HomeHome Metamath Proof Explorer
Theorem List (p. 163 of 325)
< Previous  Next >
Browser slow? Try the
Unicode version.

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

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-22404)
  Hilbert Space Explorer  Hilbert Space Explorer
(22405-23927)
  Users' Mathboxes  Users' Mathboxes
(23928-32493)
 

Theorem List for Metamath Proof Explorer - 16201-16300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlspsncv0 16201* 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 16202 Lemma for lspprat 16208. 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 16198 (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 16203 Lemma for lspprat 16208. 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 16204 Lemma for lspprat 16208. In the first case of lsppratlem1 16202, 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 16205 Lemma for lspprat 16208. In the second case of lsppratlem1 16202,  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 16206 Lemma for lspprat 16208. 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 16207 Lemma for lspprat 16208. 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 16208* 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 16209* 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 16210* 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 16211 Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 16209. (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 16212 The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 16199 and lbsacsbs 16211 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 14592. (Contributed by David Moews, 1-May-2017.)
 |-  J  =  (LBasis `  W )   =>    |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  S  ~~  T )
 
Theoremlbsextlem1 16213* Lemma for lbsext 16218. 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 16214* Lemma for lbsext 16218. 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 16215* Lemma for lbsext 16218. 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 16216* Lemma for lbsext 16218. lbsextlem3 16215 satisfies the conditions for the application of Zorn's lemma zorn 8371 (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 16217* 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 16218* 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 16219 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 16220 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 16221* If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 16222 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 16222* 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 16223 Extend class notation with the subring algebra generator.
 class subringAlg
 
Syntaxcrglmod 16224 Extend class notation with the left module induced by a ring over itself.
 class ringLMod
 
Syntaxclidl 16225 Ring left-ideal function.
 class LIdeal
 
Syntaxcrsp 16226 Ring span function.
 class RSpan
 
Definitiondf-sra 16227* 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 16228 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 16229 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 16230 Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |- RSpan  =  ( LSpan  o. ringLMod )
 
Theoremsraval 16231 Lemma for srabase 16233 through sravsca 16237. (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 16232 Lemma for srabase 16233 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 16233 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 16234 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 16235 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 16236 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 16237 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 16238 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 16239 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 16240 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 16241 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 16242 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 16243* 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 16244* 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 16245* 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 16246 ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |- ringLMod  Fn  _V
 
Theoremrlmval 16247 Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (ringLMod `  W )  =  ( ( subringAlg  `  W ) `
  ( Base `  W ) )
 
Theoremlidlval 16248 Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (LIdeal `  W )  =  ( LSubSp `  (ringLMod `  W ) )
 
Theoremrspval 16249 Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  (RSpan `  W )  =  ( LSpan `  (ringLMod `  W ) )
 
Theoremrlmbas 16250 Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( Base `  R )  =  ( Base `  (ringLMod `  R ) )
 
Theoremrlmplusg 16251 Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( +g  `  R )  =  ( +g  `  (ringLMod `  R )
 )
 
Theoremrlm0 16252 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 16253 Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( .r `  R )  =  ( .r `  (ringLMod `  R )
 )
 
Theoremrlmsca 16254 Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |-  ( R  e.  X  ->  R  =  (Scalar `  (ringLMod `  R ) ) )
 
Theoremrlmsca2 16255 Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.)
 |-  (  _I  `  R )  =  (Scalar `  (ringLMod `  R ) )
 
Theoremrlmvsca 16256 Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  ( .r `  R )  =  ( .s `  (ringLMod `  R )
 )
 
Theoremrlmtopn 16257 Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( TopOpen `  R )  =  ( TopOpen `  (ringLMod `  R ) )
 
Theoremrlmds 16258 Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( dist `  R )  =  ( dist `  (ringLMod `  R ) )
 
Theoremrlmlmod 16259 The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |-  ( R  e.  Ring  ->  (ringLMod `  R )  e. 
 LMod )
 
Theoremrlmlvec 16260 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 16261 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 16262 Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
 |-  ( + f `  (mulGrp `  R ) )  =  ( .s f `  (ringLMod `  R )
 )
 
Theoremlidlss 16263 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 16264 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 16265* 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 16266 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 16267 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 16268 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 16269 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 16270 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 16271 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 16272 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 16273 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 16274 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 16275 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 16276 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 16277 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 16278 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 16279 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 16280 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 16281 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 16282* 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 16283 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 16284* 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 16285 Ring two-sided ideal function.
 class 2Ideal
 
Definitiondf-2idl 16286 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 16287 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 16288 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 16289 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 16290 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 16291* 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 16292 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 16293 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 16294 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 16295 Ring left-principal-ideal function.
 class LPIdeal
 
Syntaxclpir 16296 Class of left principal ideal rings.
 class LPIR
 
Definitiondf-lpidl 16297* 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 16298 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 16299* 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 16300* 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 } ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32493
  Copyright terms: Public domain < Previous  Next >