π))) | ||
Theorem | nalfal 35823 | Not all sets hold β₯ as true. (Contributed by Anthony Hart, 13-Sep-2011.) |
β’ Β¬ βπ₯β₯ | ||
Theorem | nexntru 35824 | There does not exist a set such that β€ is not true. (Contributed by Anthony Hart, 13-Sep-2011.) |
β’ Β¬ βπ₯ Β¬ β€ | ||
Theorem | nexfal 35825 | There does not exist a set such that β₯ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
β’ Β¬ βπ₯β₯ | ||
Theorem | neufal 35826 | There does not exist exactly one set such that β₯ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
β’ Β¬ β!π₯β₯ | ||
Theorem | neutru 35827 | There does not exist exactly one set such that β€ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
β’ Β¬ β!π₯β€ | ||
Theorem | nmotru 35828 | There does not exist at most one set such that β€ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
β’ Β¬ β*π₯β€ | ||
Theorem | mofal 35829 | There exist at most one set such that β₯ is true. (Contributed by Anthony Hart, 13-Sep-2011.) |
β’ β*π₯β₯ | ||
Theorem | nrmo 35830 | "At most one" restricted existential quantifier for a statement which is never true. (Contributed by Thierry Arnoux, 27-Nov-2023.) |
β’ (π₯ β π΄ β Β¬ π) β β’ β*π₯ β π΄ π | ||
Theorem | meran1 35831 | A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.) |
β’ (Β¬ (Β¬ (Β¬ π β¨ π) β¨ (π β¨ (π β¨ π))) β¨ (Β¬ (Β¬ π β¨ π) β¨ (π β¨ (π β¨ π)))) | ||
Theorem | meran2 35832 | A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.) |
β’ (Β¬ (Β¬ (Β¬ π β¨ π) β¨ (π β¨ (π β¨ π))) β¨ (Β¬ (Β¬ π β¨ π) β¨ (π β¨ (π β¨ π)))) | ||
Theorem | meran3 35833 | A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.) |
β’ (Β¬ (Β¬ (Β¬ π β¨ π) β¨ (π β¨ (π β¨ π))) β¨ (Β¬ (Β¬ π β¨ π) β¨ (π β¨ (π β¨ π)))) | ||
Theorem | waj-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.) |
β’ ((π βΌ (π βΌ π)) βΌ (((π βΌ π) βΌ ((π βΌ π) βΌ (π βΌ π))) βΌ (π βΌ (π βΌ π)))) | ||
Theorem | lukshef-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.) |
β’ ((π βΌ (π βΌ π)) βΌ ((π βΌ (π βΌ π)) βΌ ((π βΌ π) βΌ ((π βΌ π) βΌ (π βΌ π))))) | ||
Theorem | arg-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.) |
β’ ((π βΌ (π βΌ π)) βΌ ((π βΌ (π βΌ π)) βΌ ((π βΌ π) βΌ ((π βΌ π) βΌ (π βΌ π))))) | ||
Theorem | negsym1 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 "
means that
"something is true of π". The expression "delta
π
" can be
substituted with Β¬ π, π β§ π, βπ₯π, etc.
Later on, Meredith discovered a single axiom, in the form of ( delta delta β₯ β delta π ). 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¬ Β¬ β₯ β Β¬ π) | ||
Theorem | imsym1 35838 |
A symmetry with β.
See negsym1 35837 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
β’ ((π β (π β β₯)) β (π β π)) | ||
Theorem | bisym1 35839 |
A symmetry with β.
See negsym1 35837 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
β’ ((π β (π β β₯)) β (π β π)) | ||
Theorem | consym1 35840 |
A symmetry with β§.
See negsym1 35837 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
β’ ((π β§ (π β§ β₯)) β (π β§ π)) | ||
Theorem | dissym1 35841 |
A symmetry with β¨.
See negsym1 35837 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
β’ ((π β¨ (π β¨ β₯)) β (π β¨ π)) | ||
Theorem | nandsym1 35842 |
A symmetry with βΌ.
See negsym1 35837 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
β’ ((π βΌ (π βΌ β₯)) β (π βΌ π)) | ||
Theorem | unisym1 35843 |
A symmetry with β.
See negsym1 35837 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
β’ (βπ₯βπ₯β₯ β βπ₯π) | ||
Theorem | exisym1 35844 |
A symmetry with β.
See negsym1 35837 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) |
β’ (βπ₯βπ₯β₯ β βπ₯π) | ||
Theorem | unqsym1 35845 |
A symmetry with β!.
See negsym1 35837 for more information. (Contributed by Anthony Hart, 6-Sep-2011.) |
β’ (β!π₯β!π₯β₯ β β!π₯π) | ||
Theorem | amosym1 35846 |
A symmetry with β*.
See negsym1 35837 for more information. (Contributed by Anthony Hart, 13-Sep-2011.) |
β’ (β*π₯β*π₯β₯ β β*π₯π) | ||
Theorem | subsym1 35847 |
A symmetry with [π₯ / π¦].
See negsym1 35837 for more information. (Contributed by Anthony Hart, 11-Sep-2011.) |
β’ ([π¦ / π₯][π¦ / π₯]β₯ β [π¦ / π₯]π) | ||
Theorem | ontopbas 35848 | An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.) |
β’ (π΅ β On β π΅ β TopBases) | ||
Theorem | onsstopbas 35849 | The class of ordinal numbers is a subclass of the class of topological bases. (Contributed by Chen-Pang He, 8-Oct-2015.) |
β’ On β TopBases | ||
Theorem | onpsstopbas 35850 | The class of ordinal numbers is a proper subclass of the class of topological bases. (Contributed by Chen-Pang He, 9-Oct-2015.) |
β’ On β TopBases | ||
Theorem | ontgval 35851 | The topology generated from an ordinal number π΅ is suc βͺ π΅. (Contributed by Chen-Pang He, 10-Oct-2015.) |
β’ (π΅ β On β (topGenβπ΅) = suc βͺ π΅) | ||
Theorem | ontgsucval 35852 | The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.) |
β’ (π΄ β On β (topGenβsuc π΄) = suc π΄) | ||
Theorem | onsuctop 35853 | A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.) |
β’ (π΄ β On β suc π΄ β Top) | ||
Theorem | onsuctopon 35854 | One of the topologies on an ordinal number is its successor. (Contributed by Chen-Pang He, 7-Nov-2015.) |
β’ (π΄ β On β suc π΄ β (TopOnβπ΄)) | ||
Theorem | ordtoplem 35855 | Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.) |
β’ (βͺ π΄ β On β suc βͺ π΄ β π) β β’ (Ord π΄ β (π΄ β βͺ π΄ β π΄ β π)) | ||
Theorem | ordtop 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.) |
β’ (Ord π½ β (π½ β Top β π½ β βͺ π½)) | ||
Theorem | onsucconni 35857 | A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.) |
β’ π΄ β On β β’ suc π΄ β Conn | ||
Theorem | onsucconn 35858 | A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.) |
β’ (π΄ β On β suc π΄ β Conn) | ||
Theorem | ordtopconn 35859 | An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.) |
β’ (Ord π½ β (π½ β Top β π½ β Conn)) | ||
Theorem | onintopssconn 35860 | An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.) |
β’ (On β© Top) β Conn | ||
Theorem | onsuct0 35861 | A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.) |
β’ (π΄ β On β suc π΄ β Kol2) | ||
Theorem | ordtopt0 35862 | An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.) |
β’ (Ord π½ β (π½ β Top β π½ β Kol2)) | ||
Theorem | onsucsuccmpi 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.) |
β’ π΄ β On β β’ suc suc π΄ β Comp | ||
Theorem | onsucsuccmp 35864 | The successor of a successor ordinal number is a compact topology. (Contributed by Chen-Pang He, 18-Oct-2015.) |
β’ (π΄ β On β suc suc π΄ β Comp) | ||
Theorem | limsucncmpi 35865 | The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.) |
β’ Lim π΄ β β’ Β¬ suc π΄ β Comp | ||
Theorem | limsucncmp 35866 | The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.) |
β’ (Lim π΄ β Β¬ suc π΄ β Comp) | ||
Theorem | ordcmp 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.) |
β’ (Ord π΄ β (π΄ β Comp β (βͺ π΄ = βͺ βͺ π΄ β π΄ = 1o))) | ||
Theorem | ssoninhaus 35868 | The ordinal topologies 1o and 2o are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.) |
β’ {1o, 2o} β (On β© Haus) | ||
Theorem | onint1 35869 | The ordinal T1 spaces are 1o and 2o, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 9-Nov-2015.) |
β’ (On β© Fre) = {1o, 2o} | ||
Theorem | oninhaus 35870 | The ordinal Hausdorff spaces are 1o and 2o. (Contributed by Chen-Pang He, 10-Nov-2015.) |
β’ (On β© Haus) = {1o, 2o} | ||
Theorem | fveleq 35871 | Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.) |
β’ (π΄ = π΅ β ((π β (πΉβπ΄) β π) β (π β (πΉβπ΅) β π))) | ||
Theorem | findfvcl 35872* | Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.) |
β’ (π β (πΉββ ) β π) & β’ (π¦ β Ο β (π β ((πΉβπ¦) β π β (πΉβsuc π¦) β π))) β β’ (π΄ β Ο β (π β (πΉβπ΄) β π)) | ||
Theorem | findreccl 35873* | Please add description here. (Contributed by Jeff Hoffman, 19-Feb-2008.) |
β’ (π§ β π β (πΊβπ§) β π) β β’ (πΆ β Ο β (π΄ β π β (rec(πΊ, π΄)βπΆ) β π)) | ||
Theorem | findabrcl 35874* | Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.) |
β’ (π§ β π β (πΊβπ§) β π) β β’ ((πΆ β Ο β§ π΄ β π) β ((π₯ β V β¦ (rec(πΊ, π΄)βπ₯))βπΆ) β π) | ||
Theorem | nnssi2 35875 | Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
β’ β β π· & β’ (π΅ β β β π) & β’ ((π΄ β π· β§ π΅ β π· β§ π) β π) β β’ ((π΄ β β β§ π΅ β β) β π) | ||
Theorem | nnssi3 35876 | Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
β’ β β π· & β’ (πΆ β β β π) & β’ (((π΄ β π· β§ π΅ β π· β§ πΆ β π·) β§ π) β π) β β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β π) | ||
Theorem | nndivsub 35877 | Please add description here. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
β’ (((π΄ β β β§ π΅ β β β§ πΆ β β) β§ ((π΄ / πΆ) β β β§ π΄ < π΅)) β ((π΅ / πΆ) β β β ((π΅ β π΄) / πΆ) β β)) | ||
Theorem | nndivlub 35878 | A factor of a positive integer cannot exceed it. (Contributed by Jeff Hoffman, 17-Jun-2008.) |
β’ ((π΄ β β β§ π΅ β β) β ((π΄ / π΅) β β β π΅ β€ π΄)) | ||
Syntax | cgcdOLD 35879 | Extend class notation to include the gdc function. (New usage is discouraged.) |
class gcdOLD (π΄, π΅) | ||
Definition | df-gcdOLD 35880* | gcdOLD (π΄, π΅) is the largest positive integer that evenly divides both π΄ and π΅. (Contributed by Jeff Hoffman, 17-Jun-2008.) (New usage is discouraged.) |
β’ gcdOLD (π΄, π΅) = sup({π₯ β β β£ ((π΄ / π₯) β β β§ (π΅ / π₯) β β)}, β, < ) | ||
Theorem | ee7.2aOLD 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 π΄ mod π΅. 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.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ < π΅ β gcdOLD (π΄, π΅) = gcdOLD (π΄, (π΅ β π΄)))) | ||
Theorem | dnival 35882* | Value of the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) β β’ (π΄ β β β (πβπ΄) = (absβ((ββ(π΄ + (1 / 2))) β π΄))) | ||
Theorem | dnicld1 35883 | Closure theorem for the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ (π β π΄ β β) β β’ (π β (absβ((ββ(π΄ + (1 / 2))) β π΄)) β β) | ||
Theorem | dnicld2 35884* | Closure theorem for the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ (π β π΄ β β) β β’ (π β (πβπ΄) β β) | ||
Theorem | dnif 35885 | The "distance to nearest integer" function is a function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) β β’ π:ββΆβ | ||
Theorem | dnizeq0 35886* | The distance to nearest integer is zero for integers. (Contributed by Asger C. Ipsen, 15-Jun-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ (π β π΄ β β€) β β’ (π β (πβπ΄) = 0) | ||
Theorem | dnizphlfeqhlf 35887* | The distance to nearest integer is a half for half-integers. (Contributed by Asger C. Ipsen, 15-Jun-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ (π β π΄ β β€) β β’ (π β (πβ(π΄ + (1 / 2))) = (1 / 2)) | ||
Theorem | rddif2 35888 | Variant of rddif 15311. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ (π΄ β β β 0 β€ ((1 / 2) β (absβ((ββ(π΄ + (1 / 2))) β π΄)))) | ||
Theorem | dnibndlem1 35889* | Lemma for dnibnd 35902. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ (π β π΄ β β) & β’ (π β π΅ β β) β β’ (π β ((absβ((πβπ΅) β (πβπ΄))) β€ π β (absβ((absβ((ββ(π΅ + (1 / 2))) β π΅)) β (absβ((ββ(π΄ + (1 / 2))) β π΄)))) β€ π)) | ||
Theorem | dnibndlem2 35890* | Lemma for dnibnd 35902. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β (ββ(π΅ + (1 / 2))) = (ββ(π΄ + (1 / 2)))) β β’ (π β (absβ((πβπ΅) β (πβπ΄))) β€ (absβ(π΅ β π΄))) | ||
Theorem | dnibndlem3 35891 | Lemma for dnibnd 35902. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β (ββ(π΅ + (1 / 2))) = ((ββ(π΄ + (1 / 2))) + 1)) β β’ (π β (absβ(π΅ β π΄)) = (absβ((π΅ β ((ββ(π΅ + (1 / 2))) β (1 / 2))) + (((ββ(π΄ + (1 / 2))) + (1 / 2)) β π΄)))) | ||
Theorem | dnibndlem4 35892 | Lemma for dnibnd 35902. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ (π΅ β β β 0 β€ (π΅ β ((ββ(π΅ + (1 / 2))) β (1 / 2)))) | ||
Theorem | dnibndlem5 35893 | Lemma for dnibnd 35902. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ (π΄ β β β 0 < (((ββ(π΄ + (1 / 2))) + (1 / 2)) β π΄)) | ||
Theorem | dnibndlem6 35894 | Lemma for dnibnd 35902. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) β β’ (π β (absβ((absβ((ββ(π΅ + (1 / 2))) β π΅)) β (absβ((ββ(π΄ + (1 / 2))) β π΄)))) β€ (((1 / 2) β (absβ((ββ(π΅ + (1 / 2))) β π΅))) + ((1 / 2) β (absβ((ββ(π΄ + (1 / 2))) β π΄))))) | ||
Theorem | dnibndlem7 35895 | Lemma for dnibnd 35902. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ (π β π΅ β β) β β’ (π β ((1 / 2) β (absβ((ββ(π΅ + (1 / 2))) β π΅))) β€ (π΅ β ((ββ(π΅ + (1 / 2))) β (1 / 2)))) | ||
Theorem | dnibndlem8 35896 | Lemma for dnibnd 35902. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ (π β π΄ β β) β β’ (π β ((1 / 2) β (absβ((ββ(π΄ + (1 / 2))) β π΄))) β€ (((ββ(π΄ + (1 / 2))) + (1 / 2)) β π΄)) | ||
Theorem | dnibndlem9 35897* | Lemma for dnibnd 35902. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β (ββ(π΅ + (1 / 2))) = ((ββ(π΄ + (1 / 2))) + 1)) β β’ (π β (absβ((πβπ΅) β (πβπ΄))) β€ (absβ(π΅ β π΄))) | ||
Theorem | dnibndlem10 35898 | Lemma for dnibnd 35902. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β ((ββ(π΄ + (1 / 2))) + 2) β€ (ββ(π΅ + (1 / 2)))) β β’ (π β 1 β€ (π΅ β π΄)) | ||
Theorem | dnibndlem11 35899 | Lemma for dnibnd 35902. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) β β’ (π β (absβ((absβ((ββ(π΅ + (1 / 2))) β π΅)) β (absβ((ββ(π΄ + (1 / 2))) β π΄)))) β€ (1 / 2)) | ||
Theorem | dnibndlem12 35900* | Lemma for dnibnd 35902. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β ((ββ(π΄ + (1 / 2))) + 2) β€ (ββ(π΅ + (1 / 2)))) β β’ (π β (absβ((πβπ΅) β (πβπ΄))) β€ (absβ(π΅ β π΄))) |
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