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Theorem List for Metamath Proof Explorer - 15601-15700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremabvpropd 15601* 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 15602 Extend class notation with the functionalization of the *-ring involution.
 class  * r f
 
Syntaxcsr 15603 Extend class notation with class of all *-rings.
 class  *Ring
 
Definitiondf-staf 15604* 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 15605* 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 15606* 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 15607 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 15608 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 15609 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 15610 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 15611 A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( R  e.  *Ring  ->  R  e.  Ring )
 
Theoremsrngcnv 15612 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 15613 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 15614 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 15615 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 15616 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 15617 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 15618 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 15620.) (Contributed by Mario Carneiro, 6-Oct-2015.)
 |- 
 .*  =  ( * r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  *Ring  ->  (  .*  `  .1.  )  =  .1.  )
 
Theoremsrng0 15619 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 15620* 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 15621 Extend class notation with class of all left modules.
 class  LMod
 
Syntaxcscaf 15622 The functionalization of the scalar multiplication operation.
 class  .s f
 
Definitiondf-lmod 15623* 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 15624* 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 15625* 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 15626 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 15627* Properties that determine a left module. See note in isgrpd2 14499 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 15628 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 15629 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 15630 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 15631 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 15632 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 15633 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 15634 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 15635 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 15636 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 15637 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 15638 Closure of scalar product for a left module. (hvmulcl 21585 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 15639* 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 15640 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 15641 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 15642 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 15643 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 15644 Distributive law for scalar product. (ax-hvdistr1 21580 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 15645 Obsolete version of lmodvsdi 15644 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 15646 Distributive law for scalar product. (ax-hvdistr1 21580 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 15647 Obsolete version of lmodvsdir 15646 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 15648 Associative law for scalar product. (ax-hvmulass 21579 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 15649 Obsolete version of lmodvsass 15648 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 15650 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 15651 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 15652 Scalar product with ring unit. (ax-hvmulid 21578 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 15653 The zero vector is a vector. (ax-hv0cl 21575 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 15654 Left identity law for the zero vector. (hvaddid2 21594 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 15655 Right identity law for the zero vector. (ax-hvaddid 21576 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 15656 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 15657 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 21582 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 15658 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 21595 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 15659 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 15660 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 15661 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 15662 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 15663 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 15664 Closure of vector subtraction. (hvsubcl 21589 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 15665 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 15666 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 15667 A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
 |-  ( W  e.  LMod  ->  W  e. CMnd )
 
Theoremlmodnegadd 15668 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 15669 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 15670 Relationship between vector subtraction and addition. (hvsubadd 21648 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 15671 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 15672 Addition/subtraction cancellation law for vectors. (hvpncan 21610 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 15673 Cancellation law for vector subtraction (npcan 9055 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 15674 Value of vector subtraction in terms of addition. (hvsubval 21588 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 15675 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 15676 Scalar multiplication distributive law for subtraction. (hvsubdistr1 21620 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 15677 Scalar multiplication distributive law for subtraction. (hvsubdistr2 21621 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 15678 If the difference between two vectors is zero, they are equal. (hvsubeq0 21639 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 15679 Subtraction of a vector from itself. (hvsubid 21597 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 15680* 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 15681* If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 15682 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 15682* 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 15683 Extend class notation with linear subspaces of a left module or left vector space.
 class  LSubSp
 
Definitiondf-lss 15684* 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 15685* 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 15686* 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 15687* 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 15688 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 15689 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 15690 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 15691 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 15692 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 15693 The empty structure has no subspaces (for use with fvco4i 5558). (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (/)  =  ( LSubSp `  (/) )
 
Theoremlsscl 15694 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 )
 
Theoremlssvsubcl 15695 Closure of vector subtraction in a subspace. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  .-  =  ( -g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( X  e.  U  /\  Y  e.  U )
 )  ->  ( X  .-  Y )  e.  U )
 
Theoremlssvancl1 15696 Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. TODO: notice similarity to lspindp3 15883. Can it be used along with lspsnne1 15864, lspsnne2 15865 to shorten this proof? (Contributed by NM, 14-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  -.  Y  e.  U )   =>    |-  ( ph  ->  -.  ( X  .+  Y )  e.  U )
 
Theoremlssvancl2 15697 Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. (Contributed by NM, 20-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  -.  Y  e.  U )   =>    |-  ( ph  ->  -.  ( Y  .+  X )  e.  U )
 
Theoremlss0cl 15698 The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  .0.  e.  U )
 
Theoremlsssn0 15699 The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( W  e.  LMod 
 ->  {  .0.  }  e.  S )
 
Theoremlss0ss 15700 The zero subspace is included in every subspace. (sh0le 22011 analog.) (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  S )  ->  {  .0.  } 
 C_  X )
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