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Theorem List for Metamath Proof Explorer - 38401-38500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremirinitoringc 38401 The ring of integers is an initial object in the category of unital rings (within a universe containing the ring of integers). Example 7.2 (6) of [Adamek] p. 101 , and example in [Lang] p. 58. (Contributed by AV, 3-Apr-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->ring  e.  U )   &    |-  C  =  (RingCat `  U )   =>    |-  ( ph  ->ring  e.  (InitO `  C )
 )
 
Theoremzrtermoringc 38402 The zero ring is a terminal object in the category of unital rings. (Contributed by AV, 17-Apr-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RingCat `  U )   &    |-  ( ph  ->  Z  e.  ( Ring  \ NzRing ) )   &    |-  ( ph  ->  Z  e.  U )   =>    |-  ( ph  ->  Z  e.  (TermO `  C )
 )
 
Theoremzrninitoringc 38403* The zero ring is not an initial object in the category of unital rings (if the universe contains at least one unital ring different from the zero ring). (Contributed by AV, 18-Apr-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RingCat `  U )   &    |-  ( ph  ->  Z  e.  ( Ring  \ NzRing ) )   &    |-  ( ph  ->  Z  e.  U )   &    |-  ( ph  ->  E. r  e.  ( Base `  C ) r  e. NzRing )   =>    |-  ( ph  ->  Z  e/  (InitO `  C )
 )
 
Theoremnzerooringczr 38404 There is no zero object in the category of unital rings (at least in a universe which contains the zero ring and the ring of integers). Example 7.9 (3) in [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RingCat `  U )   &    |-  ( ph  ->  Z  e.  ( Ring  \ NzRing ) )   &    |-  ( ph  ->  Z  e.  U )   &    |-  ( ph  ->ring  e.  U )   =>    |-  ( ph  ->  (ZeroO `  C )  =  (/) )
 
21.33.12.10  Subcategories of the category of rings
 
Theoremsrhmsubclem1 38405* Lemma 1 for srhmsubc 38408. (Contributed by AV, 19-Feb-2020.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   =>    |-  ( X  e.  C  ->  X  e.  ( U  i^i  Ring ) )
 
Theoremsrhmsubclem2 38406* Lemma 2 for srhmsubc 38408. (Contributed by AV, 19-Feb-2020.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   =>    |-  ( ( U  e.  V  /\  X  e.  C )  ->  X  e.  ( Base `  (RingCat `  U ) ) )
 
Theoremsrhmsubclem3 38407* Lemma 3 for srhmsubc 38408. (Contributed by AV, 19-Feb-2020.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( ( U  e.  V  /\  ( X  e.  C  /\  Y  e.  C ) )  ->  ( X J Y )  =  ( X ( Hom  `  (RingCat `  U )
 ) Y ) )
 
Theoremsrhmsubc 38408* According to df-subc 15427, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15455 and subcss2 15458). Therefore, the set of special ring homomorphisms (i.e. ring homomorphisms from a special ring to another ring of that kind) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  J  e.  (Subcat `  (RingCat `  U ) ) )
 
Theoremsringcat 38409* The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  ( (RingCat `  U )  |`cat  J )  e.  Cat )
 
Theoremcrhmsubc 38410* According to df-subc 15427, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15455 and subcss2 15458). Therefore, the set of commutative ring homomorphisms (i.e. ring homomorphisms from a commutative ring to a commutative ring) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.)
 |-  C  =  ( U  i^i  CRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  J  e.  (Subcat `  (RingCat `  U ) ) )
 
Theoremcringcat 38411* The restriction of the category of (unital) rings to the set of commutative ring homomorphisms is a category, the "category of commutative rings". (Contributed by AV, 19-Feb-2020.)
 |-  C  =  ( U  i^i  CRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  ( (RingCat `  U )  |`cat  J )  e.  Cat )
 
Theoremdrhmsubc 38412* According to df-subc 15427, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15455 and subcss2 15458). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  J  e.  (Subcat `  (RingCat `  U ) ) )
 
Theoremdrngcat 38413* The restriction of the category of (unital) rings to the set of division ring homomorphisms is a category, the "category of division rings". (Contributed by AV, 20-Feb-2020.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  ( (RingCat `  U )  |`cat  J )  e.  Cat )
 
Theoremfldcat 38414* The restriction of the category of (unital) rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   &    |-  D  =  ( U  i^i Field )   &    |-  F  =  ( r  e.  D ,  s  e.  D  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  ( (RingCat `  U )  |`cat  F )  e.  Cat )
 
Theoremfldc 38415* The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   &    |-  D  =  ( U  i^i Field )   &    |-  F  =  ( r  e.  D ,  s  e.  D  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  ( ( (RingCat `  U )  |`cat  J )  |`cat  F )  e.  Cat )
 
Theoremfldhmsubc 38416* According to df-subc 15427, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15455 and subcss2 15458). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   &    |-  D  =  ( U  i^i Field )   &    |-  F  =  ( r  e.  D ,  s  e.  D  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  F  e.  (Subcat `  ( (RingCat `  U )  |`cat  J ) ) )
 
Theoremrngcrescrhm 38417 The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCat `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  ( C  |`cat  H )  =  ( ( Cs  R ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
 
Theoremrhmsubclem1 38418 Lemma 1 for rhmsubc 38422. (Contributed by AV, 2-Mar-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCat `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  H  Fn  ( R  X.  R ) )
 
Theoremrhmsubclem2 38419 Lemma 2 for rhmsubc 38422. (Contributed by AV, 2-Mar-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCat `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ( ph  /\  X  e.  R  /\  Y  e.  R )  ->  ( X H Y )  =  ( X RingHom  Y )
 )
 
Theoremrhmsubclem3 38420* Lemma 3 for rhmsubc 38422. (Contributed by AV, 2-Mar-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCat `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ( ph  /\  x  e.  R )  ->  (
 ( Id `  (RngCat `  U ) ) `  x )  e.  ( x H x ) )
 
Theoremrhmsubclem4 38421* Lemma 4 for rhmsubc 38422. (Contributed by AV, 2-Mar-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCat `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ( ( (
 ph  /\  x  e.  R )  /\  ( y  e.  R  /\  z  e.  R ) )  /\  ( f  e.  ( x H y )  /\  g  e.  ( y H z ) ) )  ->  ( g
 ( <. x ,  y >. (comp `  (RngCat `  U ) ) z ) f )  e.  ( x H z ) )
 
Theoremrhmsubc 38422 According to df-subc 15427, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15455 and subcss2 15458). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCat `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  H  e.  (Subcat `  (RngCat `  U ) ) )
 
Theoremrhmsubccat 38423 The restriction of the category of non-unital rings to the set of unital ring homomorphisms is a category. (Contributed by AV, 4-Mar-2020.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCat `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  (
 (RngCat `  U )  |`cat  H )  e.  Cat )
 
TheoremsrhmsubcALTVlem1 38424* Lemma 1 for srhmsubcALTV 38427. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   =>    |-  ( X  e.  C  ->  X  e.  ( U  i^i  Ring ) )
 
TheoremsrhmsubcALTVlem2 38425* Lemma 2 for srhmsubcALTV 38427. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   =>    |-  ( ( U  e.  V  /\  X  e.  C )  ->  X  e.  ( Base `  (RingCatALTV `  U ) ) )
 
TheoremsrhmsubcALTVlem3 38426* Lemma 3 for srhmsubcALTV 38427. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( ( U  e.  V  /\  ( X  e.  C  /\  Y  e.  C ) )  ->  ( X J Y )  =  ( X ( Hom  `  (RingCatALTV `  U ) ) Y ) )
 
TheoremsrhmsubcALTV 38427* According to df-subc 15427, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15455 and subcss2 15458). Therefore, the set of special ring homomorphisms (i.e. ring homomorphisms from a special ring to another ring of that kind) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  J  e.  (Subcat `  (RingCatALTV `  U ) ) )
 
TheoremsringcatALTV 38428* The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
 |-  A. r  e.  S  r  e.  Ring   &    |-  C  =  ( U  i^i  S )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  ( (RingCatALTV `  U )  |`cat  J )  e.  Cat )
 
TheoremcrhmsubcALTV 38429* According to df-subc 15427, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15455 and subcss2 15458). Therefore, the set of commutative ring homomorphisms (i.e. ring homomorphisms from a commutative ring to a commutative ring) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
 |-  C  =  ( U  i^i  CRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  J  e.  (Subcat `  (RingCatALTV `  U ) ) )
 
TheoremcringcatALTV 38430* The restriction of the category of (unital) rings to the set of commutative ring homomorphisms is a category, the "category of commutative rings". (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
 |-  C  =  ( U  i^i  CRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s
 ) )   =>    |-  ( U  e.  V  ->  ( (RingCatALTV `  U )  |`cat  J )  e.  Cat )
 
TheoremdrhmsubcALTV 38431* According to df-subc 15427, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15455 and subcss2 15458). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  J  e.  (Subcat `  (RingCatALTV `  U ) ) )
 
TheoremdrngcatALTV 38432* The restriction of the category of (unital) rings to the set of division ring homomorphisms is a category, the "category of division rings". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  ( (RingCatALTV `  U )  |`cat  J )  e.  Cat )
 
TheoremfldcatALTV 38433* The restriction of the category of (unital) rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   &    |-  D  =  ( U  i^i Field )   &    |-  F  =  ( r  e.  D ,  s  e.  D  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  ( (RingCatALTV `  U )  |`cat  F )  e.  Cat )
 
TheoremfldcALTV 38434* The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   &    |-  D  =  ( U  i^i Field )   &    |-  F  =  ( r  e.  D ,  s  e.  D  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  ( ( (RingCatALTV `  U )  |`cat  J )  |`cat  F )  e.  Cat )
 
TheoremfldhmsubcALTV 38435* According to df-subc 15427, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15455 and subcss2 15458). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
 |-  C  =  ( U  i^i  DivRing )   &    |-  J  =  ( r  e.  C ,  s  e.  C  |->  ( r RingHom  s ) )   &    |-  D  =  ( U  i^i Field )   &    |-  F  =  ( r  e.  D ,  s  e.  D  |->  ( r RingHom  s ) )   =>    |-  ( U  e.  V  ->  F  e.  (Subcat `  ( (RingCatALTV `  U )  |`cat  J ) ) )
 
TheoremrngcrescrhmALTV 38436 The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.) (New usage is discouraged.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCatALTV `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  ( C  |`cat  H )  =  ( ( Cs  R ) sSet  <. ( Hom  `  ndx ) ,  H >. ) )
 
TheoremrhmsubcALTVlem1 38437 Lemma 1 for rhmsubcALTV 38441. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCatALTV `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  H  Fn  ( R  X.  R ) )
 
TheoremrhmsubcALTVlem2 38438 Lemma 2 for rhmsubcALTV 38441. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCatALTV `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ( ph  /\  X  e.  R  /\  Y  e.  R )  ->  ( X H Y )  =  ( X RingHom  Y )
 )
 
TheoremrhmsubcALTVlem3 38439* Lemma 3 for rhmsubcALTV 38441. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCatALTV `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ( ph  /\  x  e.  R )  ->  (
 ( Id `  (RngCatALTV `  U ) ) `  x )  e.  ( x H x ) )
 
TheoremrhmsubcALTVlem4 38440* Lemma 4 for rhmsubcALTV 38441. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCatALTV `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ( ( (
 ph  /\  x  e.  R )  /\  ( y  e.  R  /\  z  e.  R ) )  /\  ( f  e.  ( x H y )  /\  g  e.  ( y H z ) ) )  ->  ( g
 ( <. x ,  y >. (comp `  (RngCatALTV `  U ) ) z ) f )  e.  ( x H z ) )
 
TheoremrhmsubcALTV 38441 According to df-subc 15427, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15455 and subcss2 15458). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCatALTV `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  H  e.  (Subcat `  (RngCatALTV `  U ) ) )
 
TheoremrhmsubcALTVcat 38442 The restriction of the category of non-unital rings to the set of unital ring homomorphisms is a category. (Contributed by AV, 4-Mar-2020.) (New usage is discouraged.)
 |-  ( ph  ->  U  e.  V )   &    |-  C  =  (RngCatALTV `  U )   &    |-  ( ph  ->  R  =  ( Ring  i^i  U ) )   &    |-  H  =  ( RingHom  |`  ( R  X.  R ) )   =>    |-  ( ph  ->  (
 (RngCatALTV `  U )  |`cat  H )  e.  Cat )
 
21.33.13  Basic algebraic structures (extension)
 
21.33.13.1  Auxiliary theorems
 
Theoremrabeqsn 38443* Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019.)
 |-  ( { x  e.  V  |  ph }  =  { X }  <->  A. x ( ( x  e.  V  /\  ph )  <->  x  =  X ) )
 
Theoremrabsssn 38444* Conditions for a restricted class abstraction to be a subset of a singleton, i.e. to be a singleton or the empty set. (Contributed by AV, 18-Apr-2019.)
 |-  ( { x  e.  V  |  ph }  C_  { X } 
 <-> 
 A. x  e.  V  ( ph  ->  x  =  X ) )
 
Theoremxpprsng 38445 The Cartesian product of an unordered pair and a singleton. (Contributed by AV, 20-May-2019.)
 |-  (
 ( A  e.  V  /\  B  e.  W  /\  C  e.  U )  ->  ( { A ,  B }  X.  { C } )  =  { <. A ,  C >. , 
 <. B ,  C >. } )
 
Theoremopeliun2xp 38446 Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 4877. (Contributed by AV, 30-Mar-2019.)
 |-  ( <. C ,  y >.  e.  U_ y  e.  B  ( A  X.  { y } )  <->  ( y  e.  B  /\  C  e.  A ) )
 
Theoremeliunxp2 38447* Membership in a union of Cartesian products over its second component, analogous to eliunxp 4963. (Contributed by AV, 30-Mar-2019.)
 |-  ( C  e.  U_ y  e.  B  ( A  X.  { y } )  <->  E. x E. y
 ( C  =  <. x ,  y >.  /\  ( x  e.  A  /\  y  e.  B )
 ) )
 
Theoremmpt2mptx2 38448* Express a two-argument function as a one-argument function, or vice-versa. In this version 
A ( y ) is not assumed to be constant w.r.t  y, analogous to mpt2mptx 6376. (Contributed by AV, 30-Mar-2019.)
 |-  (
 z  =  <. x ,  y >.  ->  C  =  D )   =>    |-  ( z  e.  U_ y  e.  B  ( A  X.  { y }
 )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
 
Theoremcbvmpt2x2 38449* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6359 allows  A to be a function of  y, analogous to cbvmpt2x 6358. (Contributed by AV, 30-Mar-2019.)
 |-  F/_ z A   &    |-  F/_ y D   &    |-  F/_ z C   &    |-  F/_ w C   &    |-  F/_ x E   &    |-  F/_ y E   &    |-  (
 y  =  z  ->  A  =  D )   &    |-  (
 ( y  =  z 
 /\  x  =  w )  ->  C  =  E )   =>    |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( w  e.  D ,  z  e.  B  |->  E )
 
Theoremdmmpt2ssx2 38450* The domain of a mapping is a subset of its base classes expressed as union of Cartesian products over its second component, analogous to dmmpt2ssx 6851. (Contributed by AV, 30-Mar-2019.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |- 
 dom  F  C_  U_ y  e.  B  ( A  X.  { y } )
 
Theoremmpt2exxg2 38451* Existence of an operation class abstraction (version for dependent domains, i.e. the first base class may depend on the second base class), analogous to mpt2exxg 6860. (Contributed by AV, 30-Mar-2019.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( ( B  e.  R  /\  A. y  e.  B  A  e.  S )  ->  F  e.  _V )
 
Theoremovmpt2rdxf 38452* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6411. (Contributed by AV, 30-Mar-2019.)
 |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )   &    |-  ( ( ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )   &    |-  ( ( ph  /\  y  =  B ) 
 ->  C  =  L )   &    |-  ( ph  ->  A  e.  L )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ph  ->  S  e.  X )   &    |- 
 F/ x ph   &    |-  F/ y ph   &    |-  F/_ y A   &    |-  F/_ x B   &    |-  F/_ x S   &    |-  F/_ y S   =>    |-  ( ph  ->  ( A F B )  =  S )
 
Theoremovmpt2rdx 38453* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6411. (Contributed by AV, 30-Mar-2019.)
 |-  ( ph  ->  F  =  ( x  e.  C ,  y  e.  D  |->  R ) )   &    |-  ( ( ph  /\  ( x  =  A  /\  y  =  B ) )  ->  R  =  S )   &    |-  ( ( ph  /\  y  =  B ) 
 ->  C  =  L )   &    |-  ( ph  ->  A  e.  L )   &    |-  ( ph  ->  B  e.  D )   &    |-  ( ph  ->  S  e.  X )   =>    |-  ( ph  ->  ( A F B )  =  S )
 
Theoremovmpt2x2 38454* The value of an operation class abstraction. Variant of ovmpt2ga 6415 which does not require  D and  x to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  (
 ( x  =  A  /\  y  =  B )  ->  R  =  S )   &    |-  ( y  =  B  ->  C  =  L )   &    |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )   =>    |-  ( ( A  e.  L  /\  B  e.  D  /\  S  e.  H )  ->  ( A F B )  =  S )
 
Theoremfdmdifeqresdif 38455* The restriction of a conditional mapping to function values of a function having a domain which is a difference with a singleton equals this function. (Contributed by AV, 23-Apr-2019.)
 |-  F  =  ( x  e.  D  |->  if ( x  =  Y ,  X ,  ( G `
  x ) ) )   =>    |-  ( G : ( D  \  { Y } ) --> R  ->  G  =  ( F  |`  ( D 
 \  { Y }
 ) ) )
 
Theoremoffvalfv 38456* The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  G  Fn  A )   =>    |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  A  |->  ( ( F `  x ) R ( G `  x ) ) ) )
 
Theoremofaddmndmap 38457 The function operation applied to the addition for functions (with the same domain) into a monoid is a function (with the same domain) into the monoid. (Contributed by AV, 6-Apr-2019.)
 |-  R  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   =>    |-  (
 ( M  e.  Mnd  /\  V  e.  Y  /\  ( A  e.  ( R  ^m  V )  /\  B  e.  ( R  ^m  V ) ) ) 
 ->  ( A  oF  .+  B )  e.  ( R  ^m  V ) )
 
Theoremmapsnop 38458 A singleton of an ordered pair as an element of the mapping operation. (Contributed by AV, 12-Apr-2019.)
 |-  F  =  { <. X ,  Y >. }   =>    |-  ( ( X  e.  V  /\  Y  e.  R  /\  R  e.  W ) 
 ->  F  e.  ( R 
 ^m  { X } )
 )
 
Theoremmapprop 38459 An unordered pair containing two ordered pairs as an element of the mapping operation. (Contributed by AV, 16-Apr-2019.)
 |-  F  =  { <. X ,  A >. ,  <. Y ,  B >. }   =>    |-  ( ( ( X  e.  V  /\  A  e.  R )  /\  ( Y  e.  V  /\  B  e.  R )  /\  ( X  =/=  Y  /\  R  e.  W ) )  ->  F  e.  ( R  ^m  { X ,  Y } ) )
 
Theoremztprmneprm 38460 A prime is not an integer multiple of another prime. (Contributed by AV, 23-May-2019.)
 |-  (
 ( Z  e.  ZZ  /\  A  e.  Prime  /\  B  e.  Prime )  ->  (
 ( Z  x.  A )  =  B  ->  A  =  B ) )
 
Theorem2t6m3t4e0 38461 2 times 6 minus 3 times 4 equals 0. (Contributed by AV, 24-May-2019.)
 |-  (
 ( 2  x.  6
 )  -  ( 3  x.  4 ) )  =  0
 
Theoremssnn0ssfz 38462* For any finite subset of  NN0, find a superset in the form of a set of sequential integers, analogous to ssnnssfz 28058. (Contributed by AV, 30-Sep-2019.)
 |-  ( A  e.  ( ~P NN0 
 i^i  Fin )  ->  E. n  e.  NN0  A  C_  (
 0 ... n ) )
 
Theoremnn0sumltlt 38463 If the sum of two nonnegative integers is less than a third integer, then one of the summands is already less than this third integer. (Contributed by AV, 19-Oct-2019.)
 |-  (
 ( a  e.  NN0  /\  b  e.  NN0  /\  c  e.  NN0 )  ->  (
 ( a  +  b
 )  <  c  ->  b  <  c ) )
 
21.33.13.2  The binomial coefficient operation (extension)
 
Theorembcpascm1 38464 Pascal's rule for the binomial coefficient, generalized to all integers  K, shifted down by 1. (Contributed by AV, 8-Sep-2019.)
 |-  (
 ( N  e.  NN  /\  K  e.  ZZ )  ->  ( ( ( N  -  1 )  _C  K )  +  (
 ( N  -  1
 )  _C  ( K  -  1 ) ) )  =  ( N  _C  K ) )
 
Theoremaltgsumbc 38465* The sum of binomial coefficients for a fixed positive  N with alternating signs is zero. Notice that this is not valid for  N  =  0 (since  ( ( -u
1 ^ 0 )  x.  ( 0  _C  0 ) )  =  ( 1  x.  1 )  =  1). For a proof using Pascal's rule (bcpascm1 38464) instead of the binomial theorem (binom 13795) , see altgsumbcALT 38466. (Contributed by AV, 13-Sep-2019.)
 |-  ( N  e.  NN  ->  sum_
 k  e.  ( 0
 ... N ) ( ( -u 1 ^ k
 )  x.  ( N  _C  k ) )  =  0 )
 
TheoremaltgsumbcALT 38466* Alternate proof of altgsumbc 38465, using Pascal's rule (bcpascm1 38464) instead of the binomial theorem (binom 13795). (Contributed by AV, 8-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( N  e.  NN  ->  sum_
 k  e.  ( 0
 ... N ) ( ( -u 1 ^ k
 )  x.  ( N  _C  k ) )  =  0 )
 
21.33.13.3  The ` ZZ `-module ` ZZ X. ZZ `
 
Theoremzlmodzxzlmod 38467 The  ZZ-module  ZZ  X.  ZZ is a (left) module with the ring of integers as base set. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   =>    |-  ( Z  e.  LMod  /\ring  =  (Scalar `  Z ) )
 
Theoremzlmodzxzel 38468 An element of the (base set of the) 
ZZ-module  ZZ  X.  ZZ. (Contributed by AV, 21-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   =>    |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  { <. 0 ,  A >. ,  <. 1 ,  B >. }  e.  ( Base `  Z ) )
 
Theoremzlmodzxz0 38469 The  0 of the  ZZ-module  ZZ  X.  ZZ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  .0.  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   =>    |- 
 .0.  =  ( 0g `  Z )
 
Theoremzlmodzxzscm 38470 The scalar multiplication of the 
ZZ-module  ZZ  X.  ZZ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  .xb  =  ( .s `  Z )   =>    |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A 
 .xb  { <. 0 ,  B >. ,  <. 1 ,  C >. } )  =  { <. 0 ,  ( A  x.  B ) >. , 
 <. 1 ,  ( A  x.  C ) >. } )
 
Theoremzlmodzxzadd 38471 The addition of the  ZZ-module  ZZ  X.  ZZ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  .+  =  ( +g  `  Z )   =>    |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )
 )  ->  ( { <. 0 ,  A >. , 
 <. 1 ,  C >. } 
 .+  { <. 0 ,  B >. ,  <. 1 ,  D >. } )  =  { <. 0 ,  ( A  +  B ) >. , 
 <. 1 ,  ( C  +  D ) >. } )
 
Theoremzlmodzxzsubm 38472 The subtraction of the  ZZ-module  ZZ  X.  ZZ expressed as addition. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  .-  =  ( -g `  Z )   =>    |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )
 )  ->  ( { <. 0 ,  A >. , 
 <. 1 ,  C >. } 
 .-  { <. 0 ,  B >. ,  <. 1 ,  D >. } )  =  ( { <. 0 ,  A >. ,  <. 1 ,  C >. }  ( +g  `  Z ) ( -u 1
 ( .s `  Z ) { <. 0 ,  B >. ,  <. 1 ,  D >. } ) ) )
 
Theoremzlmodzxzsub 38473 The subtraction of the  ZZ-module  ZZ  X.  ZZ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  .-  =  ( -g `  Z )   =>    |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )
 )  ->  ( { <. 0 ,  A >. , 
 <. 1 ,  C >. } 
 .-  { <. 0 ,  B >. ,  <. 1 ,  D >. } )  =  { <. 0 ,  ( A  -  B ) >. , 
 <. 1 ,  ( C  -  D ) >. } )
 
21.33.13.4  Ordered group sum operation (extension)
 
Theoremgsumpr 38474* Group sum of a pair. (Contributed by AV, 6-Dec-2018.) (Proof shortened by AV, 28-Jul-2019.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  (
 k  =  M  ->  A  =  C )   &    |-  (
 k  =  N  ->  A  =  D )   =>    |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B ) )  ->  ( G 
 gsumg  ( k  e.  { M ,  N }  |->  A ) )  =  ( C 
 .+  D ) )
 
Theoremmgpsumunsn 38475* Extract a summand/factor from the group sum for the multiplicative group of a unital ring. (Contributed by AV, 29-Dec-2018.)
 |-  M  =  (mulGrp `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  CRing
 )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  I  e.  N )   &    |-  (
 ( ph  /\  k  e.  N )  ->  A  e.  ( Base `  R )
 )   &    |-  ( ph  ->  X  e.  ( Base `  R )
 )   &    |-  ( k  =  I 
 ->  A  =  X )   =>    |-  ( ph  ->  ( M  gsumg  (
 k  e.  N  |->  A ) )  =  ( ( M  gsumg  ( k  e.  ( N  \  { I }
 )  |->  A ) ) 
 .x.  X ) )
 
Theoremmgpsumz 38476* If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the zero of the ring, the group sum itself is zero. (Contributed by AV, 29-Dec-2018.)
 |-  M  =  (mulGrp `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  CRing
 )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  I  e.  N )   &    |-  (
 ( ph  /\  k  e.  N )  ->  A  e.  ( Base `  R )
 )   &    |- 
 .0.  =  ( 0g `  R )   &    |-  ( k  =  I  ->  A  =  .0.  )   =>    |-  ( ph  ->  ( M  gsumg  ( k  e.  N  |->  A ) )  =  .0.  )
 
Theoremmgpsumn 38477* If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the one of the ring, this summand/ factor can be removed from the group sum. (Contributed by AV, 29-Dec-2018.)
 |-  M  =  (mulGrp `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  CRing
 )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  I  e.  N )   &    |-  (
 ( ph  /\  k  e.  N )  ->  A  e.  ( Base `  R )
 )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  ( k  =  I  ->  A  =  .1.  )   =>    |-  ( ph  ->  ( M  gsumg  ( k  e.  N  |->  A ) )  =  ( M  gsumg  ( k  e.  ( N  \  { I }
 )  |->  A ) ) )
 
Theoremgsumsplit2f 38478* Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.)
 |-  F/ k ph   &    |-  B  =  (
 Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |- 
 .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  k  e.  A )  ->  X  e.  B )   &    |-  ( ph  ->  ( k  e.  A  |->  X ) finSupp  .0.  )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  ( ph  ->  A  =  ( C  u.  D ) )   =>    |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  X ) )  =  ( ( G  gsumg  ( k  e.  C  |->  X ) )  .+  ( G 
 gsumg  ( k  e.  D  |->  X ) ) ) )
 
Theoremgsumdifsndf 38479* Extract a summand from a finitely supported group sum. (Contributed by AV, 4-Sep-2019.)
 |-  F/_ k Y   &    |- 
 F/ k ph   &    |-  B  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  W )   &    |-  ( ph  ->  (
 k  e.  A  |->  X ) finSupp  ( 0g `  G ) )   &    |-  (
 ( ph  /\  k  e.  A )  ->  X  e.  B )   &    |-  ( ph  ->  M  e.  A )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ( ph  /\  k  =  M )  ->  X  =  Y )   =>    |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  X ) )  =  ( ( G  gsumg  ( k  e.  ( A  \  { M } )  |->  X ) )  .+  Y ) )
 
21.33.13.5  Symmetric groups (extension)
 
Theoremnn0le2is012 38480 A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.)
 |-  (
 ( N  e.  NN0  /\  N  <_  2 )  ->  ( N  =  0  \/  N  =  1  \/  N  =  2 ) )
 
Theoremexple2lt6 38481 A nonnegative integer to the power of itself is less than 6 if it is less than or equal to 2. (Contributed by AV, 16-Mar-2019.)
 |-  (
 ( N  e.  NN0  /\  N  <_  2 )  ->  ( N ^ N )  <  6 )
 
Theorempgrple2abl 38482 Every symmetric group on a set with at most 2 elements is abelian. (Contributed by AV, 16-Mar-2019.)
 |-  G  =  ( SymGrp `  A )   =>    |-  (
 ( A  e.  V  /\  ( # `  A )  <_  2 )  ->  G  e.  Abel )
 
Theorempgrpgt2nabl 38483 Every symmetric group on a set with more than 2 elements is not abelian, see also the remark in [Rotman] p. 28. (Contributed by AV, 21-Mar-2019.)
 |-  G  =  ( SymGrp `  A )   =>    |-  (
 ( A  e.  V  /\  2  <  ( # `  A ) )  ->  G  e/  Abel )
 
21.33.13.6  Divisibility (extension)
 
Theoreminvginvrid 38484 Identity for a multiplication with additive and multiplicative inverses in a ring. (Contributed by AV, 18-May-2018.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  N  =  ( invg `  R )   &    |-  I  =  (
 invr `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  U )  ->  ( ( N `  Y )  .x.  ( ( I `  ( N `
  Y ) ) 
 .x.  X ) )  =  X )
 
21.33.13.7  The support of functions (extension)
 
Theoremrmsupp0 38485* The support of a mapping of a multiplication of zero with a function into a ring is empty. (Contributed by AV, 10-Apr-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e.  Ring  /\  V  e.  X  /\  C  =  ( 0g `  M ) )  /\  A  e.  ( R  ^m  V ) )  ->  ( ( v  e.  V  |->  ( C ( .r `  M ) ( A `  v
 ) ) ) supp  ( 0g `  M ) )  =  (/) )
 
Theoremdomnmsuppn0 38486* The support of a mapping of a multiplication of a nonzero constant with a function into a (ring theoretic) domain equals the support of the function. (Contributed by AV, 11-Apr-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e. Domn  /\  V  e.  X ) 
 /\  ( C  e.  R  /\  C  =/=  ( 0g `  M ) ) 
 /\  A  e.  ( R  ^m  V ) ) 
 ->  ( ( v  e.  V  |->  ( C ( .r `  M ) ( A `  v
 ) ) ) supp  ( 0g `  M ) )  =  ( A supp  ( 0g `  M ) ) )
 
Theoremrmsuppss 38487* The support of a mapping of a multiplication of a constant with a function into a ring is a subset of the support of the function. (Contributed by AV, 11-Apr-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R 
 ^m  V ) ) 
 ->  ( ( v  e.  V  |->  ( C ( .r `  M ) ( A `  v
 ) ) ) supp  ( 0g `  M ) ) 
 C_  ( A supp  ( 0g `  M ) ) )
 
Theoremmndpsuppss 38488 The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e.  Mnd  /\  V  e.  X ) 
 /\  ( A  e.  ( R  ^m  V ) 
 /\  B  e.  ( R  ^m  V ) ) )  ->  ( ( A  oF ( +g  `  M ) B ) supp 
 ( 0g `  M ) )  C_  ( ( A supp  ( 0g `  M ) )  u.  ( B supp  ( 0g
 `  M ) ) ) )
 
Theoremscmsuppss 38489* The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
 |-  S  =  (Scalar `  M )   &    |-  R  =  ( Base `  S )   =>    |-  (
 ( M  e.  LMod  /\  V  e.  ~P ( Base `  M )  /\  A  e.  ( R  ^m  V ) )  ->  ( ( v  e.  V  |->  ( ( A `
  v ) ( .s `  M ) v ) ) supp  ( 0g `  M ) ) 
 C_  ( A supp  ( 0g `  S ) ) )
 
21.33.13.8  Finitely supported functions (extension)
 
Theoremrmsuppfi 38490* The support of a mapping of a multiplication of a constant with a function into a ring is finite if the support of the function is finite. (Contributed by AV, 11-Apr-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R 
 ^m  V )  /\  ( A supp  ( 0g `  M ) )  e. 
 Fin )  ->  (
 ( v  e.  V  |->  ( C ( .r `  M ) ( A `
  v ) ) ) supp  ( 0g `  M ) )  e. 
 Fin )
 
Theoremrmfsupp 38491* A mapping of a multiplication of a constant with a function into a ring is finitely supported if the function is finitely supported. (Contributed by AV, 9-Jun-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e.  Ring  /\  V  e.  X  /\  C  e.  R )  /\  A  e.  ( R 
 ^m  V )  /\  A finSupp  ( 0g `  M ) )  ->  ( v  e.  V  |->  ( C ( .r `  M ) ( A `  v ) ) ) finSupp  ( 0g `  M ) )
 
Theoremmndpsuppfi 38492 The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e.  Mnd  /\  V  e.  X ) 
 /\  ( A  e.  ( R  ^m  V ) 
 /\  B  e.  ( R  ^m  V ) ) 
 /\  ( ( A supp 
 ( 0g `  M ) )  e.  Fin  /\  ( B supp  ( 0g
 `  M ) )  e.  Fin ) ) 
 ->  ( ( A  oF ( +g  `  M ) B ) supp  ( 0g
 `  M ) )  e.  Fin )
 
Theoremmndpfsupp 38493 A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
 |-  R  =  ( Base `  M )   =>    |-  (
 ( ( M  e.  Mnd  /\  V  e.  X ) 
 /\  ( A  e.  ( R  ^m  V ) 
 /\  B  e.  ( R  ^m  V ) ) 
 /\  ( A finSupp  ( 0g `  M )  /\  B finSupp  ( 0g `  M ) ) )  ->  ( A  oF
 ( +g  `  M ) B ) finSupp  ( 0g
 `  M ) )
 
Theoremscmsuppfi 38494* The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)
 |-  S  =  (Scalar `  M )   &    |-  R  =  ( Base `  S )   =>    |-  (
 ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) ) 
 /\  A  e.  ( R  ^m  V )  /\  ( A supp  ( 0g `  S ) )  e. 
 Fin )  ->  (
 ( v  e.  V  |->  ( ( A `  v ) ( .s
 `  M ) v ) ) supp  ( 0g
 `  M ) )  e.  Fin )
 
Theoremscmfsupp 38495* A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
 |-  S  =  (Scalar `  M )   &    |-  R  =  ( Base `  S )   =>    |-  (
 ( ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) ) 
 /\  A  e.  ( R  ^m  V )  /\  A finSupp  ( 0g `  S ) )  ->  ( v  e.  V  |->  ( ( A `  v ) ( .s `  M ) v ) ) finSupp  ( 0g `  M ) )
 
Theoremsuppmptcfin 38496* The support of a mapping with value 0 except of one is finite. (Contributed by AV, 27-Apr-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  F  =  ( x  e.  V  |->  if ( x  =  X ,  .1.  ,  .0.  )
 )   =>    |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V ) 
 ->  ( F supp  .0.  )  e.  Fin )
 
Theoremmptcfsupp 38497* A mapping with value 0 except of one is finitely supported. (Contributed by AV, 9-Jun-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  F  =  ( x  e.  V  |->  if ( x  =  X ,  .1.  ,  .0.  )
 )   =>    |-  ( ( M  e.  LMod  /\  V  e.  ~P B  /\  X  e.  V ) 
 ->  F finSupp  .0.  )
 
Theoremfsuppmptdmf 38498* A mapping with a finite domain is finitely supported. (Contributed by AV, 4-Sep-2019.)
 |-  F/ x ph   &    |-  F  =  ( x  e.  A  |->  Y )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  x  e.  A )  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  W )   =>    |-  ( ph  ->  F finSupp  Z )
 
21.33.13.9  Left modules (extension)
 
Theoremlmodvsmdi 38499 Multiple distributive law for scalar product (left-distributivity). (Contributed by AV, 5-Sep-2019.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   &    |-  .^  =  (.g `  W )   &    |-  E  =  (.g `  F )   =>    |-  ( ( W  e.  LMod  /\  ( R  e.  K  /\  N  e.  NN0  /\  X  e.  V ) )  ->  ( R  .x.  ( N 
 .^  X ) )  =  ( ( N E R )  .x.  X ) )
 
Theoremgsumlsscl 38500* Closure of a group sum in a linear subspace: A (finitely supported) sum of scalar multiplications of vectors 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.)
 |-  S  =  ( LSubSp `  M )   &    |-  R  =  (Scalar `  M )   &    |-  B  =  ( Base `  R )   =>    |-  (
 ( M  e.  LMod  /\  Z  e.  S  /\  V  C_  Z )  ->  ( ( F  e.  ( B  ^m  V ) 
 /\  F finSupp  ( 0g `  R ) )  ->  ( M  gsumg  ( v  e.  V  |->  ( ( F `  v ) ( .s
 `  M ) v ) ) )  e.  Z ) )
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