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

Theoremlincvalpr 32501 The linear combination over an unordered pair. (Contributed by AV, 16-Apr-2019.)
Scalar                                   linC

Theoremlincval1 32502 The linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.)
Scalar                     linC

Theoremlcosn0 32503 Properties of a linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Scalar                     finSupp linC

Theoremlincvalsc0 32504* The linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.)
Scalar                            linC

Theoremlcoc0 32505* Properties of a linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Scalar                                   finSupp linC

Theoremlinc0scn0 32506* If a set contains the zero element of a module, there is a linear combination being 0 where not all scalars are 0. (Contributed by AV, 13-Apr-2019.)
Scalar                                   linC

Theoremlincdifsn 32507 A vector is a linear combination of a set containing this vector. (Contributed by AV, 21-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Scalar                                   finSupp linC linC

Theoremlinc1 32508* A vector is a linear combination of a set containing this vector. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
Scalar                            linC

Theoremlincellss 32509 A linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Scalar finSupp Scalar linC

Theoremlco0 32510 The set of empty linear combinations over a monoid is the singleton with the identity element of the monoid. (Contributed by AV, 12-Apr-2019.)
LinCo

Theoremlcoel0 32511 The zero vector is always a linear combination. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
LinCo

Theoremlincsum 32512 The sum of two linear combinations is a linear combination, see also [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Revised by AV, 28-Jul-2019.)
linC        linC        Scalar                     finSupp finSupp linC

Theoremlincscm 32513* A linear combinations multiplied with a scalar is a linear combination, see also [Lang] p. 129. (Contributed by AV, 9-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Scalar       linC        Scalar              finSupp Scalar linC

Theoremlincsumcl 32514 The sum of two linear combinations is a linear combination, see also [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
LinCo LinCo LinCo

Theoremlincscmcl 32515 The multiplication of a linear combination with a scalar is a linear combination, see also [Lang] p. 129. (Contributed by AV, 11-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
Scalar       LinCo LinCo

Theoremlincsumscmcl 32516 The sum of a linear combination and a multiplication of a linear combination with a scalar is a linear combination. (Contributed by AV, 11-Apr-2019.)
Scalar              LinCo LinCo LinCo

Theoremlincolss 32517 According to the statement in [Lang] p. 129, the set of all linear combinations of a set of vectors V is a submodule (generated by V) of the module M. The elements of V are called generators of . (Contributed by AV, 12-Apr-2019.)
LinCo

Theoremellcoellss 32518* Every linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
LinCo

Theoremlcoss 32519 A set of vectors of a module is a subset of the set of all linear combinations of the set. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
LinCo

Theoremlspsslco 32520 Lemma for lspeqlco 32522. (Contributed by AV, 17-Apr-2019.)
LinCo

Theoremlcosslsp 32521 Lemma for lspeqlco 32522. (Contributed by AV, 20-Apr-2019.)
LinCo

Theoremlspeqlco 32522 Equivalence of a span of a set of vectors of a left module defined as the intersection of all linear subspaces which each contain every vector in that set ( see df-lsp 17489) and as the set of all linear combinations of the vectors of the set with finite support. (Contributed by AV, 20-Apr-2019.)
LinCo

21.25.9.3  Linear independency

According to the definition in [Lang] p. 129: "A subset S of a module M is said to be linearly independent (over [the ring] A) if whenever we have a linear combination ∑x ∈S axx which is equal to 0, then ax=0 for all x∈S.". This definition does not care for the finiteness of the set S (because the definition of a linear combination in [Lang] p.129 does already assure that only a finite number of coefficients can be 0 in the sum). Our definition df-lininds 32525 does also neither claim that the subset must be finite, nor that almost all coefficients within the linear combination are 0. If this is required, it must be explicitly stated as precondition in the corresponding theorems.

Usually, the linear independency is defined for vector spaces, see Wikipedia ("Linear independence", 15-Apr-2019, https://en.wikipedia.org/wiki/Linear_independence): "In the theory of vector spaces, a set of vectors is said to be linearly dependent if at least one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be linearly independent.". Furthermore, "In order to allow the number of linearly independent vectors in a vector space to be countably infinite, it is useful to define linear dependence as follows. More generally, let V be a vector space over a field K, and let {vi | i∈I} be a family of elements of V. The family is linearly dependent over K if there exists a finite family {aj | j∈J} of elements of K, all non-zero, such that ∑j∈J ajvj=0. A set X of elements of V is linearly independent if the corresponding family{x}x∈X is linearly independent".
Remark 1: There are already definitions of (linearly) independent families (df-lindf 18710) and (linearly) independent sets (df-linds 18711). These definitions are based on the principle "of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements" or (see lbsind2 17598) "every element is not in the span of the remainder of the [set]". The equivalence of the definitions df-linds 18711 and df-lininds 32525 for (linear) independency for (left) modules is shown in lindslininds 32547.
Remark 2: Subsets of the base set of a (left) module are linearly dependent if they are not linearly indepent (see df-lindeps 32527) or, according to Wikipedia, "if at least one of the vectors in the set can be defined as a linear combination of the others", see islindeps2 32566. The reversed implication is not valid for arbitrary modules (but for arbitrary vector spaces), because it requires a division by a coefficient. Therefore, the definition of Wikipedia is equivalent with our definition for (left) vector spaces (see isldepslvec2 32568) and not for (left) modules in general.

Syntaxclininds 32523 Extend class notation with the relation between a module and its linearly independent subsets.
linIndS

Syntaxclindeps 32524 Extend class notation with the relation between a module and its linearly dependent subsets.
linDepS

Definitiondf-lininds 32525* Define the relation between a module and its linearly independent subsets. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.)
linIndS Scalar finSupp Scalar linC Scalar

Theoremrellininds 32526 The class defining the relation between a module and its linearly independent subsets is a relation. (Contributed by AV, 13-Apr-2019.)
linIndS

Definitiondf-lindeps 32527* Define the relation between a module and its linearly dependent subsets. (Contributed by AV, 26-Apr-2019.)
linDepS linIndS

Theoremlinindsv 32528 The classes of the module and its linearly independent subsets are sets. (Contributed by AV, 13-Apr-2019.)
linIndS

Theoremislininds 32529* The property of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                     linIndS finSupp linC

Theoremlinindsi 32530* The implications of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                     linIndS finSupp linC

Theoremlinindslinci 32531* The implications of being a linearly independent subset and a linear combination of this subset being 0. (Contributed by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                     linIndS finSupp linC

Theoremislinindfis 32532* The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.)
Scalar                     linIndS linC

Theoremislinindfiss 32533* The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.)
Scalar                     linIndS linC

Theoremlinindscl 32534 A linearly independent set is a subset of (the base set of) a module. (Contributed by AV, 13-Apr-2019.)
linIndS

Theoremlindepsnlininds 32535 A linearly dependent subset is not a linearly independent subset. (Contributed by AV, 26-Apr-2019.)
linDepS linIndS

Theoremislindeps 32536* The property of being a linearly dependent subset. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                     linDepS finSupp linC

Theoremlincext1 32537* Property 1 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 29-Apr-2019.)
Scalar

Theoremlincext2 32538* Property 2 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                                          finSupp finSupp

Theoremlincext3 32539* Property 3 of an extension of a linear combination. (Contributed by AV, 23-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                                          finSupp linC linC

Theoremlindslinindsimp1 32540* Implication 1 for lindslininds 32547. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                            finSupp linC

Theoremlindslinindimp2lem1 32541* Lemma 1 for lindslinindsimp2 32546. (Contributed by AV, 25-Apr-2019.)
Scalar

Theoremlindslinindimp2lem2 32542* Lemma 2 for lindslinindsimp2 32546. (Contributed by AV, 25-Apr-2019.)
Scalar

Theoremlindslinindimp2lem3 32543* Lemma 3 for lindslinindsimp2 32546. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                                          finSupp finSupp

Theoremlindslinindimp2lem4 32544* Lemma 4 for lindslinindsimp2 32546. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                                          finSupp linC g

Theoremlindslinindsimp2lem5 32545* Lemma 5 for lindslinindsimp2 32546. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                            finSupp linC finSupp linC

Theoremlindslinindsimp2 32546* Implication 2 for lindslininds 32547. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Scalar                            finSupp linC

Theoremlindslininds 32547 Equivalence of definitions df-linds 18711 and df-lininds 32525 for (linear) independency for (left) modules. (Contributed by AV, 26-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
linIndS LIndS

Theoremlinds0 32548 The empty set is always a linearly independet subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
linIndS

Theoremel0ldep 32549 A set containing the zero element of a module is always linearly dependent, if the underlying ring has at least two elements. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
Scalar linDepS

Theoremel0ldepsnzr 32550 A set containing the zero element of a module over a nonzero ring is always linearly dependent. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
Scalar NzRing linDepS

Theoremlindsrng01 32551 Any subset of a module is always linearly independent if the underlying ring has at most one element. Since the underlying ring cannot be the empty set (see lmodsn0 17396), this means that the underlying ring has only one element, so it is a zero ring. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
Scalar              linIndS

Theoremlindszr 32552 Any subset of a module over a zero ring is always linearly independent. (Contributed by AV, 27-Apr-2019.)
Scalar NzRing linIndS

Theoremsnlindsntorlem 32553* Lemma for snlindsntor 32554. (Contributed by AV, 15-Apr-2019.)
Scalar                                   linC

Theoremsnlindsntor 32554* A singleton is linearly independent iff it does not contain a torsion element. According to Wikipedia ("Torsion (algebra)", 15-Apr-2019, https://en.wikipedia.org/wiki/Torsion_(algebra)): "An element m of a module M over a ring R is called a torsion element of the module if there exists a regular element r of the ring (an element that is neither a left nor a right zero divisor) that annihilates m, i.e., . In an integral domain (a commutative ring without zero divisors), every non-zero element is regular, so a torsion element of a module over an integral domain is one annihilated by a non-zero element of the integral domain." Analogously, the definition in [Lang] p. 147 states that "An element x of [a module] E [over a ring R] is called a torsion element if there exists , , such that . This definition includes the zero element of the module. Some authors, however, exclude the zero element from the definition of torsion elements. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
Scalar                                   linIndS

Theoremldepsprlem 32555 Lemma for ldepspr 32556. (Contributed by AV, 16-Apr-2019.)
Scalar

Theoremldepspr 32556 If a vector is a scalar multiple of another vector, the (unordered pair containing the) two vectors are linearly dependent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
Scalar                                   linDepS

Theoremlincresunit3lem3 32557 Lemma 3 for lincresunit3 32564. (Contributed by AV, 18-May-2019.)
Scalar              Unit

Theoremlincresunitlem1 32558 Lemma 1 for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
Scalar              Unit

Theoremlincresunitlem2 32559 Lemma for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
Scalar              Unit

Theoremlincresunit1 32560* Property 1 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
Scalar              Unit

Theoremlincresunit2 32561* Property 2 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
Scalar              Unit                                                 finSupp finSupp

Theoremlincresunit3lem1 32562* Lemma 1 for lincresunit3 32564. (Contributed by AV, 17-May-2019.)
Scalar              Unit

Theoremlincresunit3lem2 32563* Lemma 2 for lincresunit3 32564. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
Scalar              Unit                                                 finSupp g linC

Theoremlincresunit3 32564* Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
Scalar              Unit                                                 finSupp linC linC

Theoremlincreslvec3 32565* Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
Scalar              Unit                                                 finSupp linC linC

Theoremislindeps2 32566* Conditions for being a linearly dependent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
Scalar                     NzRing finSupp linC linDepS

Theoremislininds2 32567* Implication of being a linearly independent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
Scalar                     NzRing linIndS finSupp linC

Theoremisldepslvec2 32568* Alternative definition of being a linearly dependent subset of a (left) vector space. In this case, the reverse implication of islindeps2 32566 holds, so that both definitions are equivalent (see theorem 1.6 in [Roman] p. 46 and the note in [Roman] p. 112: if a nontrivial linear combination of elements (where not all of the coefficients are 0) in an R-vector space is 0, then and only then each of the elements is a linear combination of the others. (Contributed by AV, 30-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
Scalar                     finSupp linC linDepS

Theoremlindssnlvec 32569 A singleton not containing the zero element of a vector space is always linearly independent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 28-Apr-2019.)
linIndS

21.25.9.4  Simple left modules and the ` ZZ `-module

Theoremlmod1lem1 32570* Lemma 1 for lmod1 32575. (Contributed by AV, 28-Apr-2019.)
Scalar

Theoremlmod1lem2 32571* Lemma 2 for lmod1 32575. (Contributed by AV, 28-Apr-2019.)
Scalar

Theoremlmod1lem3 32572* Lemma 3 for lmod1 32575. (Contributed by AV, 29-Apr-2019.)
Scalar        Scalar

Theoremlmod1lem4 32573* Lemma 4 for lmod1 32575. (Contributed by AV, 29-Apr-2019.)
Scalar        Scalar

Theoremlmod1lem5 32574* Lemma 5 for lmod1 32575. (Contributed by AV, 28-Apr-2019.)
Scalar        Scalar

Theoremlmod1 32575* The (smallest) structure representing a zero module over an arbitrary ring. (Contributed by AV, 29-Apr-2019.)
Scalar

Theoremlmod1zr 32576 The (smallest) structure representing a zero module over a zero ring. (Contributed by AV, 29-Apr-2019.)
Scalar

Theoremlmod1zrnlvec 32577 There is a (left) module (a zero module) which is not a (left) vector space. (Contributed by AV, 29-Apr-2019.)
Scalar

Theoremlmodn0 32578 Left modules exist. (Contributed by AV, 29-Apr-2019.)

Theoremzlmodzxzequa 32579 Example of an equation within the -module (see example for a linearly dependent set in [Roman] p. 113). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
ring freeLMod

Theoremzlmodzxznm 32580 Example of a linearly dependent set whose elements are not linear combinations of the others, see note in [Roman] p. 113). (Contributed by AV, 23-May-2019.) (Revised by AV, 10-Jun-2019.)
ring freeLMod

Theoremzlmodzxzldeplem 32581 A and B are not equal. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
ring freeLMod

Theoremzlmodzxzequap 32582 Example of an equation within the -module (see example for a linearly dependent set in [Roman] p. 113), written as a sum. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
ring freeLMod

Theoremzlmodzxzldeplem1 32583 Lemma 1 for zlmodzxzldep 32587. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
ring freeLMod

Theoremzlmodzxzldeplem2 32584 Lemma 2 for zlmodzxzldep 32587. (Contributed by AV, 24-May-2019.) (Revised by AV, 30-Jul-2019.)
ring freeLMod                             finSupp

Theoremzlmodzxzldeplem3 32585 Lemma 3 for zlmodzxzldep 32587. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
ring freeLMod                             linC

Theoremzlmodzxzldeplem4 32586* Lemma 4 for zlmodzxzldep 32587. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
ring freeLMod

Theoremzlmodzxzldep 32587 { A , B } is a linearly dependent set within the -module (see [Roman] p. 113). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
ring freeLMod                      linDepS

Theoremldepsnlinclem1 32588 Lemma 1 for ldepsnlinc 32591. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.)
ring freeLMod                      ring linC

Theoremldepsnlinclem2 32589 Lemma 2 for ldepsnlinc 32591. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.)
ring freeLMod                      ring linC

21.25.9.5  Differences between (left) modules and (left) vector spaces

Theoremlvecpsslmod 32590 The class of all (left) vector spaces is a proper subclass of the class of all (left) modules. Although it is obvious (and proven by lveclmod 17623) that every left vector space is a left module, there is (at least) one left module which is no left vector space, for example the zero module over the zero ring, see lmod1zrnlvec 32577. (Contributed by AV, 29-Apr-2019.)

Theoremldepsnlinc 32591* The reverse implication of islindeps2 32566 does not hold for arbitrary (left) modules, see note in [Roman] p. 112: "... if a nontrivial linear combination of the elements ... in an R-module M is 0, ... where not all of the coefficients are 0, then we cannot conclude ... that one of the elements ... is a linear combination of the others." This means that there is at least one left module having a linearly dependent subset in which there is at least one element which is not a linear combinantion of the other elements of this subset. Such a left module can be constructed by using zlmodzxzequa 32579 and zlmodzxznm 32580 (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.)
linDepS Scalar finSupp Scalar linC

Theoremldepslinc 32592* For (left) vector spaces, isldepslvec2 32568 provides an alternative definition of being a linearly dependent subset, whereas ldepsnlinc 32591 indicates that there is not an analogous alternative definition for arbitrary (left) modules. (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.)
linDepS Scalar finSupp Scalar linC linDepS Scalar finSupp Scalar linC

21.26  Mathbox for David A. Wheeler

This is the mathbox of David A. Wheeler, dwheeler at dwheeler dot com. Among other things, I have added a number of formal definitions for widely-used functions, e.g., those defined in ISO 80000-2:2009(E) Quantities and units - Part 2: Mathematical signs and symbols used in the natural sciences and technology and the NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/.

21.26.1  Natural deduction

Theorem19.8ad 32593 If a wff is true, it is true for at least one instance. Deductive form of 19.8a 1806. (Contributed by DAW, 13-Feb-2017.)

Theoremsbidd 32594 An identity theorem for substitution. See sbid 1965. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)

Theoremsbidd-misc 32595 An identity theorem for substitution. See sbid 1965. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)

21.26.2  Greater than, greater than or equal to.

As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Here we formally define the widely-used relations 'greater than' and 'greater than or equal to', so that we have formal definitions of them, as well as a few related theorems.

Syntaxcge-real 32596 Extend wff notation to include the 'greater than or equal to' relation, see df-gte 32598.

Syntaxcgt 32597 Extend wff notation to include the 'greater than' relation, see df-gt 32599.

Definitiondf-gte 32598 Define the 'greater than or equal' predicate over the reals. Defined in ISO 80000-2:2009(E) operation 2-7.10. It is used as a primitive in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. This relation is merely the converse of the 'less than or equal to' relation defined by df-le 9646.

We do not write this as , and similarly we do not write ` > ` as , because these are not definitional axioms as understood by mmj2 (those definitions will be flagged as being "potentially non-conservative"). We could write them this way: and but these are very complicated. This definition of , and the similar one for (df-gt 32599), are a bit strange when you see them for the first time, but these definitions are much simpler for us to process and are clearly conservative definitions. (My thanks to Mario Carneiro for pointing out this simpler approach.) See gte-lte 32600 for a more conventional expression of the relationship between and . As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)

Definitiondf-gt 32599 The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 9517. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 32598 for a discussion on why this approach is used for the definition. See gt-lt 32601 and gt-lth 32603 for more conventional expression of the relationship between and .

As a stylistic issue, set.mm prefers 'less than or equal' instead of 'greater than or equal' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)

Theoremgte-lte 32600 Simple relationship between and . (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)

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