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Theorem List for Metamath Proof Explorer - 15701-15800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlvecinv 15701 Invert coefficient of scalar product. (Contributed by NM, 11-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .0.  =  ( 0g `  F )   &    |-  I  =  ( invr `  F )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  A  e.  ( K  \  {  .0.  } ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( X  =  ( A 
 .x.  Y )  <->  Y  =  (
 ( I `  A )  .x.  X ) ) )
 
Theoremlspsnvs 15702 A non-zero scalar product does not change the span of a singleton. (spansncol 21977 analog.) (Contributed by NM, 23-Apr-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   &    |-  .0.  =  ( 0g `  F )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( ( W  e.  LVec  /\  ( R  e.  K  /\  R  =/=  .0.  )  /\  X  e.  V ) 
 ->  ( N `  { ( R  .x.  X ) }
 )  =  ( N `
  { X }
 ) )
 
Theoremlspsneleq 15703 Membership relation that implies equality of spans. (spansneleq 21979 analog.) (Contributed by NM, 4-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  ( N `  { X }
 ) )   &    |-  ( ph  ->  Y  =/=  .0.  )   =>    |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { X }
 ) )
 
Theoremlspsncmp 15704 Comparable spans of nonzero singletons are equal. (Contributed by NM, 27-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  (
 ( N `  { X } )  C_  ( N `
  { Y }
 ) 
 <->  ( N `  { X } )  =  ( N `  { Y }
 ) ) )
 
Theoremlspsnne1 15705 Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )
 
Theoremlspsnne2 15706 Two ways to express that vectors have different spans. (Contributed by NM, 20-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )   =>    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )
 
Theoremlspsnnecom 15707 Swap two vectors with different spans. (Contributed by NM, 20-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y }
 ) )   =>    |-  ( ph  ->  -.  Y  e.  ( N `  { X } ) )
 
Theoremlspabs2 15708 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =  ( N `  { ( X 
 .+  Y ) }
 ) )   =>    |-  ( ph  ->  ( N `  { X }
 )  =  ( N `
  { Y }
 ) )
 
Theoremlspabs3 15709 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( X  .+  Y )  =/= 
 .0.  )   &    |-  ( ph  ->  ( N `  { X } )  =  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  ( N `  { X }
 )  =  ( N `
  { ( X 
 .+  Y ) }
 ) )
 
Theoremlspsneq 15710* Equal spans of singletons must have proportional vectors. See lspsnss2 15597 for comparable span version. TODO: can proof be shortened? (Contributed by NM, 21-Mar-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (Scalar `  W )   &    |-  K  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  S )   &    |- 
 .x.  =  ( .s `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y }
 ) 
 <-> 
 E. k  e.  ( K  \  {  .0.  }
 ) X  =  ( k  .x.  Y )
 ) )
 
Theoremlspsneu 15711* Nonzero vectors with equal singleton spans have a unique proportionality constant. (Contributed by NM, 31-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (Scalar `  W )   &    |-  K  =  (
 Base `  S )   &    |-  O  =  ( 0g `  S )   &    |- 
 .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  (
 ( N `  { X } )  =  ( N `  { Y }
 ) 
 <->  E! k  e.  ( K  \  { O }
 ) X  =  ( k  .x.  Y )
 ) )
 
Theoremlspsnel4 15712 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 21982 analog.) (Contributed by NM, 4-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  ( N `  { X }
 ) )   &    |-  ( ph  ->  Y  =/=  .0.  )   =>    |-  ( ph  ->  ( X  e.  U  <->  Y  e.  U ) )
 
Theoremlspdisj 15713 The span of a vector not in a subspace is disjoint with the subspace. (Contributed by NM, 6-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  U )   =>    |-  ( ph  ->  (
 ( N `  { X } )  i^i  U )  =  {  .0.  }
 )
 
Theoremlspdisjb 15714 The a nonzero vector is not in a subspace iff its span is disjoint with the subspace. (Contributed by NM, 23-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( -.  X  e.  U  <->  ( ( N `
  { X }
 )  i^i  U )  =  {  .0.  } )
 )
 
Theoremlspdisj2 15715 Unequal spans are disjoint (share only the zero vector). (Contributed by NM, 22-Mar-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { Y } )
 )   =>    |-  ( ph  ->  (
 ( N `  { X } )  i^i  ( N `
  { Y }
 ) )  =  {  .0.  } )
 
Theoremlspfixed 15716* Show membership in the span of the sum of two vectors, one of which ( Y) is fixed in advance. (Contributed by NM, 27-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y }
 ) )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Z }
 ) )   &    |-  ( ph  ->  X  e.  ( N `  { Y ,  Z }
 ) )   =>    |-  ( ph  ->  E. z  e.  ( ( N `  { Z } )  \  {  .0.  } ) X  e.  ( N `  { ( Y  .+  z ) } )
 )
 
Theoremlspexch 15717 Exchange property for span of a pair. TODO: see if a version with Y,Z and X,Z reversed will shorten proofs (analogous to lspexchn1 15718 vs. lspexchn2 15719); look for lspexch 15717 and prcom 3609 in same proof. TODO: would a hypothesis of  -.  X  e.  ( N `  { Z } ) instead of  ( N `  { X } )  =/=  ( N { Z } ) ` be better overall? This would be shorter and also satisfy the 
X  =/=  .0. condition. Here and also lspindp* and all proofs affected by them (all in NM's mathbox); there are 58 hypotheses with the 
=/= pattern as of 24-May-2015. (Contributed by NM, 11-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )   &    |-  ( ph  ->  X  e.  ( N `  { Y ,  Z } ) )   =>    |-  ( ph  ->  Y  e.  ( N `  { X ,  Z }
 ) )
 
Theoremlspexchn1 15718 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 15717 to see if this will shorten proofs. (Contributed by NM, 20-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  -.  Y  e.  ( N `  { Z } ) )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   =>    |-  ( ph  ->  -.  Y  e.  ( N `  { X ,  Z } ) )
 
Theoremlspexchn2 15719 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 15717 to see if this will shorten proofs. (Contributed by NM, 24-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  -.  Y  e.  ( N `  { Z } ) )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Z ,  Y } ) )   =>    |-  ( ph  ->  -.  Y  e.  ( N `  { Z ,  X } ) )
 
Theoremlspindpi 15720 Partial independence property. (Contributed by NM, 23-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   =>    |-  ( ph  ->  ( ( N `  { X }
 )  =/=  ( N ` 
 { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z }
 ) ) )
 
Theoremlspindp1 15721 Alternate way to say 3 vectors are mutually independent (swap 1st and 2nd). (Contributed by NM, 11-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  -.  Z  e.  ( N `  { X ,  Y } ) )   =>    |-  ( ph  ->  ( ( N `  { Z }
 )  =/=  ( N ` 
 { Y } )  /\  -.  X  e.  ( N `  { Z ,  Y } ) ) )
 
Theoremlspindp2l 15722 Alternate way to say 3 vectors are mutually independent (rotate left). (Contributed by NM, 10-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  -.  Z  e.  ( N `  { X ,  Y } ) )   =>    |-  ( ph  ->  ( ( N `  { Y }
 )  =/=  ( N ` 
 { Z } )  /\  -.  X  e.  ( N `  { Y ,  Z } ) ) )
 
Theoremlspindp2 15723 Alternate way to say 3 vectors are mutually independent (rotate right). (Contributed by NM, 12-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  -.  Z  e.  ( N `  { X ,  Y } ) )   =>    |-  ( ph  ->  ( ( N `  { Z }
 )  =/=  ( N ` 
 { X } )  /\  -.  Y  e.  ( N `  { Z ,  X } ) ) )
 
Theoremlspindp3 15724 Independence of 2 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { ( X  .+  Y ) } )
 )
 
Theoremlspindp4 15725 (Partial) independence of 3 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  -.  Z  e.  ( N `  { X ,  Y } ) )   =>    |-  ( ph  ->  -.  Z  e.  ( N `  { X ,  ( X  .+  Y ) } ) )
 
Theoremlvecindp 15726 Compute the  X coefficient in a sum with an independent vector  X (first conjunct), which can then be removed to continue with the remaining vectors summed in expressions  Y and 
Z (second conjunct). Typically  U is the span of the remaining vectors. (Contributed by NM, 5-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  ( ph  ->  Z  e.  U )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  ( ( A 
 .x.  X )  .+  Y )  =  ( ( B  .x.  X )  .+  Z ) )   =>    |-  ( ph  ->  ( A  =  B  /\  Y  =  Z )
 )
 
Theoremlvecindp2 15727 Sums of independent vectors must have equal coefficients. (Contributed by NM, 22-Mar-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  K )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   &    |-  ( ph  ->  ( ( A  .x.  X )  .+  ( B  .x.  Y ) )  =  ( ( C  .x.  X )  .+  ( D  .x.  Y ) ) )   =>    |-  ( ph  ->  ( A  =  C  /\  B  =  D )
 )
 
Theoremlspsnsubn0 15728 Unequal singleton spans imply nonzero vector subtraction. (Contributed by NM, 19-Mar-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  .-  =  ( -g `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { Y } )
 )   =>    |-  ( ph  ->  ( X  .-  Y )  =/= 
 .0.  )
 
Theoremlsmcv 15729 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 22079 analog.) TODO: ugly proof; can it be shortened? (Contributed by NM, 2-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ( ph  /\  T  C.  U  /\  U  C_  ( T  .(+)  ( N `
  { X }
 ) ) )  ->  U  =  ( T  .(+) 
 ( N `  { X } ) ) )
 
Theoremlspsolvlem 15730* Lemma for lspsolv 15731. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  F  =  (Scalar `  W )   &    |-  B  =  ( Base `  F )   &    |-  .+  =  ( +g  `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  Q  =  { z  e.  V  |  E. r  e.  B  ( z  .+  ( r  .x.  Y ) )  e.  ( N `
  A ) }   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  A  C_  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  e.  ( N `  ( A  u.  { Y } ) ) )   =>    |-  ( ph  ->  E. r  e.  B  ( X  .+  ( r  .x.  Y ) )  e.  ( N `
  A ) )
 
Theoremlspsolv 15731 If  X is in the span of  A  u.  { Y } but not  A, then  Y is in the span of  A  u.  { X }. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  (
 ( W  e.  LVec  /\  ( A  C_  V  /\  Y  e.  V  /\  X  e.  ( ( N `  ( A  u.  { Y } ) ) 
 \  ( N `  A ) ) ) )  ->  Y  e.  ( N `  ( A  u.  { X }
 ) ) )
 
Theoremlspsnat 15732 There is no subspace strictly between the zero subspace and the span of a vector (i.e. a 1-dimensional subspace is an atom). (h1datomi 21990 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `  { X } ) )  ->  ( U  =  ( N `  { X }
 )  \/  U  =  {  .0.  } ) )
 
Theoremlspsncv0 15733* The span of a singleton covers the zero subspace, using Definition 3.2.18 of [PtakPulmannova] p. 68 for "covers".) (Contributed by NM, 12-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  X  =/=  .0.  )   =>    |-  ( ph  ->  -. 
 E. y  e.  S  ( {  .0.  }  C.  y  /\  y  C.  ( N `  { X }
 ) ) )
 
Theoremlsppratlem1 15734 Lemma for lspprat 15740. Let  x  e.  ( U  \  { 0 } ) (if there is no such  x then  U is the zero subspace), and let  y  e.  ( U  \  ( N `
 { x }
) ) (assuming the conclusion is false). The goal is to write  X,  Y in terms of  x,  y, which would normally be done by solving the system of linear equations. The span equivalent of this process is lspsolv 15731 (hence the name), which we use extensively below. In this lemma, we show that since  x  e.  ( N `  { X ,  Y } ), either  x  e.  ( N `  { Y } ) or  X  e.  ( N `  { x ,  Y } ). (Contributed by NM, 29-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  U  C.  ( N `  { X ,  Y } ) )   &    |-  .0.  =  ( 0g `  W )   &    |-  ( ph  ->  x  e.  ( U  \  {  .0.  } ) )   &    |-  ( ph  ->  y  e.  ( U  \  ( N `
  { x }
 ) ) )   =>    |-  ( ph  ->  ( x  e.  ( N `
  { Y }
 )  \/  X  e.  ( N `  { x ,  Y } ) ) )
 
Theoremlsppratlem2 15735 Lemma for lspprat 15740. Show that if  X and 
Y are both in  ( N `  { x ,  y } ) (which will be our goal for each of the two cases above), then  ( N `  { X ,  Y }
)  C_  U, contradicting the hypothesis for  U. (Contributed by NM, 29-Aug-2014.) (Revised by Mario Carneiro, 5-Sep-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  U  C.  ( N `  { X ,  Y } ) )   &    |-  .0.  =  ( 0g `  W )   &    |-  ( ph  ->  x  e.  ( U  \  {  .0.  } ) )   &    |-  ( ph  ->  y  e.  ( U  \  ( N `
  { x }
 ) ) )   &    |-  ( ph  ->  X  e.  ( N `  { x ,  y } ) )   &    |-  ( ph  ->  Y  e.  ( N `  { x ,  y } ) )   =>    |-  ( ph  ->  ( N `  { X ,  Y } )  C_  U )
 
Theoremlsppratlem3 15736 Lemma for lspprat 15740. In the first case of lsppratlem1 15734, since  x  e/  ( N `  (/) ), also  Y  e.  ( N `  {
x } ), and since  y  e.  ( N `  { X ,  Y } )  C_  ( N `  { X ,  x } ) and  y  e/  ( N `  { x } ), we have  X  e.  ( N `  { x ,  y } ) as desired. (Contributed by NM, 29-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  U  C.  ( N `  { X ,  Y } ) )   &    |-  .0.  =  ( 0g `  W )   &    |-  ( ph  ->  x  e.  ( U  \  {  .0.  } ) )   &    |-  ( ph  ->  y  e.  ( U  \  ( N `
  { x }
 ) ) )   &    |-  ( ph  ->  x  e.  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  ( X  e.  ( N ` 
 { x ,  y } )  /\  Y  e.  ( N `  { x ,  y } ) ) )
 
Theoremlsppratlem4 15737 Lemma for lspprat 15740. In the second case of lsppratlem1 15734,  y  e.  ( N `  { X ,  Y } )  C_  ( N `  { x ,  Y } ) and  y  e/  ( N `  { x } ) implies  Y  e.  ( N `  { x ,  y } ) and thus  X  e.  ( N `  { x ,  Y } )  C_  ( N `  { x ,  y } ) as well. (Contributed by NM, 29-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  U  C.  ( N `  { X ,  Y } ) )   &    |-  .0.  =  ( 0g `  W )   &    |-  ( ph  ->  x  e.  ( U  \  {  .0.  } ) )   &    |-  ( ph  ->  y  e.  ( U  \  ( N `
  { x }
 ) ) )   &    |-  ( ph  ->  X  e.  ( N `  { x ,  Y } ) )   =>    |-  ( ph  ->  ( X  e.  ( N `
  { x ,  y } )  /\  Y  e.  ( N `  { x ,  y } ) ) )
 
Theoremlsppratlem5 15738 Lemma for lspprat 15740. Combine the two cases and show a contradiction to  U  C.  ( N `  { X ,  Y } ) under the assumptions on  x and  y. (Contributed by NM, 29-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  U  C.  ( N `  { X ,  Y } ) )   &    |-  .0.  =  ( 0g `  W )   &    |-  ( ph  ->  x  e.  ( U  \  {  .0.  } ) )   &    |-  ( ph  ->  y  e.  ( U  \  ( N `
  { x }
 ) ) )   =>    |-  ( ph  ->  ( N `  { X ,  Y } )  C_  U )
 
Theoremlsppratlem6 15739 Lemma for lspprat 15740. Negating the assumption on  y, we arrive close to the desired conclusion. (Contributed by NM, 29-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  U  C.  ( N `  { X ,  Y } ) )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( ph  ->  ( x  e.  ( U  \  {  .0.  } )  ->  U  =  ( N `
  { x }
 ) ) )
 
Theoremlspprat 15740* A proper subspace of the span of a pair of vectors is the span of a singleton (an atom) or the zero subspace (if  z is zero). Proof suggested by Mario Carneiro, 28-Aug-2014. (Contributed by NM, 29-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  U  C.  ( N `  { X ,  Y } ) )   =>    |-  ( ph  ->  E. z  e.  V  U  =  ( N `  { z } ) )
 
Theoremislbs2 15741* An equivalent formulation of the basis predicate in a vector space: a subset is a basis iff no element is in the span of the rest of the set. (Contributed by Mario Carneiro, 14-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( W  e.  LVec 
 ->  ( B  e.  J  <->  ( B  C_  V  /\  ( N `  B )  =  V  /\  A. x  e.  B  -.  x  e.  ( N `  ( B  \  { x } ) ) ) ) )
 
Theoremislbs3 15742* An equivalent formulation of the basis predicate: a subset is a basis iff it is a minimal spanning set. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( W  e.  LVec 
 ->  ( B  e.  J  <->  ( B  C_  V  /\  ( N `  B )  =  V  /\  A. s ( s  C.  B  ->  ( N `  s )  C.  V ) ) ) )
 
Theoremlbsextlem1 15743* Lemma for lbsext 15748. The set  S is the set of all linearly independent sets containing 
C; we show here that it is nonempty. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  C  C_  V )   &    |-  ( ph  ->  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x }
 ) ) )   &    |-  S  =  { z  e.  ~P V  |  ( C  C_  z  /\  A. x  e.  z  -.  x  e.  ( N `  (
 z  \  { x } ) ) ) }   =>    |-  ( ph  ->  S  =/= 
 (/) )
 
Theoremlbsextlem2 15744* Lemma for lbsext 15748. Since  A is a chain (actually, we only need it to be closed under binary union), the union  T of the spans of each individual element of 
A is a subspace, and it contains all of  U. A (except for our target vector  x- we are trying to make  x a linear combination of all the other vectors in some set from  A). (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  C  C_  V )   &    |-  ( ph  ->  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x }
 ) ) )   &    |-  S  =  { z  e.  ~P V  |  ( C  C_  z  /\  A. x  e.  z  -.  x  e.  ( N `  (
 z  \  { x } ) ) ) }   &    |-  P  =  (
 LSubSp `  W )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  -> [ C.] 
 Or  A )   &    |-  T  =  U_ u  e.  A  ( N `  ( u 
 \  { x }
 ) )   =>    |-  ( ph  ->  ( T  e.  P  /\  ( U. A  \  { x } )  C_  T ) )
 
Theoremlbsextlem3 15745* Lemma for lbsext 15748. A chain in  S has an upper bound in  S. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  C  C_  V )   &    |-  ( ph  ->  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x }
 ) ) )   &    |-  S  =  { z  e.  ~P V  |  ( C  C_  z  /\  A. x  e.  z  -.  x  e.  ( N `  (
 z  \  { x } ) ) ) }   &    |-  P  =  (
 LSubSp `  W )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  -> [ C.] 
 Or  A )   &    |-  T  =  U_ u  e.  A  ( N `  ( u 
 \  { x }
 ) )   =>    |-  ( ph  ->  U. A  e.  S )
 
Theoremlbsextlem4 15746* Lemma for lbsext 15748. lbsextlem3 15745 satisfies the conditions for the application of Zorn's lemma zorn 8018 (thus invoking AC), and so there is a maximal linearly independent set extending  C. Here we prove that such a set is a basis. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  C  C_  V )   &    |-  ( ph  ->  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x }
 ) ) )   &    |-  S  =  { z  e.  ~P V  |  ( C  C_  z  /\  A. x  e.  z  -.  x  e.  ( N `  (
 z  \  { x } ) ) ) }   &    |-  ( ph  ->  ~P V  e.  dom  card )   =>    |-  ( ph  ->  E. s  e.  J  C  C_  s
 )
 
Theoremlbsextg 15747* For any linearly independent subset 
C of  V, there is a basis containing the vectors in 
C. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  J  =  (LBasis `  W )   &    |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( ( W  e.  LVec  /\  ~P V  e.  dom  card )  /\  C  C_  V  /\  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x } ) ) ) 
 ->  E. s  e.  J  C  C_  s )
 
Theoremlbsext 15748* For any linearly independent subset 
C of  V, there is a basis containing the vectors in 
C. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  J  =  (LBasis `  W )   &    |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LVec  /\  C  C_  V  /\  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x }
 ) ) )  ->  E. s  e.  J  C  C_  s )
 
Theoremlbsexg 15749 Every vector space has a basis. This theorem is an AC equivalent; this is the forward implication. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  J  =  (LBasis `  W )   =>    |-  ( (CHOICE 
 /\  W  e.  LVec ) 
 ->  J  =/=  (/) )
 
Theoremlbsex 15750 Every vector space has a basis. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  J  =  (LBasis `  W )   =>    |-  ( W  e.  LVec  ->  J  =/=  (/) )
 
Theoremlvecprop2d 15751* If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 15752 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.  LVec  <->  L  e.  LVec )
 )
 
Theoremlvecpropd 15752* If two structures have the same components (properties), one is a left vector space iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ph  ->  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.  LVec  <->  L  e.  LVec )
 )
 
10.8  Ideals
 
10.8.1  The subring algebra; ideals
 
Syntaxcsra 15753 Extend class notation with the subring algebra generator.
 class subringAlg
 
Syntaxcrglmod 15754 Extend class notation with the left module induced by a ring over itself.
 class ringLMod
 
Syntaxclidl 15755 Ring left-ideal function.
 class LIdeal
 
Syntaxcrsp 15756 Ring span function.
 class RSpan
 
Definitiondf-sra 15757* Given any subring of a ring, we can construct a left-algebra by regarding the elements of the subring as scalars and the ring itself as a set of vectors. (Contributed by Mario Carneiro, 27-Nov-2014.)
 |- subringAlg  =  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |->  ( ( w sSet  <. (Scalar `  ndx ) ,  ( ws  s ) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  w ) >. ) ) )
 
Definitiondf-rgmod 15758 Every ring can be viewed as a left module over itself. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |- ringLMod  =  ( w  e.  _V  |->  ( ( subringAlg  `  w ) `
  ( Base `  w ) ) )
 
Definitiondf-lidl 15759 Define the class of left ideals of a given ring. An ideal is a submodule of the ring viewed as a module over itself. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |- LIdeal  =  ( LSubSp  o. ringLMod )
 
Definitiondf-rsp 15760 Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |- RSpan  =  ( LSpan  o. ringLMod )
 
Theoremsraval 15761 Lemma for srabase 15763 through sravsca 15767. (Contributed by Mario Carneiro, 27-Nov-2014.)
 |-  ( ( W  e.  V  /\  S  C_  ( Base `  W ) ) 
 ->  ( ( subringAlg  `  W ) `
  S )  =  ( ( W sSet  <. (Scalar `  ndx ) ,  ( Ws  S ) >. ) sSet  <. ( .s
 `  ndx ) ,  ( .r `  W ) >. ) )
 
Theoremsralem 15762 Lemma for srabase 15763 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  ( N  <  5  \/  6  <  N )   =>    |-  ( ph  ->  ( E `  W )  =  ( E `  A ) )
 
Theoremsrabase 15763 Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  ( Base `  W )  =  (
 Base `  A ) )
 
Theoremsraaddg 15764 Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  ( +g  `  W )  =  (
 +g  `  A )
 )
 
Theoremsramulr 15765 Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  ( .r `  W )  =  ( .r `  A ) )
 
Theoremsrasca 15766 The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  ( Ws  S )  =  (Scalar `  A ) )
 
Theoremsravsca 15767 The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  ( .r `  W )  =  ( .s `  A ) )
 
Theoremsratset 15768 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  (TopSet `  W )  =  (TopSet `  A ) )
 
Theoremsratopn 15769 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  ( TopOpen `  W )  =  ( TopOpen `  A ) )
 
Theoremsrads 15770 Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  S  C_  ( Base `  W ) )   =>    |-  ( ph  ->  ( dist `  W )  =  (
 dist `  A ) )
 
Theoremsralmod 15771 The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.)
 |-  A  =  ( ( subringAlg  `  W ) `  S )   =>    |-  ( S  e.  (SubRing `  W )  ->  A  e.  LMod )
 
Theoremsralmod0 15772 The subring module inherits a zero from its ring. (Contributed by Stefan O'Rear, 27-Dec-2014.)
 |-  ( ph  ->  A  =  ( ( subringAlg  `  W ) `
  S ) )   &    |-  ( ph  ->  .0.  =  ( 0g `  W ) )   &    |-  ( ph  ->  S 
 C_  ( Base `  W ) )   =>    |-  ( ph  ->  .0.  =  ( 0g `  A ) )
 
Theoremissubgrpd2 15773* Prove a subgroup by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
 |-  ( ph  ->  S  =  ( Is  D ) )   &    |-  ( ph  ->  .0.  =  ( 0g `  I ) )   &    |-  ( ph  ->  .+  =  (
 +g  `  I )
 )   &    |-  ( ph  ->  D  C_  ( Base `  I )
 )   &    |-  ( ph  ->  .0.  e.  D )   &    |-  ( ( ph  /\  x  e.  D  /\  y  e.  D )  ->  ( x  .+  y
 )  e.  D )   &    |-  ( ( ph  /\  x  e.  D )  ->  (
 ( inv g `  I
 ) `  x )  e.  D )   &    |-  ( ph  ->  I  e.  Grp )   =>    |-  ( ph  ->  D  e.  (SubGrp `  I
 ) )
 
Theoremissubgrpd 15774* Prove a subgroup by closure. (Contributed by Stefan O'Rear, 7-Dec-2014.)
 |-  ( ph  ->  S  =  ( Is  D ) )   &    |-  ( ph  ->  .0.  =  ( 0g `  I ) )   &    |-  ( ph  ->  .+  =  (
 +g  `  I )
 )   &    |-  ( ph  ->  D  C_  ( Base `  I )
 )   &    |-  ( ph  ->  .0.  e.  D )   &    |-  ( ( ph  /\  x  e.  D  /\  y  e.  D )  ->  ( x  .+  y
 )  e.  D )   &    |-  ( ( ph  /\  x  e.  D )  ->  (
 ( inv g `  I
 ) `  x )  e.  D )   &    |-  ( ph  ->  I  e.  Grp )   =>    |-  ( ph  ->  S  e.  Grp )
 
Theoremissubrngd2 15775* Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
 |-  ( ph  ->  S  =  ( Is  D ) )   &    |-  ( ph  ->  .0.  =  ( 0g `  I ) )   &    |-  ( ph  ->  .+  =  (
 +g  `  I )
 )   &    |-  ( ph  ->  D  C_  ( Base `  I )
 )   &    |-  ( ph  ->  .0.  e.  D )   &    |-  ( ( ph  /\  x  e.  D  /\  y  e.  D )  ->  ( x  .+  y
 )  e.  D )   &    |-  ( ( ph  /\  x  e.  D )  ->  (
 ( inv g `  I
 ) `  x )  e.  D )   &    |-  ( ph  ->  .1. 
 =  ( 1r `  I ) )   &    |-  ( ph  ->  .x.  =  ( .r `  I ) )   &    |-  ( ph  ->  .1.  e.  D )   &    |-  ( ( ph  /\  x  e.  D  /\  y  e.  D )  ->  ( x  .x.  y
 )  e.  D )   &    |-  ( ph  ->  I  e.  Ring
 )   =>    |-  ( ph  ->  D  e.  (SubRing `  I )
 )
 
Theoremrlmfn 15776 ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |- ringLMod  Fn  _V
 
Theoremrlmval 15777 Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (ringLMod `  W )  =  ( ( subringAlg  `  W ) `
  ( Base `  W ) )
 
Theoremlidlval 15778 Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (LIdeal `  W )  =  ( LSubSp `  (ringLMod `  W ) )
 
Theoremrspval 15779 Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  (RSpan `  W )  =  ( LSpan `  (ringLMod `  W ) )
 
Theoremrlmbas 15780 Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( Base `  R )  =  ( Base `  (ringLMod `  R ) )
 
Theoremrlmplusg 15781 Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( +g  `  R )  =  ( +g  `  (ringLMod `  R )
 )
 
Theoremrlm0 15782 Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  ( 0g `  R )  =  ( 0g `  (ringLMod `  R )
 )
 
Theoremrlmmulr 15783 Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( .r `  R )  =  ( .r `  (ringLMod `  R )
 )
 
Theoremrlmsca 15784 Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |-  ( R  e.  X  ->  R  =  (Scalar `  (ringLMod `  R ) ) )
 
Theoremrlmsca2 15785 Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  (  _I  `  R )  =  (Scalar `  (ringLMod `  R ) )
 
Theoremrlmvsca 15786 Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( .r `  R )  =  ( .s `  (ringLMod `  R )
 )
 
Theoremrlmtopn 15787 Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( TopOpen `  R )  =  ( TopOpen `  (ringLMod `  R ) )
 
Theoremrlmds 15788 Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( dist `  R )  =  ( dist `  (ringLMod `  R ) )
 
Theoremrlmlmod 15789 The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |-  ( R  e.  Ring  ->  (ringLMod `  R )  e. 
 LMod )
 
Theoremrlmlvec 15790 The ring module over a division ring is a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( R  e.  DivRing  ->  (ringLMod `  R )  e. 
 LVec )
 
Theoremrlmvneg 15791 Vector negation in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 5-Jun-2015.)
 |-  ( inv g `  R )  =  ( inv g `  (ringLMod `  R ) )
 
Theoremrlmscaf 15792 Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( + f `  (mulGrp `  R ) )  =  ( .s f `  (ringLMod `  R )
 )
 
Theoremlidlss 15793 An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  B  =  ( Base `  W )   &    |-  I  =  (LIdeal `  W )   =>    |-  ( U  e.  I  ->  U  C_  B )
 
TheoremlidlssOLD 15794 An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  B  =  ( Base `  W )   &    |-  I  =  (LIdeal `  W )   =>    |-  ( ( W  e.  V  /\  U  e.  I
 )  ->  U  C_  B )
 
Theoremislidl 15795* Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( I  e.  U  <->  ( I  C_  B  /\  I  =/=  (/)  /\  A. x  e.  B  A. a  e.  I  A. b  e.  I  ( ( x 
 .x.  a )  .+  b )  e.  I
 ) )
 
Theoremlidl0cl 15796 An ideal contains 0. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  .0.  e.  I
 )
 
Theoremlidlacl 15797 An ideal is closed under addition. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |- 
 .+  =  ( +g  `  R )   =>    |-  ( ( ( R  e.  Ring  /\  I  e.  U )  /\  ( X  e.  I  /\  Y  e.  I )
 )  ->  ( X  .+  Y )  e.  I
 )
 
Theoremlidlnegcl 15798 An ideal contains negatives. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  N  =  ( inv
 g `  R )   =>    |-  (
 ( R  e.  Ring  /\  I  e.  U  /\  X  e.  I )  ->  ( N `  X )  e.  I )
 
Theoremlidlsubg 15799 An ideal is a subgroup of the additive group. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  U  =  (LIdeal `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  I  e.  (SubGrp `  R ) )
 
Theoremlidlsubcl 15800 An ideal is closed under subtraction. (Contributed by Stefan O'Rear, 28-Mar-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  .-  =  ( -g `  R )   =>    |-  ( ( ( R  e.  Ring  /\  I  e.  U )  /\  ( X  e.  I  /\  Y  e.  I )
 )  ->  ( X  .-  Y )  e.  I
 )
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