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Theorem List for Metamath Proof Explorer - 25301-25400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcprd3 25301 Extend class notation to include finite supports products/sums.
prod3

Definitiondf-prod2 25302* Definition of a sum or product operator to be used with generic structures defined by extensible structures. is the set of indices, the operation, an expression, is normally a free variable in . may be any extensible structure with a base set. Its base set may be infinite provided that the "support" is finite. The support is the set: . The base set of may be empty. must be an extensible structure with a law commutative, associative with a neutral element. (Contributed by FL, 17-Oct-2011.)
prod2

Definitiondf-prod3 25303* Definition of a sum or product operator to be used with generic structures defined by extensible structures. is the set of indices, the operation, an expression, is normally a free variable in . must be a total order. Its base set may be infinite provided that the "support" is finite. The support is the set: . The base set of may be empty. must be an associative law with a neutral element. (Contributed by FL, 17-Oct-2011.)
prod3

Theoremprodex 25304 A finite composite is a set. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremprod0 25305 The value of . (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)
GId

Theoremprodeq1 25306 Equality theorem for a composite. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremprodeq2 25307 Equality theorem for a composite. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremprodeq3ii 25308* Equality theorem for a composite. (Contributed by Mario Carneiro, 26-May-2014.)

Theoremprodeq3 25309* Equality theorem for a composite. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2014.)

Theoremnfprod1 25310* Bound-variable hypothesis builder for . (Contributed by FL, 14-Sep-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremnfprod 25311* Bound-variable hypothesis builder for . If is (effectively) not free in , and , it is not free in . (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremcbvprodi 25312 Change bound variable in a finite composite. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremprodeq1d 25313 Conditions for two composites to be equal. (Contributed by FL, 5-Sep-2010.)

Theoremprodeq2d 25314 Conditions for two composites to be equal. (Contributed by FL, 5-Sep-2010.)

Theoremprodeq3d 25315* Conditions for two composites to be equal. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremprodeq123d 25316* Conditions for two composites to be equal. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremprodeq123i 25317* Conditions for two composites to be equal. (Contributed by FL, 6-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremprodeqfv 25318* Convert a composite of function values to a composite of classes . (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremdffprod 25319 Special case of composite over a finite index set. (Contributed by FL, 5-Sep-2010.) (Proof shortened by Mario Carneiro, 26-May-2014.)

Theoremfprodser 25320* A finite composite expressed in terms of a partial composite of an infinite series. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfprodserf 25321 Version of fprodser 25320 with a bound-variable hypothesis instead of a distinct variable condition. (Contributed by FL, 5-Sep-2010.)

Theoremfprod1i 25322* The finite composite of from to (i.e. a composite with only one term) is i.e. . (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfprodp1 25323* The composite of the next term in a finite composite of is the previous term composed with . (Contributed by Mario Carneiro, 26-May-2014.)

Theoremfprodp1i 25324* The composite of the next term in a finite composite of is the previous term composed with . (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfprod1s 25325 The finite composite of a sequence from to (i.e. a composite with only one term) is . (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfprod1fi 25326* The finite composite of a term from to (i.e. a composite with only one term) is , where is effectively not free in . (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfprodp1s 25327 The composite of the next term in a finite sum of is the previous term composed with . (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfprodp1fi 25328* The composite of the next term in a finite composite of is the previous term composed with . (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfsumprd 25329* Relation between and . (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremfprod2 25330* The finite composite of from to (i.e. a composite with two terms) is i.e. . (Contributed by FL, 22-Nov-2010.) (Revised by Mario Carneiro, 26-May-2014.)

18.13.15  Operation properties

Syntaxccm1 25331 Extend class notation with a class that adds commutativity to semi-groups, monoids and so on.

Definitiondf-com1 25332* A device to add commutativity to magmas, semi-groups, monoids and so on. A commutative group is composed of 5 properties (internal operation, commutativity, associativity, existence of a neutral element and an inverse). If we switch on or off those four properties we get 32 structures. Instead of giving a name to those 32 structures, I suggest we use intersected classes and speak of or . (Contributed by FL, 5-Sep-2010.)

Theoremiscom 25333* The predicate "is a commutative operation". (Contributed by FL, 5-Sep-2010.)

Theoremiscomb 25334 The predicate "is a commutative operation". (Contributed by FL, 14-Sep-2010.)

18.13.16  Groups and related structures

Theoremridlideq 25335* If a magma has a left identity element and a right identity element, they are equal. (Contributed by FL, 25-Sep-2011.)

Theoremrzrlzreq 25336* If a magma has a left zero element and a right zero element, they are equal. (Contributed by FL, 25-Dec-2011.)

Theoremmgmlion 25337* If a magma has a left identity element, it is onto. (Contributed by FL, 25-Sep-2011.)

Theoremrrisgrp 25338 is a group for addition. (Contributed by FL, 22-Dec-2008.)

Theoremdmrngrp 25339 A way to express the domain of a group. (Contributed by FL, 9-Jan-2011.)

Theorembsmgrli 25340 The base set of an operation with a right and left identity element is not empty. (Contributed by FL, 18-Feb-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)

Theoremsmgrpass2 25341 A semi-group is associative. (Contributed by FL, 12-Dec-2009.)

Theoremablocomgrp 25342 An abelian group is a commutative group. (Contributed by FL, 14-Sep-2010.)

Theoremreacomsmgrp1 25343 Rearrangement of three terms in a commutative semi-group. (Contributed by FL, 14-Sep-2010.)

Theoremreacomsmgrp2 25344 Rearrangement of three terms in a commutative semi-group. (Contributed by FL, 14-Sep-2010.)

Theoremreacomsmgrp3 25345 Rearrangement of three terms in a commutative semi-group. (Contributed by FL, 14-Sep-2010.)

Theoremreacomsmgrp4 25346 Rearrangement of terms in a commutative semi-group. (Contributed by FL, 18-Sep-2010.)

Theoremclfsebs 25347* Closure of a finite composite of elements of the base set of an internal operation. (Contributed by FL, 14-Sep-2010.) (Proof shortened by Mario Carneiro, 26-May-2014.)

Theoremclfsebsg 25348* Closure of a finite composite of elements of the base set of an internal operation. (Closed version.) (Contributed by FL, 14-Sep-2010.)

Theoremclfsebsr 25349* Closure of a finite composite of elements of the base set of an internal operation. (Case of a magma with a right and left identity element.) (Contributed by FL, 14-Sep-2010.)

Theoremfincmpzer 25350* Finite composite of identity elements. (Contributed by FL, 14-Sep-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2014.)
GId

Theoremresgcom 25351 Rearrangement of four terms in a commutative, associative magma. (Contributed by FL, 14-Sep-2010.)

Theoremfprodadd 25352* The composite of two finite composites. (Contributed by FL, 14-Sep-2010.) (Proof shortened by Mario Carneiro, 30-May-2014.)

Theoremabloinvop 25353 The inverse of the abelian group operation doesn't reverse the arguments. cf grpoinvop 20908. (Contributed by FL, 14-Sep-2010.)

Theoremisppm 25354 The sequence of partial composites of elements of a magma is a sequence of elements of this magma. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 30-May-2014.)

Theoremseqzp2 25355 Value of the arbitrary-based recursive sequence builder at a successor value when the operation is associative. Compare with seqp1 11061. (Contributed by FL, 24-Jan-2010.)

Theoremmndoisass 25356 A monoid is associative. (Contributed by FL, 2-Nov-2009.)
MndOp

Theoremmndoid 25357* A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 23-Dec-2013.)
MndOp

Theoremmndoio 25358 A monoid is an internal operation. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 23-Dec-2013.)
MndOp

Theoremmndoass 25359* A monoid is associative. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 23-Dec-2013.)
MndOp

Theoremmndoass2 25360 A monoid is associative. (Contributed by FL, 12-Dec-2009.)
MndOp

Syntaxclsg 25361 Extend class notation with monoid exponentiation.

Definitiondf-expsg 25362* Define the exponentiation of an element of a monoid. Experimental. I define exponentiation on a monoid (and not on a semi-group or a magma ) because I need an identity element for the basis hypothesis and associativity for interesting properties such as the composite of two exponentiated elements. is used in df-gx 20862 here I used because the inverse is not defined in a monoid. (Contributed by FL, 2-Sep-2013.)
MndOp GId

Theoremexpmiz 25363 Value of a member of a monoid (or any other structure where GId is defined ) raised to the 0th power. (Contributed by FL, 12-Dec-2009.)
GId        GId

Theoremexpm 25364* Exponentiation of a monoid. The value at a successor. What I am calculating is . (Contributed by FL, 12-Dec-2009.)
GId

Theoremexpus 25365* The exponentiation of a member of a monoid belongs to the underlying set. (Contributed by FL, 12-Dec-2009.)
GId        MndOp

Theoremmgmrddd 25366 The range of the domain of a magma equals the domain of the domain. (Contributed by FL, 17-May-2010.)

Theoremunsgrp 25367 The underlying set of a group is a set. (Contributed by FL, 17-May-2010.)

Theoremsymgfo 25368 The operation of a symetry group is onto. (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)

Theoremgapm2 25369 The action of a particular group element is a permutation of the base set. gapm 14760 expressed with the currying operator. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 3-May-2015.)

Theoremrngapm 25370 The range of the action of a particular group element equals the range of the action. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 3-May-2015.)

Theoremfnopabco2b 25371* Composition of a function with a function abstraction. Adapted from fnopabco 26388. (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Theoremcurgrpact 25372 The currying of a group action is a group homomorphism between the group and the symmetry group . (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)

Theoremgrpodivone 25373 "Division" by the neutral element of a group. (Contributed by FL, 21-Jun-2010.)
GId

Theoremgrpodivfo 25374 A "division" maps onto the group's underlying set. (Contributed by FL, 21-Jun-2010.)

Theoremgrpodrcan 25375 Right cancellation law for group "subtraction" (or "division"). (Contributed by FL, 14-Sep-2010.)

Theoremgrpodlcan 25376 Left cancellation law for group "subtraction" (or "division"). (Contributed by FL, 14-Sep-2010.)

Theoremgrpodivzer 25377 Condition for a "subtraction" (or "division") value to be equal to the identity element. (Contributed by FL, 14-Sep-2010.)
GId

Theoremfprodneg 25378* The inverse of a finite composite in the case of an abelian group. (Contributed by FL, 14-Sep-2010.) (Proof shortened by Mario Carneiro, 30-May-2014.)

Theoremfprodsub 25379* The "difference" (or "quotient") of two finite composites. (Contributed by FL, 14-Sep-2010.) (Revised by Mario Carneiro, 30-May-2014.)

Theoremclfsebs3 25380* Closure of a finite composite of elements of the base set of an internal operation. (When the operation is a monoid.) (Contributed by FL, 22-Nov-2010.)
MndOp

Theoremclfsebs4 25381* Closure of a finite composite of elements of the base set of an internal operation. (When the operation is a group.) (Contributed by FL, 22-Nov-2010.)

Theoremclfsebs5 25382* Closure of a finite composite of elements of the base set of an internal operation. (When the operation is an abelian group.) (Contributed by FL, 22-Nov-2010.)

18.13.17  Free structures

Syntaxcsubsmg 25383 Extend class notation to include the class of subsemigroups.

Definitiondf-subsmg 25384* Define the set of subsemigroups of . Experimental. (Contributed by FL, 2-Sep-2013.)

Syntaxcsbsgrg 25385 Extend class notation with a function that returns the subsemigroup of a group generated by a set.

Definitiondf-sggen 25386* the subsemigroup of generated by . Experimental. (Contributed by FL, 15-Jul-2012.)

Syntaxcsmhom 25387 Extend class notation to include the class of semigroup homomorphisms.

Definitiondf-gsmhom 25388* Define the set of semigroup homomorphisms from to . Experimental. (Contributed by FL, 15-Jul-2012.)

Syntaxcfsm 25389 Extend class notation to include the class of free semigroup.

Definitiondf-frsmgrp 25390* Definition of a free semigroup. The definition is somewhat cryptic. Let's say it guarantees the elements of the semigroup can be decomposed into elementary components and that the decomposition is unique. As a consequence you define the elements of the semigroup with nice recursive function by giving the value for every elementary component and the recursive equation . This is not true in every semigroup. For intance if you take the semigroup of strings generated by the elementary components "ab", "c", "a", "bc", the string "abc" is equal to "ab" "c" or to "a" "bc" and those beautiful recursive function can't exist. (See a nice explanation in Gallier p. 20.) Experimental. (Contributed by FL, 15-Jul-2012.)

18.13.18  Translations

Theoremtrdom2 25391* The domain of a right translation. The term is a constant: is not present. (Contributed by FL, 21-Jun-2010.)

Theoremtrset 25392* A right translation is a set. (Contributed by FL, 19-Sep-2010.)

Theoremtrran2 25393* The range of a right translation. The term is a constant: is not present. (Contributed by FL, 21-Jun-2010.)

Theoremtrooo 25394* A right translation is a bijection. The term is a constant. (Contributed by FL, 21-Jun-2010.)

Theoremtrinv 25395* The converse of a right translation. The term is a constant. (Contributed by FL, 21-Jun-2010.)

Theoremcmprtr 25396* Composite of two right translations. The terms and are constant. Don't use. See cmprtr2 25397. (Contributed by FL, 17-Oct-2010.)

Theoremcmprtr2 25397* Composite of two right translations. (cmprtr 25396 with a distinct variable condition relaxed.) (Contributed by FL, 1-Jan-2011.)

Theoremimtr 25398* The image of a set through a translation. (Contributed by FL, 30-Dec-2010.)

Theoremprsubrtr 25399* The product of a subset of by an element of is the image of by a right translation. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)

Theoremcaytr 25400* "It follows that if the entire group is multiplied by any one of the symbols, either as further or nearer factor, the effect is simply to reproduce the group... ." Cayley, On the theory of groups, as depending on the symbolic equation th^n = 1, 1854. (it is the original paper where the axiomatic definition of a group was given for the first time.) (Contributed by FL, 15-Oct-2012.)

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