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Theorem List for Metamath Proof Explorer - 15901-16000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremislbs2 15901* 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 15902* 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 15903 Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 15901. (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 15904 The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 15891 and lbsacsbs 15903 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 14280. (Contributed by David Moews, 1-May-2017.)
 |-  J  =  (LBasis `  W )   =>    |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  S  ~~  T )
 
Theoremlbsextlem1 15905* Lemma for lbsext 15910. 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 15906* Lemma for lbsext 15910. 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 15907* Lemma for lbsext 15910. 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 15908* Lemma for lbsext 15910. lbsextlem3 15907 satisfies the conditions for the application of Zorn's lemma zorn 8129 (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 15909* 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 15910* 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 15911 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 15912 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 15913* If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 15914 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 15914* 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 15915 Extend class notation with the subring algebra generator.
 class subringAlg
 
Syntaxcrglmod 15916 Extend class notation with the left module induced by a ring over itself.
 class ringLMod
 
Syntaxclidl 15917 Ring left-ideal function.
 class LIdeal
 
Syntaxcrsp 15918 Ring span function.
 class RSpan
 
Definitiondf-sra 15919* 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 15920 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 15921 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 15922 Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |- RSpan  =  ( LSpan  o. ringLMod )
 
Theoremsraval 15923 Lemma for srabase 15925 through sravsca 15929. (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 15924 Lemma for srabase 15925 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 15925 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 15926 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 15927 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 15928 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 15929 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 15930 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 15931 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 15932 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 15933 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 15934 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 15935* 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 15936* 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 15937* 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 15938 ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |- ringLMod  Fn  _V
 
Theoremrlmval 15939 Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (ringLMod `  W )  =  ( ( subringAlg  `  W ) `
  ( Base `  W ) )
 
Theoremlidlval 15940 Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (LIdeal `  W )  =  ( LSubSp `  (ringLMod `  W ) )
 
Theoremrspval 15941 Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  (RSpan `  W )  =  ( LSpan `  (ringLMod `  W ) )
 
Theoremrlmbas 15942 Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( Base `  R )  =  ( Base `  (ringLMod `  R ) )
 
Theoremrlmplusg 15943 Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( +g  `  R )  =  ( +g  `  (ringLMod `  R )
 )
 
Theoremrlm0 15944 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 15945 Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( .r `  R )  =  ( .r `  (ringLMod `  R )
 )
 
Theoremrlmsca 15946 Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |-  ( R  e.  X  ->  R  =  (Scalar `  (ringLMod `  R ) ) )
 
Theoremrlmsca2 15947 Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  (  _I  `  R )  =  (Scalar `  (ringLMod `  R ) )
 
Theoremrlmvsca 15948 Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( .r `  R )  =  ( .s `  (ringLMod `  R )
 )
 
Theoremrlmtopn 15949 Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( TopOpen `  R )  =  ( TopOpen `  (ringLMod `  R ) )
 
Theoremrlmds 15950 Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( dist `  R )  =  ( dist `  (ringLMod `  R ) )
 
Theoremrlmlmod 15951 The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |-  ( R  e.  Ring  ->  (ringLMod `  R )  e. 
 LMod )
 
Theoremrlmlvec 15952 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 15953 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 15954 Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( + f `  (mulGrp `  R ) )  =  ( .s f `  (ringLMod `  R )
 )
 
Theoremlidlss 15955 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 15956 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 15957* 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 15958 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 15959 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 15960 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 15961 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 15962 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 15963 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 15964 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 15965 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 15966 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 15967 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 15968 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 15969 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 15970 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 15971 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 15972 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 15973 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 15974* 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 15975 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 15976* 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 15977 Ring two-sided ideal function.
 class 2Ideal
 
Definitiondf-2idl 15978 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 ) ) ) )
 
Theorem2idlval 15979 Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  I  =  (LIdeal `  R )   &    |-  O  =  (oppr `  R )   &    |-  J  =  (LIdeal `  O )   &    |-  T  =  (2Ideal `  R )   =>    |-  T  =  ( I  i^i  J )
 
Theorem2idlcpbl 15980 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  X  =  ( Base `  R )   &    |-  E  =  ( R ~QG 
 S )   &    |-  I  =  (2Ideal `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  S  e.  I )  ->  ( ( A E C  /\  B E D )  ->  ( A  .x.  B ) E ( C  .x.  D ) ) )
 
Theoremdivs1 15981 The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  U  =  ( R 
 /.s 
 ( R ~QG  S ) )   &    |-  I  =  (2Ideal `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  S  e.  I ) 
 ->  ( U  e.  Ring  /\ 
 [  .1.  ] ( R ~QG  S )  =  ( 1r
 `  U ) ) )
 
Theoremdivsrng 15982 If  S is a two-sided ideal in  R, then  U  =  R  /  S is a ring, called the quotient ring of 
R by  S. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  U  =  ( R 
 /.s 
 ( R ~QG  S ) )   &    |-  I  =  (2Ideal `  R )   =>    |-  (
 ( R  e.  Ring  /\  S  e.  I ) 
 ->  U  e.  Ring )
 
Theoremdivsrhm 15983* If  S is a two-sided ideal in  R, then the "natural map" from elements to their cosets is a ring homomorphism from  R to  R  /  S. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  U  =  ( R 
 /.s 
 ( R ~QG  S ) )   &    |-  I  =  (2Ideal `  R )   &    |-  X  =  ( Base `  R )   &    |-  F  =  ( x  e.  X  |->  [ x ] ( R ~QG  S ) )   =>    |-  ( ( R  e.  Ring  /\  S  e.  I ) 
 ->  F  e.  ( R RingHom  U ) )
 
Theoremcrngridl 15984 In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  I  =  (LIdeal `  R )   &    |-  O  =  (oppr `  R )   =>    |-  ( R  e.  CRing  ->  I  =  (LIdeal `  O ) )
 
Theoremcrng2idl 15985 In a commutative ring, a two-sided ideal is the same as a left ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  I  =  (LIdeal `  R )   =>    |-  ( R  e.  CRing  ->  I  =  (2Ideal `  R ) )
 
Theoremdivscrng 15986 The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  U  =  ( R 
 /.s 
 ( R ~QG  S ) )   &    |-  I  =  (LIdeal `  R )   =>    |-  (
 ( R  e.  CRing  /\  S  e.  I ) 
 ->  U  e.  CRing )
 
10.8.3  Principal ideal rings. Divisibility in the integers
 
Syntaxclpidl 15987 Ring left-principal-ideal function.
 class LPIdeal
 
Syntaxclpir 15988 Class of left principal ideal rings.
 class LPIR
 
Definitiondf-lpidl 15989* Define the class of left principal ideals of a ring, which are ideals with a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |- LPIdeal  =  ( w  e.  Ring  |->  U_ g  e.  ( Base `  w ) { (
 (RSpan `  w ) `  { g } ) } )
 
Definitiondf-lpir 15990 Define the class of left principal ideal rings, rings where every left ideal has a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |- LPIR  =  { w  e.  Ring  |  (LIdeal `  w )  =  (LPIdeal `  w ) }
 
Theoremlpival 15991* Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |-  K  =  (RSpan `  R )   &    |-  B  =  (
 Base `  R )   =>    |-  ( R  e.  Ring 
 ->  P  =  U_ g  e.  B  { ( K `
  { g }
 ) } )
 
Theoremislpidl 15992* Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |-  K  =  (RSpan `  R )   &    |-  B  =  (
 Base `  R )   =>    |-  ( R  e.  Ring 
 ->  ( I  e.  P  <->  E. g  e.  B  I  =  ( K `  { g } ) ) )
 
Theoremlpi0 15993 The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  {  .0.  }  e.  P )
 
Theoremlpi1 15994 The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( R  e.  Ring  ->  B  e.  P )
 
Theoremislpir 15995 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  ( R  e. LPIR  <->  ( R  e.  Ring  /\  U  =  P ) )
 
Theoremlpiss 15996 Principal ideals are a subclass of ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  ( R  e.  Ring  ->  P  C_  U )
 
Theoremislpir2 15997 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  P  =  (LPIdeal `  R )   &    |-  U  =  (LIdeal `  R )   =>    |-  ( R  e. LPIR  <->  ( R  e.  Ring  /\  U  C_  P )
 )
 
Theoremlpirrng 15998 Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  ( R  e. LPIR  ->  R  e.  Ring )
 
Theoremdrnglpir 15999 Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
 |-  ( R  e.  DivRing  ->  R  e. LPIR )
 
Theoremrspsn 16000* Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  B  =  ( Base `  R )   &    |-  K  =  (RSpan `  R )   &    |-  .||  =  ( ||r `  R )   =>    |-  ( ( R  e.  Ring  /\  G  e.  B ) 
 ->  ( K `  { G } )  =  { x  |  G  .||  x }
 )
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