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Theorem List for Metamath Proof Explorer - 40601-40700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrhmsubccat 40601 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 40602* Lemma 1 for srhmsubcALTV 40605. (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 40603* Lemma 2 for srhmsubcALTV 40605. (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 40604* Lemma 3 for srhmsubcALTV 40605. (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 40605* According to df-subc 15795, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15823 and subcss2 15826). 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 40606* 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 40607* According to df-subc 15795, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15823 and subcss2 15826). 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 40608* 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 40609* According to df-subc 15795, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15823 and subcss2 15826). 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 40610* 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 40611* 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 40612* 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 40613* According to df-subc 15795, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15823 and subcss2 15826). 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 40614 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 40615 Lemma 1 for rhmsubcALTV 40619. (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 40616 Lemma 2 for rhmsubcALTV 40619. (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 40617* Lemma 3 for rhmsubcALTV 40619. (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 40618* Lemma 4 for rhmsubcALTV 40619. (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 40619 According to df-subc 15795, the subcategories  (Subcat `  C ) of a category  C are subsets of the homomorphisms of  C ( see subcssc 15823 and subcss2 15826). 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 40620 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 )
 
21.33.14  Basic algebraic structures (extension)
 
21.33.14.1  Auxiliary theorems
 
Theoremxpprsng 40621 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 40622 Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 4891. (Contributed by AV, 30-Mar-2019.)
 |-  ( <. C ,  y >.  e.  U_ y  e.  B  ( A  X.  { y } )  <->  ( y  e.  B  /\  C  e.  A ) )
 
Theoremeliunxp2 40623* Membership in a union of Cartesian products over its second component, analogous to eliunxp 4977. (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 40624* 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 6406. (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 40625* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6389 allows  A to be a function of  y, analogous to cbvmpt2x 6388. (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 40626* 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 6877. (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 40627* 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 6886. (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 40628* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6441. (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 40629* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6441. (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 40630* The value of an operation class abstraction. Variant of ovmpt2ga 6445 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 40631* 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 40632* 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 40633 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 40634 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 40635 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 40636 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 40637 2 times 6 minus 3 times 4 equals 0. (Contributed by AV, 24-May-2019.)
 |-  (
 ( 2  x.  6
 )  -  ( 3  x.  4 ) )  =  0
 
Theoremssnn0ssfz 40638* For any finite subset of  NN0, find a superset in the form of a set of sequential integers, analogous to ssnnssfz 28442. (Contributed by AV, 30-Sep-2019.)
 |-  ( A  e.  ( ~P NN0 
 i^i  Fin )  ->  E. n  e.  NN0  A  C_  (
 0 ... n ) )
 
Theoremnn0sumltlt 40639 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 ) )
 
21.33.14.2  The binomial coefficient operation (extension)
 
Theorembcpascm1 40640 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 40641* 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 40640) instead of the binomial theorem (binom 13965) , see altgsumbcALT 40642. (Contributed by AV, 13-Sep-2019.)
 |-  ( N  e.  NN  ->  sum_
 k  e.  ( 0
 ... N ) ( ( -u 1 ^ k
 )  x.  ( N  _C  k ) )  =  0 )
 
TheoremaltgsumbcALT 40642* Alternate proof of altgsumbc 40641, using Pascal's rule (bcpascm1 40640) instead of the binomial theorem (binom 13965). (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 )
 
21.33.14.3  The ` ZZ `-module ` ZZ X. ZZ `
 
Theoremzlmodzxzlmod 40643 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 40644 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 40645 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 40646 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 40647 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 40648 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 40649 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 ) >. } )
 
21.33.14.4  Ordered group sum operation (extension)
 
Theoremgsumpr 40650* 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 40651* 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 40652* 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 40653* 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 40654* 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 40655* 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 ) )
 
21.33.14.5  Symmetric groups (extension)
 
Theoremnn0le2is012 40656 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 40657 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 40658 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 40659 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 )
 
21.33.14.6  Divisibility (extension)
 
Theoreminvginvrid 40660 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 )
 
21.33.14.7  The support of functions (extension)
 
Theoremrmsupp0 40661* 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 40662* 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 40663* 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 40664 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 40665* 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 ) ) )
 
21.33.14.8  Finitely supported functions (extension)
 
Theoremrmsuppfi 40666* 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 40667* 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 40668 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 40669 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 40670* 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 40671* 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 40672* 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 40673* 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 40674* 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 )
 
21.33.14.9  Left modules (extension)
 
Theoremlmodvsmdi 40675 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 40676* 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 ) )
 
21.33.14.10  Associative algebras (extension)
 
Theoremascl0 40677 The scalar 0 embedded into a left module corresponds to the 0 of the left module if the left module is also a ring. (Contributed by AV, 31-Jul-2019.)
 |-  A  =  (algSc `  W )   &    |-  F  =  (Scalar `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  W  e.  Ring
 )   =>    |-  ( ph  ->  ( A `  ( 0g `  F ) )  =  ( 0g `  W ) )
 
Theoremascl1 40678 The scalar 1 embedded into a left module corresponds to the 1 of the left module if the left module is also a ring. (Contributed by AV, 31-Jul-2019.)
 |-  A  =  (algSc `  W )   &    |-  F  =  (Scalar `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  W  e.  Ring
 )   =>    |-  ( ph  ->  ( A `  ( 1r `  F ) )  =  ( 1r `  W ) )
 
Theoremassaascl0 40679 The scalar 0 embedded into an associative algebra corresponds to the 0 of the associative algebra. (Contributed by AV, 31-Jul-2019.)
 |-  A  =  (algSc `  W )   &    |-  F  =  (Scalar `  W )   &    |-  ( ph  ->  W  e. AssAlg )   =>    |-  ( ph  ->  ( A `  ( 0g `  F ) )  =  ( 0g
 `  W ) )
 
Theoremassaascl1 40680 The scalar 1 embedded into an associative algebra corresponds to the 1 of the an associative algebra. (Contributed by AV, 31-Jul-2019.)
 |-  A  =  (algSc `  W )   &    |-  F  =  (Scalar `  W )   &    |-  ( ph  ->  W  e. AssAlg )   =>    |-  ( ph  ->  ( A `  ( 1r `  F ) )  =  ( 1r
 `  W ) )
 
21.33.14.11  Univariate polynomials (extension)
 
Theoremply1vr1smo 40681 The variable in a polynomial expressed as scaled monomial. (Contributed by AV, 12-Aug-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  .1.  =  ( 1r `  R )   &    |- 
 .x.  =  ( .s `  P )   &    |-  G  =  (mulGrp `  P )   &    |-  .^  =  (.g `  G )   &    |-  X  =  (var1 `  R )   =>    |-  ( R  e.  Ring  ->  (  .1.  .x.  ( 1  .^  X ) )  =  X )
 
Theoremply1ass23l 40682 Associative identity with scalar and ring multiplication for the polynomial ring. (Contributed by AV, 14-Aug-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  .X.  =  ( .r `  P )   &    |-  B  =  ( Base `  P )   &    |-  K  =  (
 Base `  R )   &    |-  .x.  =  ( .s `  P )   =>    |-  ( ( R  e.  Ring  /\  ( A  e.  K  /\  X  e.  B  /\  Y  e.  B )
 )  ->  ( ( A  .x.  X )  .X.  Y )  =  ( A 
 .x.  ( X  .X.  Y ) ) )
 
Theoremply1sclrmsm 40683 The ring multiplication of a polynomial with a scalar polynomial is equal to the scalar multiplication of the polynomial with the corresponding scalar. (Contributed by AV, 14-Aug-2019.)
 |-  K  =  ( Base `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  E  =  ( Base `  P )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  .X. 
 =  ( .r `  P )   &    |-  N  =  (mulGrp `  P )   &    |-  .^  =  (.g `  N )   &    |-  A  =  (algSc `  P )   =>    |-  ( ( R  e.  Ring  /\  F  e.  K  /\  Z  e.  E )  ->  ( ( A `  F )  .X.  Z )  =  ( F  .x.  Z ) )
 
Theoremcoe1id 40684* Coefficient vector of the unit polynomial. (Contributed by AV, 9-Aug-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  I  =  ( 1r `  P )   &    |- 
 .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  (coe1 `  I )  =  ( x  e.  NN0  |->  if ( x  =  0 ,  .1.  ,  .0.  ) ) )
 
Theoremcoe1sclmulval 40685 The value of the coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by AV, 14-Aug-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  ( Base `  R )   &    |-  A  =  (algSc `  P )   &    |-  S  =  ( .s `  P )   &    |-  .xb  =  ( .r `  P )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( Y  e.  K  /\  Z  e.  B )  /\  N  e.  NN0 )  ->  (
 (coe1 `
  ( Y S Z ) ) `  N )  =  ( Y  .x.  ( (coe1 `  Z ) `  N ) ) )
 
Theoremply1mulgsumlem1 40686* Lemma 1 for ply1mulgsum 40690. (Contributed by AV, 19-Oct-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  A  =  (coe1 `  K )   &    |-  C  =  (coe1 `  L )   &    |-  X  =  (var1 `  R )   &    |-  .X.  =  ( .r `  P )   &    |-  .x. 
 =  ( .s `  P )   &    |-  .*  =  ( .r `  R )   &    |-  M  =  (mulGrp `  P )   &    |-  .^  =  (.g `  M )   =>    |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  E. s  e.  NN0  A. n  e.  NN0  (
 s  <  n  ->  ( ( A `  n )  =  ( 0g `  R )  /\  ( C `  n )  =  ( 0g `  R ) ) ) )
 
Theoremply1mulgsumlem2 40687* Lemma 2 for ply1mulgsum 40690. (Contributed by AV, 19-Oct-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  A  =  (coe1 `  K )   &    |-  C  =  (coe1 `  L )   &    |-  X  =  (var1 `  R )   &    |-  .X.  =  ( .r `  P )   &    |-  .x. 
 =  ( .s `  P )   &    |-  .*  =  ( .r `  R )   &    |-  M  =  (mulGrp `  P )   &    |-  .^  =  (.g `  M )   =>    |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  E. s  e.  NN0  A. n  e.  NN0  (
 s  <  n  ->  ( R  gsumg  ( l  e.  (
 0 ... n )  |->  ( ( A `  l
 )  .*  ( C `  ( n  -  l
 ) ) ) ) )  =  ( 0g
 `  R ) ) )
 
Theoremply1mulgsumlem3 40688* Lemma 3 for ply1mulgsum 40690. (Contributed by AV, 20-Oct-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  A  =  (coe1 `  K )   &    |-  C  =  (coe1 `  L )   &    |-  X  =  (var1 `  R )   &    |-  .X.  =  ( .r `  P )   &    |-  .x. 
 =  ( .s `  P )   &    |-  .*  =  ( .r `  R )   &    |-  M  =  (mulGrp `  P )   &    |-  .^  =  (.g `  M )   =>    |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( k  e.  NN0  |->  ( R  gsumg  ( l  e.  (
 0 ... k )  |->  ( ( A `  l
 )  .*  ( C `  ( k  -  l
 ) ) ) ) ) ) finSupp  ( 0g
 `  R ) )
 
Theoremply1mulgsumlem4 40689* Lemma 4 for ply1mulgsum 40690. (Contributed by AV, 19-Oct-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  A  =  (coe1 `  K )   &    |-  C  =  (coe1 `  L )   &    |-  X  =  (var1 `  R )   &    |-  .X.  =  ( .r `  P )   &    |-  .x. 
 =  ( .s `  P )   &    |-  .*  =  ( .r `  R )   &    |-  M  =  (mulGrp `  P )   &    |-  .^  =  (.g `  M )   =>    |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  ( 0 ... k )  |->  ( ( A `  l )  .*  ( C `  ( k  -  l
 ) ) ) ) )  .x.  ( k  .^  X ) ) ) finSupp  ( 0g `  P ) )
 
Theoremply1mulgsum 40690* The product of two polynomials expressed as group sum of scaled monomials. (Contributed by AV, 20-Oct-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  A  =  (coe1 `  K )   &    |-  C  =  (coe1 `  L )   &    |-  X  =  (var1 `  R )   &    |-  .X.  =  ( .r `  P )   &    |-  .x. 
 =  ( .s `  P )   &    |-  .*  =  ( .r `  R )   &    |-  M  =  (mulGrp `  P )   &    |-  .^  =  (.g `  M )   =>    |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( K  .X.  L )  =  ( P  gsumg  (
 k  e.  NN0  |->  ( ( R  gsumg  ( l  e.  (
 0 ... k )  |->  ( ( A `  l
 )  .*  ( C `  ( k  -  l
 ) ) ) ) )  .x.  ( k  .^  X ) ) ) ) )
 
Theoremevl1at0 40691 Polynomial evaluation for the 0 scalar. (Contributed by AV, 10-Aug-2019.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  P )   =>    |-  ( R  e.  CRing  ->  ( ( O `  Z ) `  .0.  )  =  .0.  )
 
Theoremevl1at1 40692 Polynomial evaluation for the 1 scalar. (Contributed by AV, 10-Aug-2019.)
 |-  O  =  (eval1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  I  =  ( 1r
 `  P )   =>    |-  ( R  e.  CRing  ->  ( ( O `  I ) `  .1.  )  =  .1.  )
 
21.33.14.12  Univariate polynomials (examples)
 
Theoremlinply1 40693 A term of the form  x  -  C is a (univariate) polynomial, also called "linear polynomial". (Part of ply1remlem 23192). (Contributed by AV, 3-Jul-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  ( Base `  R )   &    |-  X  =  (var1 `  R )   &    |-  .-  =  ( -g `  P )   &    |-  A  =  (algSc `  P )   &    |-  G  =  ( X 
 .-  ( A `  C ) )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  G  e.  B )
 
Theoremlineval 40694 A term of the form  x  -  C evaluated for  x  =  V results in  V  -  C (part of ply1remlem 23192). (Contributed by AV, 3-Jul-2019.)
 |-  P  =  (Poly1 `  R )   &    |-  B  =  ( Base `  P )   &    |-  K  =  ( Base `  R )   &    |-  X  =  (var1 `  R )   &    |-  .-  =  ( -g `  P )   &    |-  A  =  (algSc `  P )   &    |-  G  =  ( X 
 .-  ( A `  C ) )   &    |-  ( ph  ->  C  e.  K )   &    |-  O  =  (eval1 `  R )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  V  e.  K )   =>    |-  ( ph  ->  ( ( O `  G ) `  V )  =  ( V ( -g `  R ) C ) )
 
Theoremzringsubgval 40695 Subtraction in the ring of integers. (Contributed by AV, 3-Aug-2019.)
 |-  .-  =  ( -g ` ring )   =>    |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( X  -  Y )  =  ( X  .-  Y ) )
 
Theoremlinevalexample 40696 The polynomial  x  -  3 over 
ZZ evaluated for  x  =  5 results in 2. (Contributed by AV, 3-Jul-2019.)
 |-  P  =  (Poly1 ` ring )   &    |-  B  =  (
 Base `  P )   &    |-  X  =  (var1 ` ring )   &    |-  .-  =  ( -g `  P )   &    |-  A  =  (algSc `  P )   &    |-  G  =  ( X  .-  ( A `  3 ) )   &    |-  O  =  (eval1 ` ring )   =>    |-  ( ( O `  ( X  .-  ( A `
  3 ) ) ) `  5 )  =  2
 
21.33.15  Linear algebra (extension)
 
21.33.15.1  The subalgebras of diagonal and scalar matrices (extension)

In the following, alternative definitions for diagonal and scalar matrices are provided. These definitions define diagonal and scalar matrices as extensible structures, whereas the definitions df-dmat 19592 and df-scmat 19593 define diagonal and scalar matrices as sets.

 
Syntaxcdmatalt 40697 Alternative notation for the algebra of diagonal matrices.
 class DMatALT
 
Syntaxcscmatalt 40698 Alternative notation for the algebra of scalar matrices.
 class ScMatALT
 
Definitiondf-dmatalt 40699* Define the set of n x n diagonal (square) matrices over a set (usually a ring) r, see definition in [Roman] p. 4 or Definition 3.12 in [Hefferon] p. 240. (Contributed by AV, 8-Dec-2019.)
 |- DMatALT  =  ( n  e.  Fin ,  r  e.  _V  |->  [_ ( n Mat  r )  /  a ]_ ( as  { m  e.  ( Base `  a )  | 
 A. i  e.  n  A. j  e.  n  ( i  =/=  j  ->  ( i m j )  =  ( 0g
 `  r ) ) } ) )
 
Definitiondf-scmatalt 40700* Define the algebra of n x n scalar matrices over a set (usually a ring) r, see definition in [Connell] p. 57: "A scalar matrix is a diagonal matrix for which all the diagonal terms are equal, i.e., a matrix of the form cIn";. (Contributed by AV, 8-Dec-2019.)
 |- ScMatALT  =  ( n  e.  Fin ,  r  e.  _V  |->  [_ ( n Mat  r )  /  a ]_ ( as  { m  e.  ( Base `  a )  | 
 E. c  e.  ( Base `  r ) A. i  e.  n  A. j  e.  n  (
 i m j )  =  if ( i  =  j ,  c ,  ( 0g `  r
 ) ) } )
 )
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