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Theorem List for Metamath Proof Explorer - 15901-16000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlspindpi 15901 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 15902 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 15903 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 15904 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 15905 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 15906 (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 15907 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 15908 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 15909 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 15910 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 22247 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 15911* Lemma for lspsolv 15912. (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 15912 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 }
 ) ) )
 
Theoremlssacsex 15913* In a vector space, subspaces form an algebraic closure system whose closure operator has the exchange property. Strengthening of lssacs 15740 by lspsolv 15912. (Contributed by David Moews, 1-May-2017.)
 |-  A  =  ( LSubSp `  W )   &    |-  N  =  (mrCls `  A )   &    |-  X  =  (
 Base `  W )   =>    |-  ( W  e.  LVec 
 ->  ( A  e.  (ACS `  X )  /\  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y } ) )  \  ( N `  s ) ) y  e.  ( N `  ( s  u. 
 { z } )
 ) ) )
 
Theoremlspsnat 15914 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 22176 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 15915* 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 15916 Lemma for lspprat 15922. 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 15912 (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 15917 Lemma for lspprat 15922. 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 15918 Lemma for lspprat 15922. In the first case of lsppratlem1 15916, 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 15919 Lemma for lspprat 15922. In the second case of lsppratlem1 15916,  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 15920 Lemma for lspprat 15922. 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 15921 Lemma for lspprat 15922. 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 15922* 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 15923* 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 15924* 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 ) ) ) )
 
Theoremlbsacsbs 15925 Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 15923. (Contributed by David Moews, 1-May-2017.)
 |-  A  =  ( LSubSp `  W )   &    |-  N  =  (mrCls `  A )   &    |-  X  =  (
 Base `  W )   &    |-  I  =  (mrInd `  A )   &    |-  J  =  (LBasis `  W )   =>    |-  ( W  e.  LVec  ->  ( S  e.  J  <->  ( S  e.  I  /\  ( N `  S )  =  X ) ) )
 
Theoremlvecdim 15926 The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 15913 and lbsacsbs 15925 to show that being a basis for a vector space is equivalent to being a basis for the associated algebraic closure system, and then using acsexdimd 14302. (Contributed by David Moews, 1-May-2017.)
 |-  J  =  (LBasis `  W )   =>    |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  S  ~~  T )
 
Theoremlbsextlem1 15927* Lemma for lbsext 15932. 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 15928* Lemma for lbsext 15932. 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 15929* Lemma for lbsext 15932. 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 15930* Lemma for lbsext 15932. lbsextlem3 15929 satisfies the conditions for the application of Zorn's lemma zorn 8150 (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 15931* 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 15932* 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 15933 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 15934 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 15935* If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 15936 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 15936* 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 15937 Extend class notation with the subring algebra generator.
 class subringAlg
 
Syntaxcrglmod 15938 Extend class notation with the left module induced by a ring over itself.
 class ringLMod
 
Syntaxclidl 15939 Ring left-ideal function.
 class LIdeal
 
Syntaxcrsp 15940 Ring span function.
 class RSpan
 
Definitiondf-sra 15941* 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 15942 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 15943 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 15944 Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |- RSpan  =  ( LSpan  o. ringLMod )
 
Theoremsraval 15945 Lemma for srabase 15947 through sravsca 15951. (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 15946 Lemma for srabase 15947 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 15947 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 15948 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 15949 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 15950 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 15951 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 15952 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 15953 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 15954 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 15955 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 15956 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 15957* 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 15958* 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 15959* 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 15960 ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |- ringLMod  Fn  _V
 
Theoremrlmval 15961 Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (ringLMod `  W )  =  ( ( subringAlg  `  W ) `
  ( Base `  W ) )
 
Theoremlidlval 15962 Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (LIdeal `  W )  =  ( LSubSp `  (ringLMod `  W ) )
 
Theoremrspval 15963 Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  (RSpan `  W )  =  ( LSpan `  (ringLMod `  W ) )
 
Theoremrlmbas 15964 Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( Base `  R )  =  ( Base `  (ringLMod `  R ) )
 
Theoremrlmplusg 15965 Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( +g  `  R )  =  ( +g  `  (ringLMod `  R )
 )
 
Theoremrlm0 15966 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 15967 Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( .r `  R )  =  ( .r `  (ringLMod `  R )
 )
 
Theoremrlmsca 15968 Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |-  ( R  e.  X  ->  R  =  (Scalar `  (ringLMod `  R ) ) )
 
Theoremrlmsca2 15969 Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  (  _I  `  R )  =  (Scalar `  (ringLMod `  R ) )
 
Theoremrlmvsca 15970 Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( .r `  R )  =  ( .s `  (ringLMod `  R )
 )
 
Theoremrlmtopn 15971 Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( TopOpen `  R )  =  ( TopOpen `  (ringLMod `  R ) )
 
Theoremrlmds 15972 Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( dist `  R )  =  ( dist `  (ringLMod `  R ) )
 
Theoremrlmlmod 15973 The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |-  ( R  e.  Ring  ->  (ringLMod `  R )  e. 
 LMod )
 
Theoremrlmlvec 15974 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 15975 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 15976 Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( + f `  (mulGrp `  R ) )  =  ( .s f `  (ringLMod `  R )
 )
 
Theoremlidlss 15977 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 15978 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 15979* 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 15980 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 15981 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 15982 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 15983 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 15984 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
 )
 
Theoremlidlmcl 15985 An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( ( R  e.  Ring  /\  I  e.  U )  /\  ( X  e.  B  /\  Y  e.  I )
 )  ->  ( X  .x.  Y )  e.  I
 )
 
Theoremlidl1el 15986 An ideal contains 1 iff it is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  B  =  ( Base `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  U )  ->  (  .1.  e.  I 
 <->  I  =  B ) )
 
Theoremlidl0 15987 Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  {  .0.  }  e.  U )
 
Theoremlidl1 15988 Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( R  e.  Ring  ->  B  e.  U )
 
Theoremlidlacs 15989 The ideal system is an algebraic closure system on the base set. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  B  =  ( Base `  W )   &    |-  I  =  (LIdeal `  W )   =>    |-  ( W  e.  Ring  ->  I  e.  (ACS `  B ) )
 
Theoremrspcl 15990 The span of a set of ring elements is an ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  K  =  (RSpan `  R )   &    |-  B  =  (
 Base `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  (
 ( R  e.  Ring  /\  G  C_  B )  ->  ( K `  G )  e.  U )
 
Theoremrspssid 15991 The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  K  =  (RSpan `  R )   &    |-  B  =  (
 Base `  R )   =>    |-  ( ( R  e.  Ring  /\  G  C_  B )  ->  G  C_  ( K `  G ) )
 
Theoremrsp1 15992 The span of the identity element is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  K  =  (RSpan `  R )   &    |-  B  =  (
 Base `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  ( K `  {  .1.  } )  =  B )
 
Theoremrsp0 15993 The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  K  =  (RSpan `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  ( K `  {  .0.  } )  =  {  .0.  } )
 
Theoremrspssp 15994 The ideal span of a set of elements in a ring is contained in any subring which contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  K  =  (RSpan `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  U  /\  G  C_  I )  ->  ( K `  G )  C_  I )
 
Theoremmrcrsp 15995 Moore closure generalizes ideal span. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  U  =  (LIdeal `  R )   &    |-  K  =  (RSpan `  R )   &    |-  F  =  (mrCls `  U )   =>    |-  ( R  e.  Ring  ->  K  =  F )
 
Theoremlidlnz 15996* A nonzero ideal contains a nonzero element. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  U  =  (LIdeal `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  U  /\  I  =/=  {  .0.  } )  ->  E. x  e.  I  x  =/=  .0.  )
 
Theoremdrngnidl 15997 A division ring has only the two trivial ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  ( R  e.  DivRing  ->  U  =  { {  .0.  } ,  B } )
 
Theoremlidlrsppropd 15998* The left ideals and ring span of a ring depend only on the ring components. Here  W is expected to be either 
B (when closure is available) or  _V (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  B  C_  W )   &    |-  ( ( ph  /\  ( x  e.  W  /\  y  e.  W ) )  ->  ( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( x ( .r `  K ) y )  e.  W )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( x ( .r `  K ) y )  =  ( x ( .r
 `  L ) y ) )   =>    |-  ( ph  ->  (
 (LIdeal `  K )  =  (LIdeal `  L )  /\  (RSpan `  K )  =  (RSpan `  L )
 ) )
 
10.8.2  Two-sided ideals and quotient rings
 
Syntaxc2idl 15999 Ring two-sided ideal function.
 class 2Ideal
 
Definitiondf-2idl 16000 Define the class of two-sided ideals of a ring. A two-sided ideal is a left ideal which is also a right ideal (or a left ideal over the opposite ring). (Contributed by Mario Carneiro, 14-Jun-2015.)
 |- 2Ideal  =  ( r  e.  _V  |->  ( (LIdeal `  r )  i^i  (LIdeal `  (oppr `  r ) ) ) )
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