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Theorem List for Metamath Proof Explorer - 15601-15700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremabvrec 15601 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 15602 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 15603 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 15604 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 15605* 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 15606* 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 15603 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 15607* 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 15608 Extend class notation with the functionalization of the *-ring involution.
 class  * r f
 
Syntaxcsr 15609 Extend class notation with class of all *-rings.
 class  *Ring
 
Definitiondf-staf 15610* 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 15611* 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 15612* 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 15613 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 15614 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 15615 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 15616 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 15617 A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( R  e.  *Ring  ->  R  e.  Ring )
 
Theoremsrngcnv 15618 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 15619 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 15620 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 15621 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 15622 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 15623 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 15624 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 15626.) (Contributed by Mario Carneiro, 6-Oct-2015.)
 |- 
 .*  =  ( * r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  *Ring  ->  (  .*  `  .1.  )  =  .1.  )
 
Theoremsrng0 15625 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 15626* 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 15627 Extend class notation with class of all left modules.
 class  LMod
 
Syntaxcscaf 15628 The functionalization of the scalar multiplication operation.
 class  .s f
 
Definitiondf-lmod 15629* 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 15630* 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 15631* 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 15632 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 15633* Properties that determine a left module. See note in isgrpd2 14505 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 15634 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 15635 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 15636 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 15637 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 15638 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 )
 
Theoremlmodmcl 15639 Closure of ring multiplication for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .r `  F )   =>    |-  ( ( W  e.  LMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .x.  Y )  e.  K )
 
Theoremlmodsn0 15640 The set of scalars in a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  F  =  (Scalar `  W )   &    |-  B  =  ( Base `  F )   =>    |-  ( W  e.  LMod  ->  B  =/=  (/) )
 
Theoremlmodvacl 15641 Closure of vector addition for a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .+  Y )  e.  V )
 
Theoremlmodass 15642 Left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   =>    |-  ( ( W  e.  LMod  /\  ( X  e.  V  /\  Y  e.  V  /\  Z  e.  V ) )  ->  ( ( X  .+  Y )  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) )
 
Theoremlmodlcan 15643 Left cancellation law for vector sum. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   =>    |-  ( ( W  e.  LMod  /\  ( X  e.  V  /\  Y  e.  V  /\  Z  e.  V ) )  ->  ( ( Z  .+  X )  =  ( Z  .+  Y )  <->  X  =  Y ) )
 
Theoremlmodvscl 15644 Closure of scalar product for a left module. (hvmulcl 21593 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   =>    |-  (
 ( W  e.  LMod  /\  R  e.  K  /\  X  e.  V )  ->  ( R  .x.  X )  e.  V )
 
Theoremscaffval 15645* The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  B  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  (
 Base `  F )   &    |-  .xb  =  ( .s f `  W )   &    |- 
 .x.  =  ( .s `  W )   =>    |-  .xb  =  ( x  e.  K ,  y  e.  B  |->  ( x  .x.  y ) )
 
Theoremscafval 15646 The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  B  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  (
 Base `  F )   &    |-  .xb  =  ( .s f `  W )   &    |- 
 .x.  =  ( .s `  W )   =>    |-  ( ( X  e.  K  /\  Y  e.  B )  ->  ( X  .xb  Y )  =  ( X 
 .x.  Y ) )
 
Theoremscafeq 15647 If the scalar multiplication operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  B  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  (
 Base `  F )   &    |-  .xb  =  ( .s f `  W )   &    |- 
 .x.  =  ( .s `  W )   =>    |-  (  .x.  Fn  ( K  X.  B )  ->  .xb 
 =  .x.  )
 
Theoremscaffn 15648 The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  B  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  (
 Base `  F )   &    |-  .xb  =  ( .s f `  W )   =>    |-  .xb  Fn  ( K  X.  B )
 
Theoremlmodscaf 15649 The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  B  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  (
 Base `  F )   &    |-  .xb  =  ( .s f `  W )   =>    |-  ( W  e.  LMod  ->  .xb 
 : ( K  X.  B ) --> B )
 
Theoremlmodvsdi 15650 Distributive law for scalar product. (ax-hvdistr1 21588 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e.  LMod  /\  ( R  e.  K  /\  X  e.  V  /\  Y  e.  V )
 )  ->  ( R  .x.  ( X  .+  Y ) )  =  (
 ( R  .x.  X )  .+  ( R  .x.  Y ) ) )
 
Theoremlmodvsdi1OLD 15651 Obsolete version of lmodvsdi 15650 as of 22-Sep-2015. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e.  LMod  /\  R  e.  K  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  ( R  .x.  ( X  .+  Y ) )  =  (
 ( R  .x.  X )  .+  ( R  .x.  Y ) ) )
 
Theoremlmodvsdir 15652 Distributive law for scalar product. (ax-hvdistr1 21588 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   &    |-  .+^  =  ( +g  `  F )   =>    |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K  /\  X  e.  V )
 )  ->  ( ( Q  .+^  R )  .x.  X )  =  ( ( Q  .x.  X )  .+  ( R  .x.  X ) ) )
 
Theoremlmodvsdi2OLD 15653 Obsolete version of lmodvsdir 15652 as of 22-Sep-2015. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   &    |-  .+^  =  ( +g  `  F )   =>    |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K ) 
 /\  X  e.  V )  ->  ( ( Q  .+^  R )  .x.  X )  =  ( ( Q  .x.  X )  .+  ( R  .x.  X ) ) )
 
Theoremlmodvsass 15654 Associative law for scalar product. (ax-hvmulass 21587 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   &    |-  .X.  =  ( .r `  F )   =>    |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K  /\  X  e.  V )
 )  ->  ( ( Q  .X.  R )  .x.  X )  =  ( Q 
 .x.  ( R  .x.  X ) ) )
 
TheoremlmodvsassOLD 15655 Obsolete version of lmodvsass 15654 as of 22-Sep-2015. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   &    |-  .X.  =  ( .r `  F )   =>    |-  ( ( W  e.  LMod  /\  ( Q  e.  K  /\  R  e.  K ) 
 /\  X  e.  V )  ->  ( ( Q 
 .X.  R )  .x.  X )  =  ( Q  .x.  ( R  .x.  X ) ) )
 
Theoremlmod0cl 15656 The ring zero in a left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .0.  =  ( 0g `  F )   =>    |-  ( W  e.  LMod  ->  .0. 
 e.  K )
 
Theoremlmod1cl 15657 The ring unit in a left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .1.  =  ( 1r `  F )   =>    |-  ( W  e.  LMod  ->  .1. 
 e.  K )
 
Theoremlmodvs1 15658 Scalar product with ring unit. (ax-hvmulid 21586 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  .1.  =  ( 1r `  F )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V ) 
 ->  (  .1.  .x.  X )  =  X )
 
Theoremlmod0vcl 15659 The zero vector is a vector. (ax-hv0cl 21583 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( W  e.  LMod  ->  .0. 
 e.  V )
 
Theoremlmod0vlid 15660 Left identity law for the zero vector. (hvaddid2 21602 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V ) 
 ->  (  .0.  .+  X )  =  X )
 
Theoremlmod0vrid 15661 Right identity law for the zero vector. (ax-hvaddid 21584 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( X  .+  .0.  )  =  X )
 
Theoremlmod0vid 15662 Identity equivalent to the value of the zero vector. Provides a convenient way to compute the value. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( ( X  .+  X )  =  X  <->  .0. 
 =  X ) )
 
Theoremlmod0vs 15663 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 21590 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  O  =  ( 0g `  F )   &    |- 
 .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( O  .x.  X )  =  .0.  )
 
Theoremlmodvs0 15664 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 21603 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  K  =  (
 Base `  F )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  K ) 
 ->  ( X  .x.  .0.  )  =  .0.  )
 
Theoremlmodvnegcl 15665 Closure of vector negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( inv g `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( N `  X )  e.  V )
 
Theoremlmodvnegid 15666 Addition of a vector with its negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( inv
 g `  W )   =>    |-  (
 ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( X  .+  ( N `  X ) )  =  .0.  )
 
Theoremlmodvneg1 15667 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( inv g `  W )   &    |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  .1.  =  ( 1r `  F )   &    |-  M  =  ( inv g `
  F )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( ( M `  .1.  )  .x.  X )  =  ( N `  X ) )
 
TheoremlmodvsnegOLD 15668 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  B  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( inv g `  W )   &    |-  M  =  ( inv g `  F )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  ( ( M `  R )  .x.  X )  =  ( N `  ( R  .x.  X ) ) )
 
Theoremlmodvsneg 15669 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( inv g `  W )   &    |-  K  =  (
 Base `  F )   &    |-  M  =  ( inv g `  F )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  R  e.  K )   =>    |-  ( ph  ->  ( N `  ( R  .x.  X ) )  =  ( ( M `  R )  .x.  X ) )
 
Theoremlmodvsubcl 15670 Closure of vector subtraction. (hvsubcl 21597 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .-  Y )  e.  V )
 
Theoremlmodcom 15671 Left module vector sum is commutative. (Contributed by Gérard Lang, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremlmodabl 15672 A left module is an abelian group (of vectors, under addition). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)
 |-  ( W  e.  LMod  ->  W  e.  Abel )
 
Theoremlmodcmn 15673 A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
 |-  ( W  e.  LMod  ->  W  e. CMnd )
 
Theoremlmodnegadd 15674 Distribute negation through addition of scalar products. (Contributed by NM, 9-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( inv g `
  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  I  =  ( inv g `  R )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( N `  ( ( A  .x.  X )  .+  ( B  .x.  Y ) ) )  =  ( ( ( I `
  A )  .x.  X )  .+  ( ( I `  B ) 
 .x.  Y ) ) )
 
Theoremlmod4 15675 Commutative/associative law for left module vector sum. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   =>    |-  ( ( W  e.  LMod  /\  ( X  e.  V  /\  Y  e.  V )  /\  ( Z  e.  V  /\  U  e.  V )
 )  ->  ( ( X  .+  Y )  .+  ( Z  .+  U ) )  =  ( ( X  .+  Z ) 
 .+  ( Y  .+  U ) ) )
 
Theoremlmodvsubadd 15676 Relationship between vector subtraction and addition. (hvsubadd 21656 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  LMod  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( ( A  .-  B )  =  C  <->  ( B  .+  C )  =  A ) )
 
Theoremlmodvaddsub4 15677 Vector addition/subtraction law. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  LMod  /\  ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) )  ->  ( ( A  .+  B )  =  ( C  .+  D )  <->  ( A  .-  C )  =  ( D  .-  B ) ) )
 
Theoremlmodvpncan 15678 Addition/subtraction cancellation law for vectors. (hvpncan 21618 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A  .+  B )  .-  B )  =  A )
 
Theoremlmodvnpcan 15679 Cancellation law for vector subtraction (npcan 9060 analog). (Contributed by NM, 19-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A  .-  B )  .+  B )  =  A )
 
Theoremlmodvsubval2 15680 Value of vector subtraction in terms of addition. (hvsubval 21596 analog.) (Contributed by NM, 31-Mar-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   &    |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  N  =  ( inv g `  F )   &    |- 
 .1.  =  ( 1r `  F )   =>    |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .-  B )  =  ( A  .+  ( ( N `  .1.  )  .x.  B )
 ) )
 
Theoremlmodsubvs 15681 Subtraction of a scalar product in terms of addition. (Contributed by NM, 9-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   &    |-  .x. 
 =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  (
 Base `  F )   &    |-  N  =  ( inv g `  F )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( X  .-  ( A  .x.  Y ) )  =  ( X  .+  ( ( N `  A ) 
 .x.  Y ) ) )
 
Theoremlmodsubdi 15682 Scalar multiplication distributive law for subtraction. (hvsubdistr1 21628 analog, with longer proof since our scalar multiplication is not commutative.) (Contributed by NM, 2-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .-  =  ( -g `  W )   &    |-  ( ph  ->  W  e.  LMod
 )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( A  .x.  ( X  .-  Y ) )  =  ( ( A  .x.  X )  .-  ( A  .x.  Y ) ) )
 
Theoremlmodsubdir 15683 Scalar multiplication distributive law for subtraction. (hvsubdistr2 21629 analog.) (Contributed by NM, 2-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .-  =  ( -g `  W )   &    |-  S  =  ( -g `  F )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  (
 ( A S B )  .x.  X )  =  ( ( A  .x.  X )  .-  ( B  .x.  X ) ) )
 
Theoremlmodsubeq0 15684 If the difference between two vectors is zero, they are equal. (hvsubeq0 21647 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A  .-  B )  =  .0.  <->  A  =  B ) )
 
Theoremlmodsubid 15685 Subtraction of a vector from itself. (hvsubid 21605 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  LMod  /\  A  e.  V ) 
 ->  ( A  .-  A )  =  .0.  )
 
Theoremlmodvsghm 15686* Scalar multiplication of the vector space by a fixed scalar is an automorphism of the addiive group of vectors. (Contributed by Mario Carneiro, 5-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   =>    |-  (
 ( W  e.  LMod  /\  R  e.  K ) 
 ->  ( x  e.  V  |->  ( R  .x.  x ) )  e.  ( W 
 GrpHom  W ) )
 
Theoremlmodprop2d 15687* If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 15688 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  F  =  (Scalar `  K )   &    |-  G  =  (Scalar `  L )   &    |-  ( ph  ->  P  =  ( Base `  F )
 )   &    |-  ( ph  ->  P  =  ( Base `  G )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  P )
 )  ->  ( x ( +g  `  F )
 y )  =  ( x ( +g  `  G ) y ) )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  P )
 )  ->  ( x ( .r `  F ) y )  =  ( x ( .r `  G ) y ) )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  B ) )  ->  ( x ( .s `  K ) y )  =  ( x ( .s
 `  L ) y ) )   =>    |-  ( ph  ->  ( K  e.  LMod  <->  L  e.  LMod )
 )
 
Theoremlmodpropd 15688* If two structures have the same components (properties), one is a left module iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised 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  ->  F  =  (Scalar `  K ) )   &    |-  ( ph  ->  F  =  (Scalar `  L ) )   &    |-  P  =  ( Base `  F )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  B ) )  ->  ( x ( .s `  K ) y )  =  ( x ( .s
 `  L ) y ) )   =>    |-  ( ph  ->  ( K  e.  LMod  <->  L  e.  LMod )
 )
 
10.6.2  Subspaces and spans in a left module
 
Syntaxclss 15689 Extend class notation with linear subspaces of a left module or left vector space.
 class  LSubSp
 
Definitiondf-lss 15690* Define the set of linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
 |-  LSubSp  =  ( w  e. 
 _V  |->  { s  e.  ( ~P ( Base `  w )  \  { (/) } )  | 
 A. x  e.  ( Base `  (Scalar `  w ) ) A. a  e.  s  A. b  e.  s  ( ( x ( .s `  w ) a ) (
 +g  `  w )
 b )  e.  s } )
 
Theoremlssset 15691* The set of all (not necessarily closed) linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 15-Jul-2014.)
 |-  F  =  (Scalar `  W )   &    |-  B  =  ( Base `  F )   &    |-  V  =  (
 Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  S  =  ( LSubSp `  W )   =>    |-  ( W  e.  X  ->  S  =  { s  e.  ( ~P V  \  { (/) } )  | 
 A. x  e.  B  A. a  e.  s  A. b  e.  s  (
 ( x  .x.  a
 )  .+  b )  e.  s } )
 
Theoremislss 15692* The predicate "is a subspace" (of a left module or left vector space). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
 |-  F  =  (Scalar `  W )   &    |-  B  =  ( Base `  F )   &    |-  V  =  (
 Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  S  =  ( LSubSp `  W )   =>    |-  ( U  e.  S  <->  ( U  C_  V  /\  U  =/=  (/)  /\  A. x  e.  B  A. a  e.  U  A. b  e.  U  ( ( x 
 .x.  a )  .+  b )  e.  U ) )
 
Theoremislssd 15693* Properties that determine a subspace of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
 |-  ( ph  ->  F  =  (Scalar `  W )
 )   &    |-  ( ph  ->  B  =  ( Base `  F )
 )   &    |-  ( ph  ->  V  =  ( Base `  W )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  W )
 )   &    |-  ( ph  ->  .x.  =  ( .s `  W ) )   &    |-  ( ph  ->  S  =  ( LSubSp `  W ) )   &    |-  ( ph  ->  U 
 C_  V )   &    |-  ( ph  ->  U  =/=  (/) )   &    |-  (
 ( ph  /\  ( x  e.  B  /\  a  e.  U  /\  b  e.  U ) )  ->  ( ( x  .x.  a )  .+  b )  e.  U )   =>    |-  ( ph  ->  U  e.  S )
 
Theoremlssss 15694 A subspace is a set of vectors. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( U  e.  S  ->  U  C_  V )
 
Theoremlssel 15695 A subspace member is a vector. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( U  e.  S  /\  X  e.  U )  ->  X  e.  V )
 
Theoremlss1 15696 The set of vectors in a left module is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( W  e.  LMod 
 ->  V  e.  S )
 
Theoremlssuni 15697 The union of all subspaces is the vector space. (Contributed by NM, 13-Mar-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  U. S  =  V )
 
Theoremlssn0 15698 A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   =>    |-  ( U  e.  S  ->  U  =/=  (/) )
 
Theorem00lss 15699 The empty structure has no subspaces (for use with fvco4i 5597). (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (/)  =  ( LSubSp `  (/) )
 
Theoremlsscl 15700 Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
 |-  F  =  (Scalar `  W )   &    |-  B  =  ( Base `  F )   &    |-  .+  =  ( +g  `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  S  =  ( LSubSp `  W )   =>    |-  ( ( U  e.  S  /\  ( Z  e.  B  /\  X  e.  U  /\  Y  e.  U ) )  ->  ( ( Z  .x.  X )  .+  Y )  e.  U )
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