HomeHome Metamath Proof Explorer
Theorem List (p. 160 of 325)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-22374)
  Hilbert Space Explorer  Hilbert Space Explorer
(22375-23897)
  Users' Mathboxes  Users' Mathboxes
(23898-32447)
 

Theorem List for Metamath Proof Explorer - 15901-16000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsrngmul 15901 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 15902 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 15904.) (Contributed by Mario Carneiro, 6-Oct-2015.)
 |- 
 .*  =  ( * r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  *Ring  ->  (  .*  `  .1.  )  =  .1.  )
 
Theoremsrng0 15903 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 15904* 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 15905 Extend class notation with class of all left modules.
 class  LMod
 
Syntaxcscaf 15906 The functionalization of the scalar multiplication operation.
 class  .s f
 
Definitiondf-lmod 15907* 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 15908* 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 15909* 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 15910 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 15911* Properties that determine a left module. See note in isgrpd2 14783 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 15912 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 15913 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 15914 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 15915 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 15916 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 15917 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 15918 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 15919 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 15920 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 15921 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 15922 Closure of scalar product for a left module. (hvmulcl 22469 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 15923* 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 15924 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 15925 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 15926 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 15927 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 15928 Distributive law for scalar product. (ax-hvdistr1 22464 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 ) ) )
 
Theoremlmodvsdir 15929 Distributive law for scalar product. (ax-hvdistr1 22464 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 ) ) )
 
Theoremlmodvsass 15930 Associative law for scalar product. (ax-hvmulass 22463 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 ) ) )
 
Theoremlmod0cl 15931 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 15932 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 15933 Scalar product with ring unit. (ax-hvmulid 22462 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 15934 The zero vector is a vector. (ax-hv0cl 22459 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 15935 Left identity law for the zero vector. (hvaddid2 22478 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 15936 Right identity law for the zero vector. (ax-hvaddid 22460 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 15937 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 15938 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 22466 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 15939 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 22479 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 15940 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 15941 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 15942 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 ) )
 
Theoremlmodvsneg 15943 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 15944 Closure of vector subtraction. (hvsubcl 22473 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 15945 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 15946 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 15947 A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
 |-  ( W  e.  LMod  ->  W  e. CMnd )
 
Theoremlmodnegadd 15948 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 15949 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 15950 Relationship between vector subtraction and addition. (hvsubadd 22532 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 15951 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 15952 Addition/subtraction cancellation law for vectors. (hvpncan 22494 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 15953 Cancellation law for vector subtraction (npcan 9270 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 15954 Value of vector subtraction in terms of addition. (hvsubval 22472 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 15955 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 15956 Scalar multiplication distributive law for subtraction. (hvsubdistr1 22504 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 15957 Scalar multiplication distributive law for subtraction. (hvsubdistr2 22505 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 15958 If the difference between two vectors is zero, they are equal. (hvsubeq0 22523 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 15959 Subtraction of a vector from itself. (hvsubid 22481 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 15960* 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 15961* If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 15962 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 15962* 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 15963 Extend class notation with linear subspaces of a left module or left vector space.
 class  LSubSp
 
Definitiondf-lss 15964* 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 15965* 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 15966* 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 15967* 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 15968 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 15969 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 15970 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 15971 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 15972 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 15973 The empty structure has no subspaces (for use with fvco4i 5760). (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (/)  =  ( LSubSp `  (/) )
 
Theoremlsscl 15974 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 15975 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 15976 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 16163. Can it be used along with lspsnne1 16144, lspsnne2 16145 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 15977 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 15978 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 15979 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 15980 The zero subspace is included in every subspace. (sh0le 22895 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 )
 
Theoremlssle0 15981 No subspace is smaller than the zero subspace. (shle0 22897 analog.) (Contributed by NM, 20-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  S )  ->  ( X 
 C_  {  .0.  }  <->  X  =  {  .0.  } ) )
 
Theoremlssne0 15982* A nonzero subspace has a nonzero vector. (shne0i 22903 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( X  e.  S  ->  ( X  =/=  {  .0.  }  <->  E. y  e.  X  y  =/=  .0.  ) )
 
Theoremlssneln0 15983 A vector which doesn't belong to a subspace is nonzero. (Contributed by NM, 14-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  U )   =>    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )
 
Theoremlssssr 15984* Conclude subspace ordering from nonzero vector membership. (ssrdv 3314 analog.) (Contributed by NM, 17-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T 
 C_  V )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ( ph  /\  x  e.  ( V  \  {  .0.  } ) )  ->  ( x  e.  T  ->  x  e.  U ) )   =>    |-  ( ph  ->  T  C_  U )
 
Theoremlssvacl 15985 Closure of vector addition in a subspace. (Contributed by NM, 11-Jan-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 )
 
Theoremlssvscl 15986 Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  B  =  (
 Base `  F )   &    |-  S  =  ( LSubSp `  W )   =>    |-  (
 ( ( W  e.  LMod  /\  U  e.  S ) 
 /\  ( X  e.  B  /\  Y  e.  U ) )  ->  ( X 
 .x.  Y )  e.  U )
 
Theoremlssvnegcl 15987 Closure of negative vectors in a subspace. (Contributed by Stefan O'Rear, 11-Dec-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  ( inv g `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  ( N `  X )  e.  U )
 
Theoremlsssubg 15988 All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.)
 |-  S  =  ( LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S ) 
 ->  U  e.  (SubGrp `  W ) )
 
Theoremlsssssubg 15989 All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  S  =  ( LSubSp `  W )   =>    |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
 
Theoremislss3 15990 A linear subspace of a module is a subset which is a module in its own right. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  X  =  ( Ws  U )   &    |-  V  =  (
 Base `  W )   &    |-  S  =  ( LSubSp `  W )   =>    |-  ( W  e.  LMod  ->  ( U  e.  S  <->  ( U  C_  V  /\  X  e.  LMod ) ) )
 
Theoremlsslmod 15991 A submodule is a module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  LMod
 )
 
Theoremlsslss 15992 The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   &    |-  T  =  ( LSubSp `  X )   =>    |-  (
 ( W  e.  LMod  /\  U  e.  S ) 
 ->  ( V  e.  T  <->  ( V  e.  S  /\  V  C_  U ) ) )
 
Theoremislss4 15993* A linear subspace is a subgroup which respects scalar multiplication. (Contributed by Stefan O'Rear, 11-Dec-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  F  =  (Scalar `  W )   &    |-  B  =  ( Base `  F )   &    |-  V  =  (
 Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  S  =  ( LSubSp `  W )   =>    |-  ( W  e.  LMod  ->  ( U  e.  S  <->  ( U  e.  (SubGrp `  W )  /\  A. a  e.  B  A. b  e.  U  ( a  .x.  b )  e.  U ) ) )
 
Theoremlss1d 15994* One-dimensional subspace (or zero-dimensional if  X is the zero vector). (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   &    |-  S  =  ( LSubSp `  W )   =>    |-  (
 ( W  e.  LMod  /\  X  e.  V ) 
 ->  { v  |  E. k  e.  K  v  =  ( k  .x.  X ) }  e.  S )
 
Theoremlssintcl 15995 The intersection of a nonempty set of subspaces is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  S  =  ( LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  |^| A  e.  S )
 
Theoremlssincl 15996 The intersection of two subspaces is a subspace. (Contributed by NM, 7-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  S  =  ( LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  i^i  U )  e.  S )
 
Theoremlssmre 15997 The subspaces of a module comprise a Moore system on the vectors of the module. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  B  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( W  e.  LMod 
 ->  S  e.  (Moore `  B ) )
 
Theoremlssacs 15998 Submodules are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  B  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( W  e.  LMod 
 ->  S  e.  (ACS `  B ) )
 
Theoremprdsvscacl 15999* Pointwise scalar multiplication is closed in products of modules. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  .x.  =  ( .s `  Y )   &    |-  K  =  ( Base `  S )   &    |-  ( ph  ->  S  e.  Ring )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R : I --> LMod )   &    |-  ( ph  ->  F  e.  K )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ( ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x ) )  =  S )   =>    |-  ( ph  ->  ( F  .x.  G )  e.  B )
 
Theoremprdslmodd 16000* The product of a family of left modules is a left module. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  Ring )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R : I --> LMod )   &    |-  ( ( ph  /\  y  e.  I ) 
 ->  (Scalar `  ( R `  y ) )  =  S )   =>    |-  ( ph  ->  Y  e.  LMod )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32447
  Copyright terms: Public domain < Previous  Next >