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Theorem List for Metamath Proof Explorer - 15501-15600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdrngpropd 15501* If two structures have the same group components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 27-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  ->  ( K  e.  DivRing  <->  L  e.  DivRing ) )
 
Theoremfldpropd 15502* If two structures have the same group components (properties), one is a field iff the other one is. (Contributed by Mario Carneiro, 8-Feb-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  ->  ( K  e. Field  <->  L  e. Field ) )
 
10.5.2  Subrings of a ring
 
Syntaxcsubrg 15503 Extend class notation with all subrings of a ring.
 class SubRing
 
Syntaxcrgspn 15504 Extend class notation with span of a set of elements over a ring.
 class RingSpan
 
Definitiondf-subrg 15505* Define a subring of a ring as a set of elements that is a ring in its own right and contains the multiplicative identity.

The additional constraint is necessary because the multiplicative identity of a ring, unlike the additive identity of a ring/group or the multiplicative identity of a field, cannot be identified by a local property. Thus it is possible for a subset of a ring to be a ring while not containing the true identity if it contains a false identity. For instance, the subset  ( ZZ  X.  {
0 } ) of  ( ZZ  X.  ZZ ) (where multiplication is component-wise) contains the false identity  <. 1 ,  0 >. which preserves every element of the subset and thus appears to be the identity of the subset, but is not the identity of the larger ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)

 |- SubRing  =  ( w  e.  Ring  |->  { s  e.  ~P ( Base `  w )  |  ( ( ws  s )  e.  Ring  /\  ( 1r
 `  w )  e.  s ) } )
 
Definitiondf-rgspn 15506* The ring-span of a set of elements in a ring is the smallest subring which contains all of them. (Contributed by Stefan O'Rear, 7-Dec-2014.)
 |- RingSpan  =  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |-> 
 |^| { t  e.  (SubRing `  w )  |  s 
 C_  t } )
 )
 
Theoremissubrg 15507 The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  B  =  ( Base `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( A  e.  (SubRing `  R )  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  B  /\  .1.  e.  A ) ) )
 
Theoremsubrgss 15508 A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  B  =  ( Base `  R )   =>    |-  ( A  e.  (SubRing `  R )  ->  A  C_  B )
 
Theoremsubrgid 15509 Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
 |-  B  =  ( Base `  R )   =>    |-  ( R  e.  Ring  ->  B  e.  (SubRing `  R ) )
 
Theoremsubrgrng 15510 A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  S  =  ( Rs  A )   =>    |-  ( A  e.  (SubRing `  R )  ->  S  e.  Ring )
 
Theoremsubrgcrng 15511 A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  S  =  ( Rs  A )   =>    |-  ( ( R  e.  CRing  /\  A  e.  (SubRing `  R ) )  ->  S  e.  CRing
 )
 
Theoremsubrgrcl 15512 Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
 |-  ( A  e.  (SubRing `  R )  ->  R  e.  Ring )
 
Theoremsubrgsubg 15513 A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
 |-  ( A  e.  (SubRing `  R )  ->  A  e.  (SubGrp `  R )
 )
 
Theoremsubrg0 15514 A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  S  =  ( Rs  A )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( A  e.  (SubRing `  R )  ->  .0.  =  ( 0g `  S ) )
 
Theoremsubrg1cl 15515 A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |- 
 .1.  =  ( 1r `  R )   =>    |-  ( A  e.  (SubRing `  R )  ->  .1.  e.  A )
 
Theoremsubrgbas 15516 Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  S  =  ( Rs  A )   =>    |-  ( A  e.  (SubRing `  R )  ->  A  =  ( Base `  S )
 )
 
Theoremsubrg1 15517 A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  S  =  ( Rs  A )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( A  e.  (SubRing `  R )  ->  .1.  =  ( 1r `  S ) )
 
Theoremsubrgacl 15518 A subring is closed under addition. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- 
 .+  =  ( +g  `  R )   =>    |-  ( ( A  e.  (SubRing `  R )  /\  X  e.  A  /\  Y  e.  A )  ->  ( X  .+  Y )  e.  A )
 
Theoremsubrgmcl 15519 A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- 
 .x.  =  ( .r `  R )   =>    |-  ( ( A  e.  (SubRing `  R )  /\  X  e.  A  /\  Y  e.  A )  ->  ( X  .x.  Y )  e.  A )
 
Theoremsubrgsubm 15520 A subring is a submonoid of the multiplicative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  M  =  (mulGrp `  R )   =>    |-  ( A  e.  (SubRing `  R )  ->  A  e.  (SubMnd `  M )
 )
 
Theoremsubrgdvds 15521 If an element divides another in a subring, then it also divides the other in the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  S  =  ( Rs  A )   &    |-  .||  =  ( ||r `  R )   &    |-  E  =  ( ||r `  S )   =>    |-  ( A  e.  (SubRing `  R )  ->  E  C_  .||  )
 
Theoremsubrguss 15522 A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  S  =  ( Rs  A )   &    |-  U  =  (Unit `  R )   &    |-  V  =  (Unit `  S )   =>    |-  ( A  e.  (SubRing `  R )  ->  V  C_  U )
 
Theoremsubrginv 15523 A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  S  =  ( Rs  A )   &    |-  I  =  (
 invr `  R )   &    |-  U  =  (Unit `  S )   &    |-  J  =  ( invr `  S )   =>    |-  (
 ( A  e.  (SubRing `  R )  /\  X  e.  U )  ->  ( I `  X )  =  ( J `  X ) )
 
Theoremsubrgdv 15524 A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  S  =  ( Rs  A )   &    |-  ./  =  (/r `  R )   &    |-  U  =  (Unit `  S )   &    |-  E  =  (/r `  S )   =>    |-  ( ( A  e.  (SubRing `  R )  /\  X  e.  A  /\  Y  e.  U )  ->  ( X  ./  Y )  =  ( X E Y ) )
 
Theoremsubrgunit 15525 An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  S  =  ( Rs  A )   &    |-  U  =  (Unit `  R )   &    |-  V  =  (Unit `  S )   &    |-  I  =  (
 invr `  R )   =>    |-  ( A  e.  (SubRing `  R )  ->  ( X  e.  V  <->  ( X  e.  U  /\  X  e.  A  /\  ( I `  X )  e.  A ) ) )
 
Theoremsubrgugrp 15526 The units of a subring form a subgroup of the unit group of the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  S  =  ( Rs  A )   &    |-  U  =  (Unit `  R )   &    |-  V  =  (Unit `  S )   &    |-  G  =  ( (mulGrp `  R )s  U )   =>    |-  ( A  e.  (SubRing `  R )  ->  V  e.  (SubGrp `  G )
 )
 
Theoremissubrg2 15527* Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( R  e.  Ring  ->  ( A  e.  (SubRing `  R )  <->  ( A  e.  (SubGrp `  R )  /\  .1.  e.  A  /\  A. x  e.  A  A. y  e.  A  ( x  .x.  y )  e.  A ) ) )
 
Theoremopprsubrg 15528 Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
 |-  O  =  (oppr `  R )   =>    |-  (SubRing `  R )  =  (SubRing `  O )
 
Theoremsubrgint 15529 The intersection of a nonempty collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
 |-  ( ( S  C_  (SubRing `  R )  /\  S  =/=  (/) )  ->  |^| S  e.  (SubRing `  R )
 )
 
Theoremsubrgin 15530 The intersection of two subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
 |-  ( ( A  e.  (SubRing `  R )  /\  B  e.  (SubRing `  R ) )  ->  ( A  i^i  B )  e.  (SubRing `  R )
 )
 
Theoremsubrgmre 15531 The subrings of a ring are a Moore system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  B  =  ( Base `  R )   =>    |-  ( R  e.  Ring  ->  (SubRing `  R )  e.  (Moore `  B )
 )
 
Theoremissubdrg 15532* Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)
 |-  S  =  ( Rs  A )   &    |-  .0.  =  ( 0g `  R )   &    |-  I  =  ( invr `  R )   =>    |-  ( ( R  e.  DivRing  /\  A  e.  (SubRing `  R ) )  ->  ( S  e.  DivRing 
 <-> 
 A. x  e.  ( A  \  {  .0.  }
 ) ( I `  x )  e.  A ) )
 
Theoremsubsubrg 15533 A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  S  =  ( Rs  A )   =>    |-  ( A  e.  (SubRing `  R )  ->  ( B  e.  (SubRing `  S ) 
 <->  ( B  e.  (SubRing `  R )  /\  B  C_  A ) ) )
 
Theoremsubsubrg2 15534 The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  S  =  ( Rs  A )   =>    |-  ( A  e.  (SubRing `  R )  ->  (SubRing `  S )  =  ( (SubRing `  R )  i^i  ~P A ) )
 
Theoremissubrg3 15535 A subring is an additive subgroup which is also a multiplicative submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  M  =  (mulGrp `  R )   =>    |-  ( R  e.  Ring  ->  ( S  e.  (SubRing `  R )  <->  ( S  e.  (SubGrp `  R )  /\  S  e.  (SubMnd `  M ) ) ) )
 
Theoremresrhm 15536 Restriction of a ring homomorphism to a subring is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  U  =  ( Ss  X )   =>    |-  ( ( F  e.  ( S RingHom  T )  /\  X  e.  (SubRing `  S ) )  ->  ( F  |`  X )  e.  ( U RingHom  T ) )
 
Theoremrhmeql 15537 The equalizer of two ring homomorphisms is a subring. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  ( ( F  e.  ( S RingHom  T )  /\  G  e.  ( S RingHom  T ) )  ->  dom  (  F  i^i  G )  e.  (SubRing `  S )
 )
 
Theoremrhmima 15538 The homomorphic image of a subring is a subring. (Contributed by Stefan O'Rear, 10-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  ( ( F  e.  ( M RingHom  N )  /\  X  e.  (SubRing `  M ) )  ->  ( F
 " X )  e.  (SubRing `  N )
 )
 
Theoremcntzsubr 15539 Centralizers in a ring are subrings. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  R )   &    |-  M  =  (mulGrp `  R )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( ( R  e.  Ring  /\  S  C_  B )  ->  ( Z `  S )  e.  (SubRing `  R ) )
 
Theorempwsdiagrhm 15540* Diagonal homomorphism into a structure power (Rings). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  R )   &    |-  F  =  ( x  e.  B  |->  ( I  X.  { x } ) )   =>    |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  F  e.  ( R RingHom  Y )
 )
 
Theoremsubrgpropd 15541* If two structures have the same group components (properties), they have the same set of subrings. (Contributed by Mario Carneiro, 9-Feb-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  ->  (SubRing `  K )  =  (SubRing `  L ) )
 
Theoremrhmpropd 15542* Ring homomorphism depends only on the ring attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  J )
 )   &    |-  ( ph  ->  C  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  C  =  ( Base `  M )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  J )
 y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  C )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  M ) y ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  J ) y )  =  ( x ( .r `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  C ) )  ->  ( x ( .r `  K ) y )  =  ( x ( .r
 `  M ) y ) )   =>    |-  ( ph  ->  ( J RingHom  K )  =  ( L RingHom  M ) )
 
10.5.3  Absolute value (abstract algebra)
 
Syntaxcabv 15543 The set of absolute values on a ring.
 class AbsVal
 
Definitiondf-abv 15544* Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 11686 to arbitrary rings. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |- AbsVal  =  ( r  e.  Ring  |->  { f  e.  ( ( 0 [,)  +oo )  ^m  ( Base `  r )
 )  |  A. x  e.  ( Base `  r )
 ( ( ( f `
  x )  =  0  <->  x  =  ( 0g `  r ) ) 
 /\  A. y  e.  ( Base `  r ) ( ( f `  ( x ( .r `  r ) y ) )  =  ( ( f `  x )  x.  ( f `  y ) )  /\  ( f `  ( x ( +g  `  r
 ) y ) ) 
 <_  ( ( f `  x )  +  (
 f `  y )
 ) ) ) }
 )
 
Theoremabvfval 15545* Value of the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  A  =  { f  e.  ( ( 0 [,)  +oo )  ^m  B )  |  A. x  e.  B  ( ( ( f `  x )  =  0  <->  x  =  .0.  )  /\  A. y  e.  B  ( ( f `
  ( x  .x.  y ) )  =  ( ( f `  x )  x.  (
 f `  y )
 )  /\  ( f `  ( x  .+  y
 ) )  <_  (
 ( f `  x )  +  ( f `  y ) ) ) ) } )
 
Theoremisabv 15546* Elementhood in the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  ( F  e.  A  <->  ( F : B --> ( 0 [,)  +oo )  /\  A. x  e.  B  (
 ( ( F `  x )  =  0  <->  x  =  .0.  )  /\  A. y  e.  B  ( ( F `  ( x  .x.  y ) )  =  ( ( F `
  x )  x.  ( F `  y
 ) )  /\  ( F `  ( x  .+  y ) )  <_  ( ( F `  x )  +  ( F `  y ) ) ) ) ) ) )
 
Theoremisabvd 15547* Properties that determine an absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 4-Dec-2014.)
 |-  ( ph  ->  A  =  (AbsVal `  R )
 )   &    |-  ( ph  ->  B  =  ( Base `  R )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  R )
 )   &    |-  ( ph  ->  .x.  =  ( .r `  R ) )   &    |-  ( ph  ->  .0. 
 =  ( 0g `  R ) )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  F : B
 --> RR )   &    |-  ( ph  ->  ( F `  .0.  )  =  0 )   &    |-  (
 ( ph  /\  x  e.  B  /\  x  =/= 
 .0.  )  ->  0  <  ( F `  x ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  ( y  e.  B  /\  y  =/=  .0.  )
 )  ->  ( F `  ( x  .x.  y
 ) )  =  ( ( F `  x )  x.  ( F `  y ) ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  ( y  e.  B  /\  y  =/=  .0.  )
 )  ->  ( F `  ( x  .+  y
 ) )  <_  (
 ( F `  x )  +  ( F `  y ) ) )   =>    |-  ( ph  ->  F  e.  A )
 
Theoremabvrcl 15548 Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   =>    |-  ( F  e.  A  ->  R  e.  Ring )
 
Theoremabvfge0 15549 An absolute value is a function from the ring to the nonnegative real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( F  e.  A  ->  F : B --> ( 0 [,)  +oo ) )
 
Theoremabvf 15550 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( F  e.  A  ->  F : B --> RR )
 
Theoremabvcl 15551 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( F `  X )  e.  RR )
 
Theoremabvge0 15552 The absolute value of a number is greater or equal to zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( ( F  e.  A  /\  X  e.  B )  ->  0  <_  ( F `  X ) )
 
Theoremabveq0 15553 The value of an absolute value is zero iff the argument is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( ( F `
  X )  =  0  <->  X  =  .0.  ) )
 
Theoremabvne0 15554 The absolute value of a nonzero number is nonzero. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( F  e.  A  /\  X  e.  B  /\  X  =/=  .0.  )  ->  ( F `  X )  =/=  0 )
 
Theoremabvgt0 15555 The absolute value of a nonzero number is strictly positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( F  e.  A  /\  X  e.  B  /\  X  =/=  .0.  )  ->  0  <  ( F `
  X ) )
 
Theoremabvmul 15556 An absolute value distributes under multiplication. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( F  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `
  ( X  .x.  Y ) )  =  ( ( F `  X )  x.  ( F `  Y ) ) )
 
Theoremabvtri 15557 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   =>    |-  ( ( F  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `
  ( X  .+  Y ) )  <_  ( ( F `  X )  +  ( F `  Y ) ) )
 
Theoremabv0 15558 The absolute value of zero is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( F  e.  A  ->  ( F `  .0.  )  =  0 )
 
Theoremabv1z 15559 The absolute value of one is one in a non-trivial ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( F  e.  A  /\  .1.  =/=  .0.  )  ->  ( F `  .1.  )  =  1 )
 
Theoremabv1 15560 The absolute value of one is one in a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  DivRing  /\  F  e.  A ) 
 ->  ( F `  .1.  )  =  1 )
 
Theoremabvneg 15561 The absolute value of a negative is the same as that of the positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  N  =  ( inv g `  R )   =>    |-  ( ( F  e.  A  /\  X  e.  B )  ->  ( F `  ( N `  X ) )  =  ( F `
  X ) )
 
Theoremabvsubtri 15562 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .-  =  ( -g `  R )   =>    |-  ( ( F  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `
  ( X  .-  Y ) )  <_  ( ( F `  X )  +  ( F `  Y ) ) )
 
Theoremabvrec 15563 The absolute value distributes under reciprocal. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  I  =  ( invr `  R )   =>    |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  X  =/=  .0.  ) )  ->  ( F `  ( I `
  X ) )  =  ( 1  /  ( F `  X ) ) )
 
Theoremabvdiv 15564 The absolute value distributes under division. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  ./  =  (/r `  R )   =>    |-  ( ( ( R  e.  DivRing  /\  F  e.  A )  /\  ( X  e.  B  /\  Y  e.  B  /\  Y  =/=  .0.  ) ) 
 ->  ( F `  ( X  ./  Y ) )  =  ( ( F `
  X )  /  ( F `  Y ) ) )
 
Theoremabvdom 15565 Any ring with an absolute value is a domain, which is to say that it contains no zero divisors. (Contributed by Mario Carneiro, 10-Sep-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( ( F  e.  A  /\  ( X  e.  B  /\  X  =/=  .0.  )  /\  ( Y  e.  B  /\  Y  =/=  .0.  ) )  ->  ( X 
 .x.  Y )  =/=  .0.  )
 
Theoremabvres 15566 The restriction of an absolute value to a subring is an absolute value. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  A  =  (AbsVal `  R )   &    |-  S  =  ( Rs  C )   &    |-  B  =  (AbsVal `  S )   =>    |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  ->  ( F  |`  C )  e.  B )
 
Theoremabvtrivd 15567* The trivial absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  F  =  ( x  e.  B  |->  if ( x  =  .0.  ,  0 ,  1 ) )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ( ph  /\  (
 y  e.  B  /\  y  =/=  .0.  )  /\  ( z  e.  B  /\  z  =/=  .0.  )
 )  ->  ( y  .x.  z )  =/=  .0.  )   =>    |-  ( ph  ->  F  e.  A )
 
Theoremabvtriv 15568* The trivial absolute value. (This theorem is true as long as  R is a domain, but it is not true for rings with zero divisors, which violate the multiplication axiom; abvdom 15565 is the converse of this remark.) (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  A  =  (AbsVal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  F  =  ( x  e.  B  |->  if ( x  =  .0.  ,  0 ,  1 ) )   =>    |-  ( R  e.  DivRing  ->  F  e.  A )
 
Theoremabvpropd 15569* If two structures have the same ring components, they have the same collection of absolute values. (Contributed by Mario Carneiro, 4-Oct-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  ->  (AbsVal `  K )  =  (AbsVal `  L ) )
 
10.5.4  Star rings
 
Syntaxcstf 15570 Extend class notation with the functionalization of the *-ring involution.
 class  * r f
 
Syntaxcsr 15571 Extend class notation with class of all *-rings.
 class  *Ring
 
Definitiondf-staf 15572* Define the functionalization of the involution in a star ring. This is not strictly necessary but by having  * r as an actual function we can state the principal properties of an involution much more cleanly. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  * r f  =  ( f  e.  _V  |->  ( x  e.  ( Base `  f )  |->  ( ( * r `  f ) `  x ) ) )
 
Definitiondf-srng 15573* Define class of all star rings. A star ring is a ring with an involution (conjugation) function. Involution (unlike say the ring zero) is not unique and therefore must be added as a new component to the ring. For example, two possible involutions for complex numbers are the identity function and complex conjugation. Definition of involution in [Holland95] p. 204. (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
 |-  *Ring  =  { f  | 
 [. ( * r f `  f ) 
 /  i ]. (
 i  e.  ( f RingHom  (oppr `  f ) )  /\  i  =  `' i
 ) }
 
Theoremstaffval 15574* The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  B  =  ( Base `  R )   &    |-  .*  =  ( * r `  R )   &    |-  .xb  =  ( * r f `  R )   =>    |-  .xb 
 =  ( x  e.  B  |->  (  .*  `  x ) )
 
Theoremstafval 15575 The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  B  =  ( Base `  R )   &    |-  .*  =  ( * r `  R )   &    |-  .xb  =  ( * r f `  R )   =>    |-  ( A  e.  B  ->  (  .xb  `  A )  =  (  .*  `  A ) )
 
Theoremstaffn 15576 The functionalization is equal to the original function, if it is a function on the right base set. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  B  =  ( Base `  R )   &    |-  .*  =  ( * r `  R )   &    |-  .xb  =  ( * r f `  R )   =>    |-  (  .*  Fn  B  ->  .xb 
 =  .*  )
 
Theoremissrng 15577 The predicate "is a star ring." (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
 |-  O  =  (oppr `  R )   &    |- 
 .*  =  ( * r f `  R )   =>    |-  ( R  e.  *Ring  <->  (  .*  e.  ( R RingHom  O ) 
 /\  .*  =  `'  .*  ) )
 
Theoremsrngrhm 15578 The involution function in a star ring is an antiautomorphism. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  O  =  (oppr `  R )   &    |- 
 .*  =  ( * r f `  R )   =>    |-  ( R  e.  *Ring  ->  .*  e.  ( R RingHom  O ) )
 
Theoremsrngrng 15579 A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( R  e.  *Ring  ->  R  e.  Ring )
 
Theoremsrngcnv 15580 The involution function in a star ring is its own inverse function. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |- 
 .*  =  ( * r f `  R )   =>    |-  ( R  e.  *Ring  ->  .*  =  `'  .*  )
 
Theoremsrngf1o 15581 The involution function in a star ring is a bijection. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |- 
 .*  =  ( * r f `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( R  e.  *Ring  ->  .*  : B -1-1-onto-> B )
 
Theoremsrngcl 15582 The involution function in a star ring is closed in the ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |- 
 .*  =  ( * r `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( ( R  e.  *Ring  /\  X  e.  B ) 
 ->  (  .*  `  X )  e.  B )
 
Theoremsrngnvl 15583 The involution function in a star ring is an involution. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |- 
 .*  =  ( * r `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( ( R  e.  *Ring  /\  X  e.  B ) 
 ->  (  .*  `  (  .*  `  X ) )  =  X )
 
Theoremsrngadd 15584 The involution function in a star ring distributes over addition. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |- 
 .*  =  ( * r `  R )   &    |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   =>    |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (  .*  `  ( X  .+  Y ) )  =  ( (  .*  `  X )  .+  (  .*  `  Y ) ) )
 
Theoremsrngmul 15585 The involution function in a star ring distributes over multiplication, with a change in the order of the factors. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |- 
 .*  =  ( * r `  R )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  *Ring  /\  X  e.  B  /\  Y  e.  B )  ->  (  .*  `  ( X  .x.  Y ) )  =  ( (  .*  `  Y )  .x.  (  .*  `  X ) ) )
 
Theoremsrng1 15586 The conjugate of the ring identity is the identity. (This is sometimes taken as an axiom, and indeed the proof here follows because we defined  * r to be a ring homomorphism, which preserves 1; nevertheless, it is redundant, as can be seen from the proof of issrngd 15588.) (Contributed by Mario Carneiro, 6-Oct-2015.)
 |- 
 .*  =  ( * r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  *Ring  ->  (  .*  `  .1.  )  =  .1.  )
 
Theoremsrng0 15587 The conjugate of the ring zero is zero. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |- 
 .*  =  ( * r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  *Ring  ->  (  .*  `  .0.  )  =  .0.  )
 
Theoremissrngd 15588* Properties that determine a star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2015.)
 |-  ( ph  ->  K  =  ( Base `  R )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  R )
 )   &    |-  ( ph  ->  .x.  =  ( .r `  R ) )   &    |-  ( ph  ->  .*  =  ( * r `
  R ) )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ( ph  /\  x  e.  K )  ->  (  .*  `  x )  e.  K )   &    |-  ( ( ph  /\  x  e.  K  /\  y  e.  K )  ->  (  .*  `  ( x  .+  y ) )  =  ( (  .*  `  x )  .+  (  .*  `  y ) ) )   &    |-  ( ( ph  /\  x  e.  K  /\  y  e.  K )  ->  (  .*  `  ( x  .x.  y ) )  =  ( (  .*  `  y )  .x.  (  .*  `  x ) ) )   &    |-  ( ( ph  /\  x  e.  K ) 
 ->  (  .*  `  (  .*  `  x ) )  =  x )   =>    |-  ( ph  ->  R  e.  *Ring )
 
10.6  Left Modules
 
10.6.1  Definition and basic properties
 
Syntaxclmod 15589 Extend class notation with class of all left modules.
 class  LMod
 
Syntaxcscaf 15590 The functionalization of the scalar multiplication operation.
 class  .s f
 
Definitiondf-lmod 15591* Define the class of all left modules, which are generalizations of left vector spaces. A left module over a ring is an (Abelian) group (vectors) together with a ring (scalars) and a left scalar product connecting them. (Contributed by NM, 4-Nov-2013.)
 |- 
 LMod  =  { g  e.  Grp  |  [. ( Base `  g )  /  v ]. [. ( +g  `  g )  /  a ]. [. (Scalar `  g
 )  /  f ]. [. ( .s `  g
 )  /  s ]. [. ( Base `  f )  /  k ]. [. ( +g  `  f )  /  p ]. [. ( .r
 `  f )  /  t ]. ( f  e. 
 Ring  /\  A. q  e.  k  A. r  e.  k  A. x  e.  v  A. w  e.  v  ( ( ( r s w )  e.  v  /\  (
 r s ( w a x ) )  =  ( ( r s w ) a ( r s x ) )  /\  (
 ( q p r ) s w )  =  ( ( q s w ) a ( r s w ) ) )  /\  ( ( ( q t r ) s w )  =  ( q s ( r s w ) ) 
 /\  ( ( 1r
 `  f ) s w )  =  w ) ) ) }
 
Definitiondf-scaf 15592* Define the functionalization of the 
.s operator. This restricts the value of  .s to the stated domain, which is necessary when working with restricted structures, whose operations may be defined on a larger set than the true base. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |- 
 .s f  =  ( g  e.  _V  |->  ( x  e.  ( Base `  (Scalar `  g )
 ) ,  y  e.  ( Base `  g )  |->  ( x ( .s
 `  g ) y ) ) )
 
Theoremislmod 15593* The predicate "is a left module". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .+^  =  ( +g  `  F )   &    |-  .X.  =  ( .r `  F )   &    |-  .1.  =  ( 1r `  F )   =>    |-  ( W  e.  LMod  <->  ( W  e.  Grp  /\  F  e.  Ring  /\  A. q  e.  K  A. r  e.  K  A. x  e.  V  A. w  e.  V  ( ( ( r  .x.  w )  e.  V  /\  ( r 
 .x.  ( w  .+  x ) )  =  ( ( r  .x.  w )  .+  ( r 
 .x.  x ) ) 
 /\  ( ( q  .+^  r )  .x.  w )  =  ( (
 q  .x.  w )  .+  ( r  .x.  w ) ) )  /\  ( ( ( q 
 .X.  r )  .x.  w )  =  (
 q  .x.  ( r  .x.  w ) )  /\  (  .1.  .x.  w )  =  w ) ) ) )
 
Theoremlmodlema 15594 Lemma for properties of a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .+^  =  ( +g  `  F )   &    |-  .X.  =  ( .r `  F )   &    |-  .1.  =  ( 1r `  F )   =>    |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K ) 
 /\  ( X  e.  V  /\  Y  e.  V ) )  ->  ( ( ( R  .x.  Y )  e.  V  /\  ( R  .x.  ( Y 
 .+  X ) )  =  ( ( R 
 .x.  Y )  .+  ( R  .x.  X ) ) 
 /\  ( ( Q  .+^  R )  .x.  Y )  =  ( ( Q  .x.  Y )  .+  ( R  .x.  Y ) ) )  /\  (
 ( ( Q  .X.  R )  .x.  Y )  =  ( Q  .x.  ( R  .x.  Y ) ) 
 /\  (  .1.  .x.  Y )  =  Y ) ) )
 
Theoremislmodd 15595* Properties that determine a left module. See note in isgrpd2 14467 regarding the  ph on hypotheses that name structure components. (Contributed by Mario Carneiro, 22-Jun-2014.)
 |-  ( ph  ->  V  =  ( Base `  W )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  W )
 )   &    |-  ( ph  ->  F  =  (Scalar `  W )
 )   &    |-  ( ph  ->  .x.  =  ( .s `  W ) )   &    |-  ( ph  ->  B  =  ( Base `  F ) )   &    |-  ( ph  ->  .+^  =  ( +g  `  F ) )   &    |-  ( ph  ->  .X. 
 =  ( .r `  F ) )   &    |-  ( ph  ->  .1.  =  ( 1r `  F ) )   &    |-  ( ph  ->  F  e.  Ring
 )   &    |-  ( ph  ->  W  e.  Grp )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  V )  ->  ( x  .x.  y
 )  e.  V )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  V  /\  z  e.  V )
 )  ->  ( x  .x.  ( y  .+  z
 ) )  =  ( ( x  .x.  y
 )  .+  ( x  .x.  z ) ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  V )
 )  ->  ( ( x  .+^  y )  .x.  z )  =  (
 ( x  .x.  z
 )  .+  ( y  .x.  z ) ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  V )
 )  ->  ( ( x  .X.  y )  .x.  z )  =  ( x  .x.  ( y  .x.  z ) ) )   &    |-  ( ( ph  /\  x  e.  V )  ->  (  .1.  .x.  x )  =  x )   =>    |-  ( ph  ->  W  e.  LMod )
 
Theoremlmodgrp 15596 A left module is a group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)
 |-  ( W  e.  LMod  ->  W  e.  Grp )
 
Theoremlmodrng 15597 The scalar component of a left module is a ring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  LMod  ->  F  e.  Ring )
 
Theoremlmodfgrp 15598 The scalar component of a left module is an additive group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  LMod  ->  F  e.  Grp )
 
Theoremlmodbn0 15599 The base set of a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  B  =  ( Base `  W )   =>    |-  ( W  e.  LMod  ->  B  =/=  (/) )
 
Theoremlmodacl 15600 Closure of ring addition for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .+  =  ( +g  `  F )   =>    |-  ( ( W  e.  LMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y )  e.  K )
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