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Theorem List for Metamath Proof Explorer - 16401-16500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcchr 16401 Syntax for ring characteristic.
 class chr
 
Syntaxczn 16402 The ring of integers modulo  n.
 class ℤ/n
 
Definitiondf-zrh 16403 Define the unique homomorphism from the integers into a ring. This encodes the usual notation of 
n  =  1r  +  1r  +  ...  +  1r for integers (see also df-mulg 14440). (Contributed by Mario Carneiro, 13-Jun-2015.)
 |-  ZRHom  =  ( r  e.  _V  |->  U. (
 (flds  ZZ ) RingHom  r ) )
 
Definitiondf-zlm 16404 Augment an abelian group with vector space operations to turn it into a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  ZMod  =  ( g  e.  _V  |->  ( ( g sSet  <. (Scalar `  ndx ) ,  (flds  ZZ ) >. ) sSet  <. ( .s
 `  ndx ) ,  (.g `  g ) >. ) )
 
Definitiondf-chr 16405 The characteristic of a ring is the smallest positive integer which is equal to 0 when interpreted in the ring, or 0 if there is no such positive integer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- chr 
 =  ( g  e. 
 _V  |->  ( ( od
 `  g ) `  ( 1r `  g ) ) )
 
Definitiondf-zn 16406* Define the ring of integers  mod  n. This is literally the quotient ring of  ZZ by the ideal  n ZZ, but we augment it with a total order. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |- ℤ/n =  ( n  e.  NN0  |->  [_ (flds  ZZ )  /  z ]_ [_ (
 z  /.s  ( z ~QG  ( (RSpan `  z
 ) `  { n } ) ) ) 
 /  s ]_ (
 s sSet  <. ( le `  ndx ) ,  [_ ( ( ZRHom `  s )  |` 
 if ( n  =  0 ,  ZZ ,  ( 0..^ n ) ) )  /  f ]_ ( ( f  o. 
 <_  )  o.  `' f
 ) >. ) )
 
Theoremmulgghm2 16407* The powers of a group element give a homomorphism from  ZZ to a group. (Contributed by Mario Carneiro, 13-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  .x.  =  (.g `  R )   &    |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
 )   &    |-  B  =  ( Base `  R )   =>    |-  ( ( R  e.  Grp  /\  .1.  e.  B ) 
 ->  F  e.  ( Z 
 GrpHom  R ) )
 
Theoremmulgrhm 16408* The powers of the element  1 give a ring homomorphism from  ZZ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  .x.  =  (.g `  R )   &    |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
 )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  F  e.  ( Z RingHom  R ) )
 
Theoremmulgrhm2 16409* The powers of the element  1 give the unique ring homomorphism from  ZZ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  .x.  =  (.g `  R )   &    |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
 )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  ( Z RingHom  R )  =  { F } )
 
Theoremzrhval 16410 Define the unique homomorphism from the integers to a ring or field. (Contributed by Mario Carneiro, 13-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  L  =  ( ZRHom `  R )   =>    |-  L  =  U. ( Z RingHom  R )
 
Theoremzrhval2 16411* Alternate value of the  ZRHom homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  L  =  ( ZRHom `  R )   &    |-  .x.  =  (.g `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  L  =  ( n  e. 
 ZZ  |->  ( n  .x.  .1.  ) ) )
 
Theoremzrhmulg 16412 Value of the  ZRHom homomorphism. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  L  =  ( ZRHom `  R )   &    |-  .x.  =  (.g `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  N  e.  ZZ )  ->  ( L `  N )  =  ( N  .x.  .1.  ) )
 
Theoremzrhrhmb 16413 The  ZRHom homomorphism is the unique ring homomorphism from  Z. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  L  =  ( ZRHom `  R )   =>    |-  ( R  e.  Ring  ->  ( F  e.  ( Z RingHom  R )  <->  F  =  L ) )
 
Theoremzrhrhm 16414 The  ZRHom homomorphism is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  L  =  ( ZRHom `  R )   =>    |-  ( R  e.  Ring  ->  L  e.  ( Z RingHom  R )
 )
 
Theoremzrh1 16415 Interpretation of 1 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  L  =  ( ZRHom `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  ( L `  1 )  =  .1.  )
 
Theoremzrh0 16416 Interpretation of 0 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  L  =  ( ZRHom `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  ( L `  0 )  =  .0.  )
 
Theoremzrhpropd 16417* The  ZZ ring homomorphism depends only on the ring attributes of a structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )   =>    |-  ( ph  ->  ( ZRHom `  K )  =  ( ZRHom `  L ) )
 
Theoremzlmval 16418 Augment an abelian group with vector space operations to turn it into a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  Z  =  (flds  ZZ )   &    |-  .x.  =  (.g `  G )   =>    |-  ( G  e.  V  ->  W  =  ( ( G sSet  <. (Scalar `  ndx ) ,  Z >. ) sSet  <. ( .s
 `  ndx ) ,  .x.  >.
 ) )
 
Theoremzlmlem 16419 Lemma for zlmbas 16420 and zlmplusg 16421. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  <  5   =>    |-  ( E `  G )  =  ( E `  W )
 
Theoremzlmbas 16420 Base set of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  B  =  (
 Base `  G )   =>    |-  B  =  (
 Base `  W )
 
Theoremzlmplusg 16421 Group operation of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  .+  =  ( +g  `  W )
 
Theoremzlmmulr 16422 Ring operation of a  ZZ-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  .x.  =  ( .r `  G )   =>    |-  .x.  =  ( .r `  W )
 
Theoremzlmsca 16423 Scalar ring of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  Z  =  (flds  ZZ )   =>    |-  ( G  e.  V  ->  Z  =  (Scalar `  W ) )
 
Theoremzlmvsca 16424 Scalar multiplication operation of a  ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  .x.  =  (.g `  G )   =>    |- 
 .x.  =  ( .s `  W )
 
Theoremzlmlmod 16425 The  ZZ-module operation turns an arbitrary abelian group into a left module over  ZZ. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   =>    |-  ( G  e.  Abel  <->  W  e.  LMod )
 
Theoremzlmassa 16426 The  ZZ-module operation turns a ring into an associative algebra over  ZZ. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  ( ZMod `  G )   =>    |-  ( G  e.  Ring  <->  W  e. AssAlg )
 
Theoremchrval 16427 Definition substitution of the ring characteristic. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  O  =  ( od
 `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  C  =  (chr `  R )   =>    |-  ( O `  .1.  )  =  C
 
Theoremchrcl 16428 Closure of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  C  =  (chr `  R )   =>    |-  ( R  e.  Ring  ->  C  e.  NN0 )
 
Theoremchrid 16429 The canonical  ZZ ring homomorphism applied to a ring's characteristic is zero. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  C  =  (chr `  R )   &    |-  L  =  ( ZRHom `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  ( L `  C )  =  .0.  )
 
Theoremchrdvds 16430 The  ZZ ring homomorphism is zero only at multiples of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  C  =  (chr `  R )   &    |-  L  =  ( ZRHom `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  N  e.  ZZ )  ->  ( C  ||  N  <->  ( L `  N )  =  .0.  ) )
 
Theoremchrcong 16431 If two integers are congruent relative to the ring characteristic, their images in the ring are the same. (Contributed by Mario Carneiro, 24-Sep-2015.)
 |-  C  =  (chr `  R )   &    |-  L  =  ( ZRHom `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( C  ||  ( M  -  N )  <->  ( L `  M )  =  ( L `  N ) ) )
 
Theoremchrnzr 16432 Nonzero rings are precisely those with characteristic not 1. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( R  e.  Ring  ->  ( R  e. NzRing  <->  (chr `  R )  =/=  1 ) )
 
Theoremchrrhm 16433 The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( F  e.  ( R RingHom  S )  ->  (chr `  S )  ||  (chr `  R ) )
 
Theoremdomnchr 16434 The characteristic of a domain can only be zero or a prime. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  ( R  e. Domn  ->  ( (chr `  R )  =  0  \/  (chr `  R )  e.  Prime ) )
 
Theoremznlidl 16435 The set  n ZZ is an ideal in  ZZ. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   =>    |-  ( N  e.  ZZ  ->  ( S `  { N } )  e.  (LIdeal `  Z ) )
 
Theoremzncrng2 16436 The value of the ℤ/nℤ structure. It is defined as the quotient ring  ZZ  /  n ZZ, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   =>    |-  ( N  e.  ZZ  ->  U  e.  CRing )
 
Theoremznval 16437 The value of the ℤ/nℤ structure. It is defined as the quotient ring  ZZ  /  n ZZ, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  U )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   &    |-  .<_  =  ( ( F  o.  <_  )  o.  `' F )   =>    |-  ( N  e.  NN0  ->  Y  =  ( U sSet  <.
 ( le `  ndx ) ,  .<_  >. ) )
 
Theoremznle 16438 The value of the ℤ/nℤ structure. It is defined as the quotient ring  ZZ  /  n ZZ, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  U )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   &    |-  .<_  =  ( le `  Y )   =>    |-  ( N  e.  NN0  ->  .<_  =  ( ( F  o.  <_  )  o.  `' F ) )
 
Theoremznval2 16439 Self-referential expression for the ℤ/nℤ structure. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  .<_  =  ( le `  Y )   =>    |-  ( N  e.  NN0  ->  Y  =  ( U sSet  <.
 ( le `  ndx ) ,  .<_  >. ) )
 
Theoremznbaslem 16440 Lemma for znbas 16445. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  E  = Slot  K   &    |-  K  e.  NN   &    |-  K  <  10   =>    |-  ( N  e.  NN0  ->  ( E `  U )  =  ( E `  Y ) )
 
Theoremznbas2 16441 The base set of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( Base `  U )  =  ( Base `  Y )
 )
 
Theoremznadd 16442 The additive structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( +g  `  U )  =  ( +g  `  Y ) )
 
Theoremznmul 16443 The multiplicative structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( .r `  U )  =  ( .r `  Y ) )
 
Theoremznzrh 16444 The  ZZ ring homomorphism of ℤ/nℤ is inherited from the quotient ring it is based on. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  U  =  ( Z  /.s  ( Z ~QG  ( S `  { N } ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( ZRHom `  U )  =  ( ZRHom `  Y ) )
 
Theoremznbas 16445 The base set of ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  Y  =  (ℤ/n `  N )   &    |-  R  =  ( Z ~QG  ( S `  { N } ) )   =>    |-  ( N  e.  NN0 
 ->  ( ZZ /. R )  =  ( Base `  Y ) )
 
Theoremzncrng 16446 ℤ/nℤ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  Y  e.  CRing )
 
Theoremznzrh2 16447* The  ZZ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  .~  =  ( Z ~QG  ( S `  { N } ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  ( N  e.  NN0  ->  L  =  ( x  e.  ZZ  |->  [ x ]  .~  )
 )
 
Theoremznzrhval 16448 The  ZZ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  S  =  (RSpan `  Z )   &    |-  .~  =  ( Z ~QG  ( S `  { N } ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  (
 ( N  e.  NN0  /\  A  e.  ZZ )  ->  ( L `  A )  =  [ A ]  .~  )
 
Theoremznzrhfo 16449 The  ZZ ring homomorphism is a surjection onto 
ZZ  /  n ZZ. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   &    |-  B  =  ( Base `  Y )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  ( N  e.  NN0  ->  L : ZZ -onto-> B )
 
Theoremzncyg 16450 The group  ZZ  /  n ZZ is cyclic for all  n (including  n  =  0). (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  Y  e. CycGrp )
 
Theoremzndvds 16451 Express equality of equivalence classes in  ZZ 
/  n ZZ in terms of divisibility. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( L `  A )  =  ( L `  B )  <->  N  ||  ( A  -  B ) ) )
 
Theoremzndvds0 16452 Special case of zndvds 16451 when one argument is zero. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   &    |-  .0.  =  ( 0g `  Y )   =>    |-  ( ( N  e.  NN0  /\  A  e.  ZZ )  ->  ( ( L `  A )  =  .0.  <->  N  ||  A ) )
 
Theoremznf1o 16453 The function  F enumerates all equivalence classes in ℤ/nℤ for each  n. When  n  = 
0,  ZZ  /  0 ZZ  =  ZZ  /  {
0 }  ~~  ZZ so we let  W  =  ZZ; otherwise  W  =  { 0 , 
... ,  n  - 
1 } enumerates all the equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.)
 |-  Z  =  (flds  ZZ )   &    |-  Y  =  (ℤ/n `  N )   &    |-  B  =  (
 Base `  Y )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   =>    |-  ( N  e.  NN0  ->  F : W -1-1-onto-> B )
 
Theoremzzngim 16454 The  ZZ ring homomorphism is an isomorphism for 
N  =  0. (We only show group isomorphism here, but ring isomorphism follows, since it is a bijective ring homomorphism.) (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  Y  =  (ℤ/n `  0
 )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  L  e.  ( (flds  ZZ ) GrpIso  Y )
 
Theoremznle2 16455 The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   &    |-  .<_  =  ( le `  Y )   =>    |-  ( N  e.  NN0  ->  .<_  =  ( ( F  o.  <_  )  o.  `' F ) )
 
Theoremznleval 16456 The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   &    |-  .<_  =  ( le `  Y )   &    |-  X  =  ( Base `  Y )   =>    |-  ( N  e.  NN0  ->  ( A  .<_  B  <->  ( A  e.  X  /\  B  e.  X  /\  ( `' F `  A )  <_  ( `' F `  B ) ) ) )
 
Theoremznleval2 16457 The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   &    |-  .<_  =  ( le `  Y )   &    |-  X  =  ( Base `  Y )   =>    |-  ( ( N  e.  NN0  /\  A  e.  X  /\  B  e.  X )  ->  ( A  .<_  B  <->  ( `' F `  A )  <_  ( `' F `  B ) ) )
 
Theoremzntoslem 16458 Lemma for zntos 16459. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Z  =  (flds  ZZ )   &    |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   &    |-  .<_  =  ( le `  Y )   &    |-  X  =  ( Base `  Y )   =>    |-  ( N  e.  NN0  ->  Y  e. Toset )
 
Theoremzntos 16459 The ℤ/nℤ structure is a totally ordered set. (The order is not respected by the operations, except in the case  N  =  0 when it coincides with the ordering on  ZZ.) (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  Y  e. Toset )
 
Theoremznhash 16460 The ℤ/nℤ structure has  n elements. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   &    |-  B  =  ( Base `  Y )   =>    |-  ( N  e.  NN  ->  ( # `  B )  =  N )
 
Theoremznfi 16461 The ℤ/nℤ structure is a finite ring. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  Y  =  (ℤ/n `  N )   &    |-  B  =  ( Base `  Y )   =>    |-  ( N  e.  NN  ->  B  e.  Fin )
 
Theoremznfld 16462 The ℤ/nℤ structure is a finite field when  n is prime. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  Prime  ->  Y  e. Field )
 
Theoremznidomb 16463 The ℤ/nℤ structure is a domain (and hence a field) precisely when  n is prime. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN  ->  ( Y  e. IDomn  <->  N  e.  Prime ) )
 
Theoremznchr 16464 Cyclic rings are defined by their characteristic. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  (chr `  Y )  =  N )
 
Theoremznunit 16465 The units of ℤ/nℤ are the integers coprime to the base. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  Y  =  (ℤ/n `  N )   &    |-  U  =  (Unit `  Y )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  (
 ( N  e.  NN0  /\  A  e.  ZZ )  ->  ( ( L `  A )  e.  U  <->  ( A  gcd  N )  =  1 ) )
 
Theoremznunithash 16466 The size of the unit group of ℤ/nℤ. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  Y  =  (ℤ/n `  N )   &    |-  U  =  (Unit `  Y )   =>    |-  ( N  e.  NN  ->  ( # `  U )  =  ( phi `  N ) )
 
Theoremznrrg 16467 The regular elements of ℤ/nℤ are exactly the units. (This theorem fails for  N  =  0, where all nonzero integers are regular, but only  pm 1 are units.) (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  Y  =  (ℤ/n `  N )   &    |-  U  =  (Unit `  Y )   &    |-  E  =  (RLReg `  Y )   =>    |-  ( N  e.  NN  ->  E  =  U )
 
Theoremcygznlem1 16468* Lemma for cygzn 16472. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  N  =  if ( B  e.  Fin ,  ( # `  B ) ,  0 )   &    |-  Y  =  (ℤ/n `  N )   &    |-  .x.  =  (.g `  G )   &    |-  L  =  ( ZRHom `  Y )   &    |-  E  =  { x  e.  B  |  ran  (  n  e. 
 ZZ  |->  ( n  .x.  x ) )  =  B }   &    |-  ( ph  ->  G  e. CycGrp )   &    |-  ( ph  ->  X  e.  E )   =>    |-  ( ( ph  /\  ( K  e.  ZZ  /\  M  e.  ZZ )
 )  ->  ( ( L `  K )  =  ( L `  M ) 
 <->  ( K  .x.  X )  =  ( M  .x.  X ) ) )
 
Theoremcygznlem2a 16469* Lemma for cygzn 16472. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  B  =  ( Base `  G )   &    |-  N  =  if ( B  e.  Fin ,  ( # `  B ) ,  0 )   &    |-  Y  =  (ℤ/n `  N )   &    |-  .x.  =  (.g `  G )   &    |-  L  =  ( ZRHom `  Y )   &    |-  E  =  { x  e.  B  |  ran  (  n  e. 
 ZZ  |->  ( n  .x.  x ) )  =  B }   &    |-  ( ph  ->  G  e. CycGrp )   &    |-  ( ph  ->  X  e.  E )   &    |-  F  =  ran  (  m  e. 
 ZZ  |->  <. ( L `  m ) ,  ( m  .x.  X ) >. )   =>    |-  ( ph  ->  F :
 ( Base `  Y ) --> B )
 
Theoremcygznlem2 16470* Lemma for cygzn 16472. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by Mario Carneiro, 23-Dec-2016.)
 |-  B  =  ( Base `  G )   &    |-  N  =  if ( B  e.  Fin ,  ( # `  B ) ,  0 )   &    |-  Y  =  (ℤ/n `  N )   &    |-  .x.  =  (.g `  G )   &    |-  L  =  ( ZRHom `  Y )   &    |-  E  =  { x  e.  B  |  ran  (  n  e. 
 ZZ  |->  ( n  .x.  x ) )  =  B }   &    |-  ( ph  ->  G  e. CycGrp )   &    |-  ( ph  ->  X  e.  E )   &    |-  F  =  ran  (  m  e. 
 ZZ  |->  <. ( L `  m ) ,  ( m  .x.  X ) >. )   =>    |-  ( ( ph  /\  M  e.  ZZ )  ->  ( F `  ( L `  M ) )  =  ( M  .x.  X ) )
 
Theoremcygznlem3 16471* A cyclic group with  n elements is isomorphic to  ZZ  /  n ZZ. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  N  =  if ( B  e.  Fin ,  ( # `  B ) ,  0 )   &    |-  Y  =  (ℤ/n `  N )   &    |-  .x.  =  (.g `  G )   &    |-  L  =  ( ZRHom `  Y )   &    |-  E  =  { x  e.  B  |  ran  (  n  e. 
 ZZ  |->  ( n  .x.  x ) )  =  B }   &    |-  ( ph  ->  G  e. CycGrp )   &    |-  ( ph  ->  X  e.  E )   &    |-  F  =  ran  (  m  e. 
 ZZ  |->  <. ( L `  m ) ,  ( m  .x.  X ) >. )   =>    |-  ( ph  ->  G  ~=ph𝑔  Y )
 
Theoremcygzn 16472 A cyclic group with  n elements is isomorphic to  ZZ  /  n ZZ, and an infinite cyclic group is isomorphic to  ZZ 
/  0 ZZ  ~~  ZZ. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  N  =  if ( B  e.  Fin ,  ( # `  B ) ,  0 )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( G  e. CycGrp  ->  G 
 ~=ph𝑔  Y )
 
Theoremcygth 16473* The "fundamental theorem of cyclic groups". Cyclic groups are exactly the additive groups  ZZ  /  n ZZ, for 
0  <_  n (where  n  =  0 is the infinite cyclic group 
ZZ), up to isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  ( G  e. CycGrp  <->  E. n  e.  NN0  G 
 ~=ph𝑔  (ℤ/n `  n ) )
 
Theoremcyggic 16474 Cyclic groups are isomorphic precisely when they have the same order. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  (
 Base `  H )   =>    |-  ( ( G  e. CycGrp  /\  H  e. CycGrp )  ->  ( G  ~=ph𝑔 
 H 
 <->  B  ~~  C ) )
 
Theoremfrgpcyg 16475 A free group is cyclic iff it has zero or one generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  G  =  (freeGrp `  I
 )   =>    |-  ( I  ~<_  1o  <->  G  e. CycGrp )
 
10.12  Hilbert spaces
 
10.12.1  Definition and basic properties
 
Syntaxcphl 16476 Extend class notation with class all pre-Hilbert spaces.
 class  PreHil
 
Syntaxcipf 16477 Extend class notation with inner product function.
 class  .i f
 
Definitiondf-phl 16478* Define class all generalized pre-Hilbert (inner product) spaces. (Contributed by NM, 22-Sep-2011.)
 |-  PreHil  =  { g  e. 
 LVec  |  [. ( Base `  g )  /  v ]. [. ( .i `  g )  /  h ].
 [. (Scalar `  g )  /  f ]. ( f  e.  *Ring  /\  A. x  e.  v  ( ( y  e.  v  |->  ( y h x ) )  e.  ( g LMHom  (ringLMod `  f ) )  /\  ( ( x h x )  =  ( 0g `  f ) 
 ->  x  =  ( 0g `  g ) ) 
 /\  A. y  e.  v  ( ( * r `
  f ) `  ( x h y ) )  =  ( y h x ) ) ) }
 
Definitiondf-ipf 16479* Define group addition function. Usually we will use  +g directly instead of  + f, and they have the same behavior in most cases. The main advantage of  + f is that it is a guaranteed function (mndplusf 14331), while  +g only has closure (mndcl 14320). (Contributed by Mario Carneiro, 12-Aug-2015.)
 |- 
 .i f  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g ) ,  y  e.  ( Base `  g )  |->  ( x ( .i
 `  g ) y ) ) )
 
Theoremisphl 16480* The predicate "is a generalized pre-Hilbert (inner product) space". (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .,  =  ( .i `  W )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .*  =  ( * r `  F )   &    |-  Z  =  ( 0g `  F )   =>    |-  ( W  e.  PreHil  <->  ( W  e.  LVec  /\  F  e.  *Ring  /\  A. x  e.  V  ( ( y  e.  V  |->  ( y 
 .,  x ) )  e.  ( W LMHom  (ringLMod `  F ) )  /\  ( ( x  .,  x )  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (  .*  `  ( x  .,  y ) )  =  ( y  .,  x ) ) ) )
 
Theoremphllvec 16481 A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  PreHil  ->  W  e.  LVec )
 
Theoremphllmod 16482 A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  PreHil  ->  W  e.  LMod )
 
Theoremphlsrng 16483 The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  PreHil  ->  F  e.  *Ring )
 
Theoremphllmhm 16484* The inner product of a pre-Hilbert space is linear in its left argument. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  G  =  ( x  e.  V  |->  ( x  .,  A ) )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V ) 
 ->  G  e.  ( W LMHom 
 (ringLMod `  F ) ) )
 
Theoremipcl 16485 Closure of the inner product operation in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  K  =  ( Base `  F )   =>    |-  (
 ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .,  B )  e.  K )
 
Theoremipcj 16486 Conjugate of an inner product in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .*  =  ( * r `  F )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (  .*  `  ( A  .,  B ) )  =  ( B  .,  A ) )
 
Theoremiporthcom 16487 Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  Z  =  ( 0g `  F )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A  .,  B )  =  Z  <->  ( B  .,  A )  =  Z ) )
 
Theoremip0l 16488 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  Z  =  ( 0g `  F )   &    |- 
 .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V ) 
 ->  (  .0.  .,  A )  =  Z )
 
Theoremip0r 16489 Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  Z  =  ( 0g `  F )   &    |- 
 .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V ) 
 ->  ( A  .,  .0.  )  =  Z )
 
Theoremipeq0 16490 The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  Z  =  ( 0g `  F )   &    |- 
 .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V ) 
 ->  ( ( A  .,  A )  =  Z  <->  A  =  .0.  ) )
 
Theoremipdir 16491 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .+^  =  (
 +g  `  F )   =>    |-  (
 ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( ( A  .+  B )  .,  C )  =  (
 ( A  .,  C )  .+^  ( B  .,  C ) ) )
 
Theoremipdi 16492 Distributive law for inner product. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .+^  =  (
 +g  `  F )   =>    |-  (
 ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( A  .,  ( B  .+  C ) )  =  (
 ( A  .,  B )  .+^  ( A  .,  C ) ) )
 
Theoremip2di 16493 Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .+^  =  (
 +g  `  F )   &    |-  ( ph  ->  W  e.  PreHil )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ph  ->  (
 ( A  .+  B )  .,  ( C  .+  D ) )  =  ( ( ( A 
 .,  C )  .+^  ( B  .,  D ) )  .+^  ( ( A  .,  D )  .+^  ( B  .,  C ) ) ) )
 
Theoremipsubdir 16494 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  S  =  ( -g `  F )   =>    |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( ( A  .-  B )  .,  C )  =  (
 ( A  .,  C ) S ( B  .,  C ) ) )
 
Theoremipsubdi 16495 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  S  =  ( -g `  F )   =>    |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( A  .,  ( B  .-  C ) )  =  (
 ( A  .,  B ) S ( A  .,  C ) ) )
 
Theoremip2subdi 16496 Distributive law for inner product subtraction. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  S  =  ( -g `  F )   &    |-  .+  =  ( +g  `  F )   &    |-  ( ph  ->  W  e.  PreHil )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ph  ->  (
 ( A  .-  B )  .,  ( C  .-  D ) )  =  ( ( ( A 
 .,  C )  .+  ( B  .,  D ) ) S ( ( A  .,  D ) 
 .+  ( B  .,  C ) ) ) )
 
Theoremipass 16497 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .X. 
 =  ( .r `  F )   =>    |-  ( ( W  e.  PreHil  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( ( A  .x.  B )  .,  C )  =  ( A  .X.  ( B  .,  C ) ) )
 
Theoremipassr 16498 "Associative" law for second argument of inner product (compare ipass 16497). (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .X. 
 =  ( .r `  F )   &    |-  .*  =  ( * r `  F )   =>    |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
 )  ->  ( A  .,  ( C  .x.  B ) )  =  (
 ( A  .,  B )  .X.  (  .*  `  C ) ) )
 
Theoremipassr2 16499 "Associative" law for inner product. Conjugate version of ipassr 16498. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .X. 
 =  ( .r `  F )   &    |-  .*  =  ( * r `  F )   =>    |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
 )  ->  ( ( A  .,  B )  .X.  C )  =  ( A 
 .,  ( (  .*  `  C )  .x.  B ) ) )
 
Theoremipffval 16500* The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .,  =  ( .i `  W )   &    |-  .x.  =  ( .i f `  W )   =>    |- 
 .x.  =  ( x  e.  V ,  y  e.  V  |->  ( x  .,  y ) )
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