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Theorem List for Metamath Proof Explorer - 32801-32900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremusgedgnlp 32801* An edge of a graph is not a loop. (Contributed by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  C  e.  E ) 
 ->  -.  E. v  C  =  { v }
 )
 
Theoremusgvad2edg 32802* If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex, analogous to usgra2edg 24587. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   =>    |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E )
 )  ->  E. x  e.  E  E. y  e.  E  ( x  =/=  y  /\  N  e.  x  /\  N  e.  y
 ) )
 
Theoremusg2edgneu 32803* If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex, analogous to usgra2edg1 24588. Lemma for theorems about friendship graphs. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 9-Jan-2020.)
 |-  E  =  ( Edges  `  G )   =>    |-  ( ( ( G  e. USGrph  /\  B  =/=  C )  /\  ( { N ,  B }  e.  E  /\  { C ,  N }  e.  E )
 )  ->  -.  E! x  e.  E  N  e.  x )
 
Theoremusgedgvadf1lem1 32804* Lemma 1 for usgedgvadf1 32806. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( Vtx  `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   =>    |-  ( ( G  e. USGrph  /\  C  e.  A ) 
 ->  ( iota_ m  e.  V  C  =  { m ,  N } )  e.  V )
 
Theoremusgedgvadf1lem2 32805* Lemma 2 for usgedgvadf1 32806. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( Vtx  `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   =>    |-  ( ( G  e. USGrph  /\  C  e.  A ) 
 ->  ( M  =  (
 iota_ m  e.  V  C  =  { m ,  N } )  ->  C  =  { M ,  N } ) )
 
Theoremusgedgvadf1 32806* The mapping of edges containing a given vertex into the set of vertices is 1-1, analogous to usgraidx2v 24598. The edge is mapped to the other vertex of the edge containing the vertex N. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( Vtx  `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   &    |-  F  =  ( e  e.  A  |->  (
 iota_ m  e.  V  e  =  { m ,  N } ) )   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  F : A -1-1-> V )
 
Theoremusgedgvadf1ALTlem1 32807* Lemma 1 for usgedgvadf1 32806. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   =>    |-  ( ( G  e. USGrph  /\  C  e.  A ) 
 ->  ( iota_ m  e.  V  C  =  { m ,  N } )  e.  V )
 
Theoremusgedgvadf1ALTlem2 32808* Lemma 2 for usgedgvadf1 32806. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   =>    |-  ( ( G  e. USGrph  /\  C  e.  A ) 
 ->  ( M  =  (
 iota_ m  e.  V  C  =  { m ,  N } )  ->  C  =  { M ,  N } ) )
 
Theoremusgedgvadf1ALT 32809* The mapping of edges containing a given vertex into the set of vertices is 1-1, analogous to usgraidx2v 24598. The edge is mapped to the other vertex of the edge containing the vertex N. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  A  =  { e  e.  E  |  N  e.  e }   &    |-  F  =  ( e  e.  A  |->  (
 iota_ m  e.  V  e  =  { m ,  N } ) )   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  F : A -1-1-> V )
 
Theoremusgedgleord 32810* In a graph, the number of edges which contain a given vertex is not greater than the number of vertices, analogous to usgraedgleord 24599. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( Vtx  `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  ( # `  { e  e.  E  |  N  e.  e } )  <_  ( GrOrder  `  G ) )
 
TheoremusgedgleordALT 32811* Alternate version of usgedgleord 32810 with a shorter proof. In a graph, the number of edges which contain a given vertex is not greater than the order of the graph, i. e. the number of its vertices, analogous to usgraedgleord 24599. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 12-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  O  =  ( # `  V )   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  ( # `  { e  e.  E  |  N  e.  e } )  <_  O )
 
21.24.6.8  Finite undirected simple graphs without loops
 
Syntaxcfusg 32812 Extend class notation with finite graphs.
 class FinUSGrph
 
Definitiondf-fusg 32813* Define the class of all finite undirected simple graphs without loops. Such a finite graph is an undirected simple graph without loops  <. V ,  E >. of finite order, i.e. where  V is finite. (Contributed by AV, 3-Jan-2020.)
 |- FinUSGrph  =  { <. v ,  e >.  |  ( e : dom  e -1-1-> { x  e.  ~P v  |  ( # `  x )  =  2 }  /\  v  e.  Fin ) }
 
Theoremrelfusgra 32814 The class of all finite undirected simple graph without loops is a relation. (Contributed by AV, 3-Jan-2020.)
 |-  Rel FinUSGrph
 
Theoremisfusgra 32815* The property of being a finite undirected simple graph without loops. (Contributed by AV, 3-Jan-2020.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( V FinUSGrph  E  <->  ( E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x )  =  2 }  /\  V  e.  Fin )
 ) )
 
Theoremisfusgra0 32816 The property of being a finite undirected simple graph without loops. (Contributed by AV, 3-Jan-2020.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( V FinUSGrph  E  <->  ( V USGrph  E  /\  V  e.  Fin )
 ) )
 
Theoremisfusgracl 32817 The property of being a finite undirected simple graph without loops. (Contributed by AV, 3-Jan-2020.)
 |-  ( G  e.  ( W  X.  X )  ->  ( G  e. FinUSGrph  <->  ( G  e. USGrph  /\  ( GrOrder  `  G )  e. 
 NN0 ) ) )
 
Theoremfusgraimpcl 32818 The implications of a finite undirected simple graph without loops. (Contributed by AV, 4-Jan-2020.)
 |-  ( G  e. FinUSGrph  ->  ( G  e. USGrph  /\  ( GrOrder  `  G )  e.  NN0 ) )
 
TheoremisfusgraclALT 32819 The property of being a finite undirected simple graph without loops. (Contributed by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   =>    |-  ( G  e.  ( W  X.  X )  ->  ( G  e. FinUSGrph  <->  ( G  e. USGrph  /\  V  e.  Fin )
 ) )
 
TheoremfusgraimpclALT 32820 The implications of a finite undirected simple graph without loops. (Contributed by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   =>    |-  ( G  e. FinUSGrph  ->  ( G  e. USGrph  /\  V  e.  Fin ) )
 
Theoremfusgusg 32821 A finite undirected simple graph without loops is a undirected simple graph without loops. (Contributed by AV, 16-Jan-2020.)
 |-  ( G  e. FinUSGrph  ->  G  e. USGrph  )
 
TheoremfusgraimpclALT2 32822 The implications of a finite undirected simple graph without loops. (Contributed by AV, 12-Jan-2020.) (Proof modification is discouraged.)
 |-  ( G  e. FinUSGrph  ->  ( G  e. USGrph  /\  ( 1st `  G )  e.  Fin ) )
 
Theoremfiusgedgfi 32823* In a finite graph the number of edges which contain a given vertex is also finite, analogous to fiusgraedgfi 24612. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( Vtx  `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. FinUSGrph  /\  N  e.  V )  ->  { e  e.  E  |  N  e.  e }  e.  Fin )
 
TheoremfiusgedgfiALT 32824* In a finite graph the number of edges which contain a given vertex is also finite, analogous to fiusgraedgfi 24612. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   =>    |-  ( ( G  e. FinUSGrph  /\  N  e.  V )  ->  { e  e.  E  |  N  e.  e }  e.  Fin )
 
21.24.6.9  Finite undirected simple graphs (extension)
 
Theoremusgedgffibi 32825 The number of edges in a graph is finite iff its edge function is finite. (Contributed by AV, 10-Jan-2020.)
 |-  ( V USGrph  E  ->  ( E  e.  Fin  <->  ( V Edges  E )  e.  Fin ) )
 
Theoremusgo0s0 32826 The size of a graph of order 0 (i.e. with 0 vertices) is 0, analogous to usgrafisindb0 24613. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 4-Jan-2020.)
 |-  (
 ( G  e. USGrph  /\  ( GrOrder  `  G )  =  0 )  ->  ( GrSize  `  G )  =  0 )
 
Theoremusgo0s0ALT 32827 The size of a graph of order 0 (i.e. with 0 vertices) is 0, analogous to usgrafisindb0 24613. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  O  =  ( # `  ( 1st `  G ) )   &    |-  S  =  ( # `  ( 2nd `  G ) )   =>    |-  ( ( G  e. USGrph  /\  O  =  0 ) 
 ->  S  =  0 )
 
Theoremusgo1s0ALT 32828 The size of a graph of order 1 (i.e. with 1 vertex) is 0, analogous to usgrafisindb1 24614. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  O  =  ( # `  ( 1st `  G ) )   &    |-  S  =  ( # `  ( 2nd `  G ) )   =>    |-  ( ( G  e. USGrph  /\  O  =  1 ) 
 ->  S  =  0 )
 
Theoremusgo0fis 32829 A graph of order 0 (i.e. with 0 vertices) has a finite set of edges, analogous to usgrafisbase 24619. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 10-Jan-2020.)
 |-  (
 ( G  e. USGrph  /\  ( GrOrder  `  G )  =  0 )  ->  ( Edges  `  G )  e.  Fin )
 
Theoremusgo0fisALT 32830 A graph of order 0 (i.e. with 0 vertices) has a finite set of edges, analogous to usgrafisbase 24619. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  O  =  ( # `  ( 1st `  G ) )   =>    |-  ( ( G  e. USGrph  /\  O  =  0 ) 
 ->  ( Edges  `  G )  e.  Fin )
 
TheoremusgrafisbaseALT 32831 Alternate version of usgrafisbase 24619, not depending on usgrafisindb0 24613. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 10-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 ( V USGrph  E  /\  ( # `  V )  =  0 )  ->  E  e.  Fin )
 
TheoremusgrafisbaseALT2 32832 Alternate version of usgrafisbase 24619, not depending on usgrafisindb0 24613. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 10-Jan-2020.) (Proof modification is discouraged.)
 |-  (
 ( V USGrph  E  /\  ( # `  V )  =  0 )  ->  E  e.  Fin )
 
Theoremusgo1s0 32833 The size of a graph of order 1 (i.e. with 1 vertex) is 0, analogous to usgrafisindb1 24614. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 4-Jan-2020.)
 |-  (
 ( G  e. USGrph  /\  ( GrOrder  `  G )  =  1 )  ->  ( GrSize  `  G )  =  0 )
 
Theoremusgresvm1 32834* Restricting an undirected simple graph by removing one vertex (and all edges ending at this vertex), analogous to usgrares1 24615. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 10-Jan-2020.)
 |-  V  =  ( Vtx  `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  ( V  \  { N } ) USGrph  (  _I  |`  F ) )
 
Theoremusgfislem1 32835* Lemma 1 for usgfis 32837: The set of edges is the union of the edges containing a specific vertex and the edges not containing this vertex. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 10-Jan-2020.)
 |-  V  =  ( Vtx  `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  E  =  ( F  u.  { e  e.  E  |  N  e.  e } )
 
Theoremusgfislem2 32836* Lemma 2 for usgfis 32837: In a graph of finite order (i.e. with a finite number of vertices), the number of edges is finite if and only if the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
 |-  V  =  ( Vtx  `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  ( ( G  e. FinUSGrph  /\  N  e.  V )  ->  ( E  e.  Fin  <->  F  e.  Fin ) )
 
Theoremusgfis 32837 An undirected simple graph of finite order (i.e. with a finite number of vertices) is of finite size, i.e. it has also only a finite number of edges, analogous to usgrafis 24620. Remark: The proof of this theorem is very long compared with usgrafis 24620, because the theorem brfi1ind 12520 to perform the finite induction is taylored for binary relations, so that the theorem itself and the used lemmas must be transformed accordingly. Maybe a variant of brfi1ind 12520 could be provided, which is better suitable for this theorem. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Revised by AV, 11-Jan-2018.) (Proof modification is discouraged.)
 |-  ( G  e. FinUSGrph  ->  ( GrSize  `  G )  e.  NN0 )
 
Theoremusgresvm1ALT 32838* Restricting an undirected simple graph by removing one vertex (and all edges ending at this vertex), analogous to usgrares1 24615. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 15-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  ( ( G  e. USGrph  /\  N  e.  V ) 
 ->  ( V  \  { N } ) USGrph  (  _I  |`  F ) )
 
TheoremusgfisALTlem1 32839* Lemma 1 for usgfisALT 32841: The set of edges is the union of the edges containing a specific vertex and the edges not containing this vertex. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 15-Jan-2020.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  E  =  ( F  u.  { e  e.  E  |  N  e.  e } )
 
TheoremusgfisALTlem2 32840* Lemma 2 for usgfis 32837: In a graph of finite order (i.e. with a finite number of vertices), the number of edges is finite if and only if the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 15-Jan-2020.)
 |-  V  =  ( 1st `  G )   &    |-  E  =  ( Edges  `  G )   &    |-  F  =  { e  e.  E  |  N  e/  e }   =>    |-  ( ( G  e. FinUSGrph  /\  N  e.  V )  ->  ( E  e.  Fin  <->  F  e.  Fin ) )
 
TheoremusgfisALT 32841 An undirected simple graph of finite order (i.e. with a finite number of vertices) is of finite size, i.e. it has also only a finite number of edges, analogous to usgrafis 24620. Remark: The proof of this theorem is very long compared with usgrafis 24620, because the theorem brfi1ind 12520 to perform the finite induction is taylored for binary relations, so that the theorem itself and the used lemmas must be transformed accordingly. Maybe a variant of brfi1ind 12520 could be provided, which is better suitable for this theorem. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Revised by AV, 15-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( G  e. FinUSGrph  ->  ( Edges  `  G )  e.  Fin )
 
TheoremusgrafiedgALT 32842 A simple undirected graph with a finite number of vertices has also only a finite number of edges. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 4-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( G  e. FinUSGrph  ->  ( Edges  `  G )  e.  Fin )
 
21.24.7  Monoids (extension)
 
21.24.7.1  Auxiliary theorems
 
Theoremovn0dmfun 32843 If a class' operation value for two operands is not the empty set, the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 5880. (Contributed by AV, 27-Jan-2020.)
 |-  (
 ( A F B )  =/=  (/)  ->  ( <. A ,  B >.  e.  dom  F 
 /\  Fun  ( F  |` 
 { <. A ,  B >. } ) ) )
 
Theoremxpsnopab 32844* A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.)
 |-  ( { X }  X.  C )  =  { <. a ,  b >.  |  (
 a  =  X  /\  b  e.  C ) }
 
Theoremxpiun 32845* A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.)
 |-  ( B  X.  C )  = 
 U_ x  e.  B  { <. a ,  b >.  |  ( a  =  x  /\  b  e.  C ) }
 
Theoremovn0ssdmfun 32846* If a class' operation value for two operands is not the empty set, the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 5880. (Contributed by AV, 27-Jan-2020.)
 |-  ( A. a  e.  D  A. b  e.  E  ( a F b )  =/=  (/)  ->  ( ( D  X.  E )  C_  dom 
 F  /\  Fun  ( F  |`  ( D  X.  E ) ) ) )
 
Theoremfnxpdmdm 32847 The domain of the domain of a function over a Cartesian square. (Contributed by AV, 13-Jan-2020.)
 |-  ( F  Fn  ( A  X.  A )  ->  dom  dom  F  =  A )
 
Theoremcnfldsrngbas 32848 The base set of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
 |-  R  =  (flds  S )   =>    |-  ( S  C_  CC  ->  S  =  ( Base `  R ) )
 
Theoremcnfldsrngadd 32849 The group addition operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
 |-  R  =  (flds  S )   =>    |-  ( S  e.  V  ->  +  =  ( +g  `  R ) )
 
Theoremcnfldsrngmul 32850 The ring multiplication operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
 |-  R  =  (flds  S )   =>    |-  ( S  e.  V  ->  x.  =  ( .r
 `  R ) )
 
21.24.7.2  Magmas and Semigroups (extension)
 
Theoremplusfreseq 32851 If the empty set is not contained in the range of the group addition function of an extensible structure (not necessarily a magma), the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  .+^  =  ( +f `  M )   =>    |-  ( (/)  e/  ran  .+^  ->  (  .+  |`  ( B  X.  B ) )  =  .+^  )
 
Theoremmgmplusfreseq 32852 If the empty set is not contained in the base set of a magma, the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  .+^  =  ( +f `  M )   =>    |-  ( ( M  e. Mgm  /\  (/)  e/  B )  ->  (  .+  |`  ( B  X.  B ) )  =  .+^  )
 
Theorem0mgm 32853 A set with an empty base set is always a magma". (Contributed by AV, 25-Feb-2020.)
 |-  ( Base `  M )  =  (/)   =>    |-  ( M  e.  V  ->  M  e. Mgm )
 
Theoremmgmpropd 32854* If two structures have the same (nonempty) base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a magma iff the other one is. (Contributed by AV, 25-Feb-2020.)
 |-  ( ph  ->  B  =  (
 Base `  K ) )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  B  =/= 
 (/) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )   =>    |-  ( ph  ->  ( K  e. Mgm  <->  L  e. Mgm ) )
 
Theoremismgmd 32855* Deduce a magma from its properties. (Contributed by AV, 25-Feb-2020.)
 |-  ( ph  ->  B  =  (
 Base `  G ) )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  .+  =  ( +g  `  G ) )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
 )  e.  B )   =>    |-  ( ph  ->  G  e. Mgm )
 
21.24.7.3  Magma homomorphisms and submagmas
 
Syntaxcmgmhm 32856 Hom-set generator class for magmas.
 class MgmHom
 
Syntaxcsubmgm 32857 Class function taking a magma to its lattice of submagmas.
 class SubMgm
 
Definitiondf-mgmhm 32858* A magma homomorphism is a function on the base sets which preserves the binary operation. (Contributed by AV, 24-Feb-2020.)
 |- MgmHom  =  ( s  e. Mgm ,  t  e. Mgm 
 |->  { f  e.  (
 ( Base `  t )  ^m  ( Base `  s )
 )  |  A. x  e.  ( Base `  s ) A. y  e.  ( Base `  s ) ( f `  ( x ( +g  `  s
 ) y ) )  =  ( ( f `
  x ) (
 +g  `  t )
 ( f `  y
 ) ) } )
 
Definitiondf-submgm 32859* A submagma is a subset of a magma which is closed under the operation. Such subsets are themselves magmas. (Contributed by AV, 24-Feb-2020.)
 |- SubMgm  =  ( s  e. Mgm  |->  { t  e.  ~P ( Base `  s
 )  |  A. x  e.  t  A. y  e.  t  ( x (
 +g  `  s )
 y )  e.  t } )
 
Theoremmgmhmrcl 32860 Reverse closure of a magma homomorphism. (Contributed by AV, 24-Feb-2020.)
 |-  ( F  e.  ( S MgmHom  T )  ->  ( S  e. Mgm  /\  T  e. Mgm )
 )
 
Theoremsubmgmrcl 32861 Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.)
 |-  ( S  e.  (SubMgm `  M )  ->  M  e. Mgm )
 
Theoremismgmhm 32862* Property of a magma homomorphism. (Contributed by AV, 25-Feb-2020.)
 |-  B  =  ( Base `  S )   &    |-  C  =  ( Base `  T )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   =>    |-  ( F  e.  ( S MgmHom  T )  <->  ( ( S  e. Mgm  /\  T  e. Mgm ) 
 /\  ( F : B
 --> C  /\  A. x  e.  B  A. y  e.  B  ( F `  ( x  .+  y ) )  =  ( ( F `  x )  .+^  ( F `  y
 ) ) ) ) )
 
Theoremmgmhmf 32863 A magma homomorphism is a function. (Contributed by AV, 25-Feb-2020.)
 |-  B  =  ( Base `  S )   &    |-  C  =  ( Base `  T )   =>    |-  ( F  e.  ( S MgmHom  T )  ->  F : B
 --> C )
 
Theoremmgmhmpropd 32864* Magma homomorphism depends only on the operation of structures. (Contributed by AV, 25-Feb-2020.)
 |-  ( ph  ->  B  =  (
 Base `  J ) )   &    |-  ( ph  ->  C  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  C  =  ( Base `  M )
 )   &    |-  ( ph  ->  B  =/= 
 (/) )   &    |-  ( ph  ->  C  =/=  (/) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( x ( +g  `  J ) y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  C ) )  ->  ( x ( +g  `  K ) y )  =  ( x ( +g  `  M ) y ) )   =>    |-  ( ph  ->  ( J MgmHom  K )  =  ( L MgmHom  M ) )
 
Theoremmgmhmlin 32865 A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020.)
 |-  B  =  ( Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   =>    |-  (
 ( F  e.  ( S MgmHom  T )  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `
  ( X  .+  Y ) )  =  ( ( F `  X )  .+^  ( F `
  Y ) ) )
 
Theoremmgmhmf1o 32866 A magma homomorphism is bijective iff its converse is also a magma homomorphism. (Contributed by AV, 25-Feb-2020.)
 |-  B  =  ( Base `  R )   &    |-  C  =  ( Base `  S )   =>    |-  ( F  e.  ( R MgmHom  S )  ->  ( F : B -1-1-onto-> C  <->  `' F  e.  ( S MgmHom  R ) ) )
 
Theoremidmgmhm 32867 The identity homomorphism on a magma. (Contributed by AV, 27-Feb-2020.)
 |-  B  =  ( Base `  M )   =>    |-  ( M  e. Mgm  ->  (  _I  |`  B )  e.  ( M MgmHom  M ) )
 
Theoremissubmgm 32868* Expand definition of a submagma. (Contributed by AV, 25-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   =>    |-  ( M  e. Mgm  ->  ( S  e.  (SubMgm `  M ) 
 <->  ( S  C_  B  /\  A. x  e.  S  A. y  e.  S  ( x  .+  y )  e.  S ) ) )
 
Theoremissubmgm2 32869 Submagmas are subsets that are also magmas. (Contributed by AV, 25-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  H  =  ( Ms  S )   =>    |-  ( M  e. Mgm  ->  ( S  e.  (SubMgm `  M ) 
 <->  ( S  C_  B  /\  H  e. Mgm ) ) )
 
Theoremrabsubmgmd 32870* Deduction for proving that a restricted class abstraction is a submagma. (Contributed by AV, 26-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  ( ph  ->  M  e. Mgm )   &    |-  (
 ( ph  /\  ( ( x  e.  B  /\  y  e.  B )  /\  ( th  /\  ta ) ) )  ->  et )   &    |-  ( z  =  x  ->  ( ps  <->  th ) )   &    |-  ( z  =  y  ->  ( ps  <->  ta ) )   &    |-  ( z  =  ( x  .+  y
 )  ->  ( ps  <->  et ) )   =>    |-  ( ph  ->  { z  e.  B  |  ps }  e.  (SubMgm `  M )
 )
 
Theoremsubmgmss 32871 Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020.)
 |-  B  =  ( Base `  M )   =>    |-  ( S  e.  (SubMgm `  M )  ->  S  C_  B )
 
Theoremsubmgmid 32872 Every magma is trivially a submagma of itself. (Contributed by AV, 26-Feb-2020.)
 |-  B  =  ( Base `  M )   =>    |-  ( M  e. Mgm  ->  B  e.  (SubMgm `  M ) )
 
Theoremsubmgmcl 32873 Submagmas are closed under the monoid operation. (Contributed by AV, 26-Feb-2020.)
 |-  .+  =  ( +g  `  M )   =>    |-  (
 ( S  e.  (SubMgm `  M )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X 
 .+  Y )  e.  S )
 
Theoremsubmgmmgm 32874 Submagmas are themselves magmas under the given operation. (Contributed by AV, 26-Feb-2020.)
 |-  H  =  ( Ms  S )   =>    |-  ( S  e.  (SubMgm `  M )  ->  H  e. Mgm )
 
Theoremsubmgmbas 32875 The base set of a submagma. (Contributed by AV, 26-Feb-2020.)
 |-  H  =  ( Ms  S )   =>    |-  ( S  e.  (SubMgm `  M )  ->  S  =  ( Base `  H )
 )
 
Theoremsubsubmgm 32876 A submagma of a submagma is a submagma. (Contributed by AV, 26-Feb-2020.)
 |-  H  =  ( Gs  S )   =>    |-  ( S  e.  (SubMgm `  G )  ->  ( A  e.  (SubMgm `  H ) 
 <->  ( A  e.  (SubMgm `  G )  /\  A  C_  S ) ) )
 
Theoremresmgmhm 32877 Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.)
 |-  U  =  ( Ss  X )   =>    |-  ( ( F  e.  ( S MgmHom  T )  /\  X  e.  (SubMgm `  S ) )  ->  ( F  |`  X )  e.  ( U MgmHom  T ) )
 
Theoremresmgmhm2 32878 One direction of resmgmhm2b 32879. (Contributed by AV, 26-Feb-2020.)
 |-  U  =  ( Ts  X )   =>    |-  ( ( F  e.  ( S MgmHom  U )  /\  X  e.  (SubMgm `  T ) )  ->  F  e.  ( S MgmHom  T ) )
 
Theoremresmgmhm2b 32879 Restriction of the codomain of a homomorphism. (Contributed by AV, 26-Feb-2020.)
 |-  U  =  ( Ts  X )   =>    |-  ( ( X  e.  (SubMgm `  T )  /\  ran 
 F  C_  X )  ->  ( F  e.  ( S MgmHom  T )  <->  F  e.  ( S MgmHom  U ) ) )
 
Theoremmgmhmco 32880 The composition of magma homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.)
 |-  (
 ( F  e.  ( T MgmHom  U )  /\  G  e.  ( S MgmHom  T )
 )  ->  ( F  o.  G )  e.  ( S MgmHom  U ) )
 
Theoremmgmhmima 32881 The homomorphic image of a submagma is a submagma. (Contributed by AV, 27-Feb-2020.)
 |-  (
 ( F  e.  ( M MgmHom  N )  /\  X  e.  (SubMgm `  M )
 )  ->  ( F " X )  e.  (SubMgm `  N ) )
 
Theoremmgmhmeql 32882 The equalizer of two magma homomorphisms is a submagma. (Contributed by AV, 27-Feb-2020.)
 |-  (
 ( F  e.  ( S MgmHom  T )  /\  G  e.  ( S MgmHom  T )
 )  ->  dom  ( F  i^i  G )  e.  (SubMgm `  S )
 )
 
Theoremsubmgmacs 32883 Submagmas are an algebraic closure system. (Contributed by AV, 27-Feb-2020.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e. Mgm  ->  (SubMgm `  G )  e.  (ACS `  B ) )
 
Theoremismhm0 32884 Property of a monoid homomorphism, expressed by a magma homomorphism. (Contributed by AV, 17-Apr-2020.)
 |-  B  =  ( Base `  S )   &    |-  C  =  ( Base `  T )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   &    |-  .0.  =  ( 0g `  S )   &    |-  Y  =  ( 0g
 `  T )   =>    |-  ( F  e.  ( S MndHom  T )  <->  ( ( S  e.  Mnd  /\  T  e.  Mnd )  /\  ( F  e.  ( S MgmHom  T )  /\  ( F `  .0.  )  =  Y ) ) )
 
Theoremmhmismgmhm 32885 Each monoid homomorphism is a magma homomorphism. (Contributed by AV, 29-Feb-2020.)
 |-  ( F  e.  ( R MndHom  S )  ->  F  e.  ( R MgmHom  S ) )
 
21.24.7.4  Examples and counterexamples for magmas, semigroups and monoids (extension)
 
Theoremopmpt2ismgm 32886* A structure with a group addition operation in maps-to notation is a magma if the operation value is contained in the base set. (Contributed by AV, 16-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  ( +g  `  M )  =  ( x  e.  B ,  y  e.  B  |->  C )   &    |-  ( ph  ->  B  =/=  (/) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  C  e.  B )   =>    |-  ( ph  ->  M  e. Mgm )
 
Theoremcopissgrp 32887* A structure with a constant group addition operation is a semigroup if the constant is contained in the base set. (Contributed by AV, 16-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  ( +g  `  M )  =  ( x  e.  B ,  y  e.  B  |->  C )   &    |-  ( ph  ->  B  =/=  (/) )   &    |-  ( ph  ->  C  e.  B )   =>    |-  ( ph  ->  M  e. SGrp )
 
Theoremcopisnmnd 32888* A structure with a constant group addition operation and at least two elements is not a monoid. (Contributed by AV, 16-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  ( +g  `  M )  =  ( x  e.  B ,  y  e.  B  |->  C )   &    |-  ( ph  ->  C  e.  B )   &    |-  ( ph  ->  1  <  ( # `
  B ) )   =>    |-  ( ph  ->  M  e/  Mnd )
 
Theorem0nodd 32889* 0 is not an odd integer. (Contributed by AV, 3-Feb-2020.)
 |-  O  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( (
 2  x.  x )  +  1 ) }   =>    |-  0  e/  O
 
Theorem1odd 32890* 1 is an odd integer. (Contributed by AV, 3-Feb-2020.)
 |-  O  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( (
 2  x.  x )  +  1 ) }   =>    |-  1  e.  O
 
Theorem2nodd 32891* 2 is not an odd integer. (Contributed by AV, 3-Feb-2020.)
 |-  O  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( (
 2  x.  x )  +  1 ) }   =>    |-  2  e/  O
 
Theoremoddibas 32892* Lemma 1 for oddinmgm 32894: The base set of M is the set of all odd integers. (Contributed by AV, 3-Feb-2020.)
 |-  O  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( (
 2  x.  x )  +  1 ) }   &    |-  M  =  (flds  O )   =>    |-  O  =  ( Base `  M )
 
Theoremoddiadd 32893* Lemma 2 for oddinmgm 32894: The group addition operation of M is the addition of complex numbers. (Contributed by AV, 3-Feb-2020.)
 |-  O  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( (
 2  x.  x )  +  1 ) }   &    |-  M  =  (flds  O )   =>    |- 
 +  =  ( +g  `  M )
 
Theoremoddinmgm 32894* The structure of all odd integers together with the addition of complex numbers is not a magma. Remark: the structure of the complementary subset of the set of integers, the even integers, is a magma, actually an abelian group, see 2zrngaabl 33023, and even a non-unital ring, see 2zrng 33014. (Contributed by AV, 3-Feb-2020.)
 |-  O  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( (
 2  x.  x )  +  1 ) }   &    |-  M  =  (flds  O )   =>    |-  M  e/ Mgm
 
Theoremnnsgrpmgm 32895 The structure of positive integers together with the addition of complex numbers is a magma. (Contributed by AV, 4-Feb-2020.)
 |-  M  =  (flds  NN )   =>    |-  M  e. Mgm
 
Theoremnnsgrp 32896 The structure of positive integers together with the addition of complex numbers is a semigroup. (Contributed by AV, 4-Feb-2020.)
 |-  M  =  (flds  NN )   =>    |-  M  e. SGrp
 
Theoremnnsgrpnmnd 32897 The structure of positive integers together with the addition of complex numbers is not a monoid. (Contributed by AV, 4-Feb-2020.)
 |-  M  =  (flds  NN )   =>    |-  M  e/  Mnd
 
21.24.8  Magmas and internal binary operations (alternate approach)

With df-mpt2 6275, binary operations are defined by a rule, and with df-ov 6273, the value of a binary operation applied to two operands can be expressed. In both cases, the two operands can belong to different sets, and the result can be an element of a third set. However, according to Wikipedia "Binary operation", see https://en.wikipedia.org/wiki/Binary_operation (19-Jan-2020), "... a binary operation on a set  S is a mapping of the elements of the Cartesian product  S  X.  S to S:  f : ( S  X.  S
--> S ). Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S (or sometimes expressed as having the property of closure).". To distinguish this more restrictive definition (in Wikipedia and most of the literature) from the general case, we call binary operations mapping the elements of the Cartesian product  S  X.  S internal binary operations, see df-intop 32914. If, in addition, the result is also contained in the set  S, the operation is called closed internal binary operation, see df-clintop 32915. Therefore, a "binary operation on a set  S" according to Wikipedia is a "closed internal binary operation" in our terminology. If the sets are different, the operation is explicitly called external binary operation (see Wikipedia https://en.wikipedia.org/wiki/Binary_operation#External_binary_operations ).

Taking a step back, we define "laws" applicable for "binary operations" (which even need not to be functions), according to the definition in [Hall] p. 1 and [BourbakiAlg1] p. 1, p. 4 and p. 7. These laws are used, on the one hand, to specialize internal binary operations (see df-clintop 32915 and df-assintop 32916), and on the other hand to define the common algebraic structures like magmas, groups, rings, etc. Internal binary operations, which obeys these laws, are defined afterwards. Notice that in [BourbakiAlg1] p. 1, p. 4 and p. 7, these operations are called "laws" by themselves.

In the following, an alternative definition df-cllaw 32901 for an internal binary operation is provided, which does not require to be a function, but only closure. Therefore, this definition could be used as binary operation (slot 2) defined for a magma as extensible structure, see mgmplusgiopALT 32909, or for an alternative definition df-mgm2 32934 for a magma as extensible structure. Similar results are obtained for an associative operation resp. semigroups.

 
21.24.8.1  Laws for internal binary operations

In this subsection, the "laws" applicable for "binary operations" according to the definition in [Hall] p. 1 and [BourbakiAlg1] p. 1, p. 4 and p. 7 are defined. These laws are called "internal laws" in [BourbakiAlg1] p. xxi.

 
Syntaxccllaw 32898 Extend class notation for the closure law.
 class clLaw
 
Syntaxcasslaw 32899 Extend class notation for the associative law.
 class assLaw
 
Syntaxccomlaw 32900 Extend class notation for the commutative law.
 class comLaw
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38521
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