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Theorem List for Metamath Proof Explorer - 28001-28100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlfl1dim2N 28001* Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 28000 may be more compatible with lspsn 15594. (Contributed by NM, 24-Oct-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  ( ph  ->  W  e.  LVec
 )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  { g  e.  F  |  ( L `
  G )  C_  ( L `  g ) }  =  { g  e.  F  |  E. k  e.  K  g  =  ( G  o F  .x.  ( V  X.  { k } ) ) }
 )
 
16.22.5  Opposite rings and dual vector spaces
 
Syntaxcld 28002 Extend class notation with left dualvector space.
 class LDual
 
Definitiondf-ldual 28003* Define the (left) dual of a left vector space (or module) in which the vectors are functionals. In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in [Holland95] p. 218. This allows us to reuse our existing collection of left vector space theorems. The restriction on  o F ( +g  `  v
) allows it to be a set; see ofmres 5968. Note the operation reversal in the scalar product to allow us to use the original scalar ring instead of the oppr ring, for convenience. (Contributed by NM, 18-Oct-2014.)
 |- LDual  =  ( v  e.  _V  |->  ( { <. ( Base `  ndx ) ,  (LFnl `  v
 ) >. ,  <. ( +g  ` 
 ndx ) ,  (  o F ( +g  `  (Scalar `  v ) )  |`  ( (LFnl `  v )  X.  (LFnl `  v )
 ) ) >. ,  <. (Scalar `  ndx ) ,  (oppr `  (Scalar `  v ) ) >. }  u.  { <. ( .s
 `  ndx ) ,  (
 k  e.  ( Base `  (Scalar `  v )
 ) ,  f  e.  (LFnl `  v )  |->  ( f  o F
 ( .r `  (Scalar `  v ) ) ( ( Base `  v )  X.  { k } )
 ) ) >. } )
 )
 
Theoremldualset 28004* Define the (left) dual of a left vector space (or module) in which the vectors are functionals. In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in [Holland95] p. 218. This allows us to reuse our existing collection of left vector space theorems. Note the operation reversal in the scalar product to allow us to use the original scalar ring instead of the oppr ring, for convenience. (Contributed by NM, 18-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  .+b  =  (  o F  .+  |`  ( F  X.  F ) )   &    |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   &    |-  .xb  =  (
 k  e.  K ,  f  e.  F  |->  ( f  o F  .x.  ( V  X.  { k }
 ) ) )   &    |-  ( ph  ->  W  e.  X )   =>    |-  ( ph  ->  D  =  ( { <. ( Base ` 
 ndx ) ,  F >. ,  <. ( +g  `  ndx ) ,  .+b  >. ,  <. (Scalar `  ndx ) ,  O >. }  u.  { <. ( .s `  ndx ) ,  .xb  >. } ) )
 
Theoremldualvbase 28005 The vectors of a dual space are functionals of the original space. (Contributed by NM, 18-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  V  =  ( Base `  D )   &    |-  ( ph  ->  W  e.  X )   =>    |-  ( ph  ->  V  =  F )
 
Theoremldualelvbase 28006 Utility theorem for converting a functional to a vector of the dual space in order to use standard vector theorems. (Contributed by NM, 6-Jan-2015.)
 |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  V  =  ( Base `  D )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  G  e.  V )
 
Theoremldualfvadd 28007 Vector addition in the dual of a vector space. (Contributed by NM, 21-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  D  =  (LDual `  W )   &    |-  .+b  =  ( +g  `  D )   &    |-  ( ph  ->  W  e.  X )   &    |-  .+^  =  (  o F  .+  |`  ( F  X.  F ) )   =>    |-  ( ph  ->  .+b 
 =  .+^  )
 
Theoremldualvadd 28008 Vector addition in the dual of a vector space. (Contributed by NM, 21-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  D  =  (LDual `  W )   &    |-  .+b  =  ( +g  `  D )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( G  .+b  H )  =  ( G  o F  .+  H ) )
 
Theoremldualvaddcl 28009 The value of vector addition in the dual of a vector space is a functional. (Contributed by NM, 21-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( G  .+  H )  e.  F )
 
Theoremldualvaddval 28010 The value of the value of vector addition in the dual of a vector space. (Contributed by NM, 7-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  .+b  =  ( +g  `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  (
 ( G  .+b  H ) `
  X )  =  ( ( G `  X )  .+  ( H `
  X ) ) )
 
Theoremldualsca 28011 The ring of scalars of the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
 |-  F  =  (Scalar `  W )   &    |-  O  =  (oppr `  F )   &    |-  D  =  (LDual `  W )   &    |-  R  =  (Scalar `  D )   &    |-  ( ph  ->  W  e.  X )   =>    |-  ( ph  ->  R  =  O )
 
Theoremldualsbase 28012 Base set of scalar ring for the dual of a vector space. (Contributed by NM, 24-Oct-2014.)
 |-  F  =  (Scalar `  W )   &    |-  L  =  ( Base `  F )   &    |-  D  =  (LDual `  W )   &    |-  R  =  (Scalar `  D )   &    |-  K  =  ( Base `  R )   &    |-  ( ph  ->  W  e.  V )   =>    |-  ( ph  ->  K  =  L )
 
TheoremldualsaddN 28013 Scalar addition for the dual of a vector space. (Contributed by NM, 24-Oct-2014.) (New usage is discouraged.)
 |-  F  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  F )   &    |-  D  =  (LDual `  W )   &    |-  R  =  (Scalar `  D )   &    |-  .+b  =  ( +g  `  R )   &    |-  ( ph  ->  W  e.  V )   =>    |-  ( ph  ->  .+b  =  .+  )
 
Theoremldualsmul 28014 Scalar multiplication for the dual of a vector space. (Contributed by NM, 19-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .r `  F )   &    |-  D  =  (LDual `  W )   &    |-  R  =  (Scalar `  D )   &    |-  .xb  =  ( .r `  R )   &    |-  ( ph  ->  W  e.  V )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  K )   =>    |-  ( ph  ->  ( X  .xb  Y )  =  ( Y  .x.  X ) )
 
Theoremldualfvs 28015* Scalar product operation for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  D  =  (LDual `  W )   &    |-  .xb  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  Y )   &    |-  .x.  =  ( k  e.  K ,  f  e.  F  |->  ( f  o F  .X.  ( V  X.  {
 k } ) ) )   =>    |-  ( ph  ->  .xb  =  .x.  )
 
Theoremldualvs 28016 Scalar product operation value (which is a functional) for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  D  =  (LDual `  W )   &    |-  .xb  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  Y )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( X  .xb  G )  =  ( G  o F  .X.  ( V  X.  { X } ) ) )
 
Theoremldualvsval 28017 Value of scalar product operation value for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  D  =  (LDual `  W )   &    |-  .xb  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  Y )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ( ( X  .xb  G ) `
  A )  =  ( ( G `  A )  .X.  X ) )
 
Theoremldualvscl 28018 The scalar product operation value is a functional. (Contributed by NM, 18-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LMod
 )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( X  .x.  G )  e.  F )
 
Theoremldualvaddcom 28019 Commutative law for vector (functional) addition. (Contributed by NM, 17-Jan-2015.)
 |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  F )   &    |-  ( ph  ->  Y  e.  F )   =>    |-  ( ph  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremldualvsass 28020 Associative law for scalar product operation. (Contributed by NM, 20-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  D  =  (LDual `  W )   &    |- 
 .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  (
 ( Y  .X.  X )  .x.  G )  =  ( X  .x.  ( Y  .x.  G ) ) )
 
Theoremldualvsass2 28021 Associative law for scalar product operation, using operations from the dual space. (Contributed by NM, 20-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  D  =  (LDual `  W )   &    |-  Q  =  (Scalar `  D )   &    |-  .X.  =  ( .r `  Q )   &    |-  .x. 
 =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  (
 ( X  .X.  Y )  .x.  G )  =  ( X  .x.  ( Y  .x.  G ) ) )
 
Theoremldualvsdi1 28022 Distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  D  =  (LDual `  W )   &    |-  .+  =  ( +g  `  D )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LMod
 )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( X  .x.  ( G  .+  H ) )  =  ( ( X  .x.  G )  .+  ( X 
 .x.  H ) ) )
 
Theoremldualvsdi2 28023 Reverse distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  K  =  ( Base `  R )   &    |-  D  =  (LDual `  W )   &    |-  .+b  =  ( +g  `  D )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LMod
 )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  (
 ( X  .+  Y )  .x.  G )  =  ( ( X  .x.  G )  .+b  ( Y  .x.  G ) ) )
 
Theoremldualgrplem 28024 Lemma for ldualgrp 28025. (Contributed by NM, 22-Oct-2014.)
 |-  D  =  (LDual `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  o F ( +g  `  W )   &    |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  (
 Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   &    |-  .x.  =  ( .s `  D )   =>    |-  ( ph  ->  D  e.  Grp )
 
Theoremldualgrp 28025 The dual of a vector space is a group. (Contributed by NM, 21-Oct-2014.)
 |-  D  =  (LDual `  W )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  D  e.  Grp )
 
Theoremldual0 28026 The zero scalar of the dual of a vector space. (Contributed by NM, 28-Dec-2014.)
 |-  R  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  (LDual `  W )   &    |-  S  =  (Scalar `  D )   &    |-  O  =  ( 0g `  S )   &    |-  ( ph  ->  W  e.  LMod
 )   =>    |-  ( ph  ->  O  =  .0.  )
 
Theoremldual1 28027 The unit scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015.)
 |-  R  =  (Scalar `  W )   &    |-  .1.  =  ( 1r `  R )   &    |-  D  =  (LDual `  W )   &    |-  S  =  (Scalar `  D )   &    |-  I  =  ( 1r `  S )   &    |-  ( ph  ->  W  e.  LMod
 )   =>    |-  ( ph  ->  I  =  .1.  )
 
Theoremldualneg 28028 The negative of a scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015.)
 |-  R  =  (Scalar `  W )   &    |-  M  =  ( inv g `  R )   &    |-  D  =  (LDual `  W )   &    |-  S  =  (Scalar `  D )   &    |-  N  =  ( inv g `  S )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  N  =  M )
 
Theoremldual0v 28029 The zero vector of the dual of a vector space. (Contributed by NM, 24-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  (LDual `  W )   &    |-  O  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LMod
 )   =>    |-  ( ph  ->  O  =  ( V  X.  {  .0.  } ) )
 
Theoremldual0vcl 28030 The dual zero vector is a functional. (Contributed by NM, 5-Mar-2015.)
 |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  .0.  e.  F )
 
Theoremlduallmodlem 28031 Lemma for lduallmod 28032. (Contributed by NM, 22-Oct-2014.)
 |-  D  =  (LDual `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  o F ( +g  `  W )   &    |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  (
 Base `  R )   &    |-  .X.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   &    |-  .x.  =  ( .s `  D )   =>    |-  ( ph  ->  D  e.  LMod )
 
Theoremlduallmod 28032 The dual of a left module is also a left module. (Contributed by NM, 22-Oct-2014.)
 |-  D  =  (LDual `  W )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  D  e.  LMod
 )
 
Theoremlduallvec 28033 The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 15240; otherwise, the dual would be a right vector space as is sometimes the case in the literature. (Contributed by NM, 22-Oct-2014.)
 |-  D  =  (LDual `  W )   &    |-  ( ph  ->  W  e.  LVec )   =>    |-  ( ph  ->  D  e.  LVec
 )
 
Theoremldualvsub 28034 The value of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
 |-  R  =  (Scalar `  W )   &    |-  N  =  ( inv g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  .+  =  ( +g  `  D )   &    |-  .x.  =  ( .s `  D )   &    |-  .-  =  ( -g `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( G  .-  H )  =  ( G  .+  (
 ( N `  .1.  )  .x.  H ) ) )
 
Theoremldualvsubcl 28035 Closure of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
 |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  .-  =  ( -g `  D )   &    |-  ( ph  ->  W  e.  LMod
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( G  .-  H )  e.  F )
 
Theoremldualvsubval 28036 The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 28034? (Requires  D to oppr conversion.) (Contributed by NM, 26-Feb-2015.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  S  =  ( -g `  R )   &    |-  F  =  (LFnl `  W )   &    |-  D  =  (LDual `  W )   &    |-  .-  =  ( -g `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  (
 ( G  .-  H ) `  X )  =  ( ( G `  X ) S ( H `  X ) ) )
 
Theoremldualssvscl 28037 Closure of scalar product in a dual subspace.) (Contributed by NM, 5-Feb-2015.)
 |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  S  =  ( LSubSp `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( X  .x.  Y )  e.  U )
 
Theoremldualssvsubcl 28038 Closure of vector subtraction in a dual subspace.) (Contributed by NM, 9-Mar-2015.)
 |-  D  =  (LDual `  W )   &    |-  .-  =  ( -g `  D )   &    |-  S  =  ( LSubSp `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( X  .-  Y )  e.  U )
 
Theoremldual0vs 28039 Scalar zero times a functional is the zero functional. (Contributed by NM, 17-Feb-2015.)
 |-  F  =  (LFnl `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  O  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  (  .0.  .x.  G )  =  O )
 
Theoremlkr0f2 28040 The kernel of the zero functional is the set of all vectors. (Contributed by NM, 4-Feb-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( ( K `  G )  =  V  <->  G  =  .0.  ) )
 
Theoremlduallkr3 28041 The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 22-Feb-2015.)
 |-  H  =  (LSHyp `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( ( K `  G )  e.  H  <->  G  =/=  .0.  )
 )
 
TheoremlkrpssN 28042 Proper subset relation between kernels. (Contributed by NM, 16-Feb-2015.) (New usage is discouraged.)
 |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  (
 ( K `  G )  C.  ( K `  H )  <->  ( G  =/=  .0.  /\  H  =  .0.  )
 ) )
 
Theoremlkrin 28043 Intersection of the kernels of 2 functionals is included in the kernel of their sum. (Contributed by NM, 7-Jan-2015.)
 |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( ( K `  G )  i^i  ( K `  H ) )  C_  ( K `  ( G 
 .+  H ) ) )
 
Theoremeqlkr4 28044* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 4-Feb-2015.)
 |-  S  =  (Scalar `  W )   &    |-  R  =  ( Base `  S )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LVec
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  ( K `  G )  =  ( K `  H ) )   =>    |-  ( ph  ->  E. r  e.  R  H  =  ( r  .x.  G )
 )
 
Theoremldual1dim 28045* Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  N  =  ( LSpan `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( N `  { G }
 )  =  { g  e.  F  |  ( L `
  G )  C_  ( L `  g ) } )
 
Theoremldualkrsc 28046 The kernel of a non-zero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 28-Dec-2014.)
 |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  X  =/=  .0.  )   =>    |-  ( ph  ->  ( L `  ( X  .x.  G ) )  =  ( L `  G ) )
 
Theoremlkrss 28047 The kernel of a scalar product of a functional includes the kernel of the functional. (Contributed by NM, 27-Jan-2015.)
 |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LVec
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  K )   =>    |-  ( ph  ->  ( L `  G ) 
 C_  ( L `  ( X  .x.  G ) ) )
 
Theoremlkrss2N 28048* Two functionals with kernels in a subset relationship. (Contributed by NM, 17-Feb-2015.) (New usage is discouraged.)
 |-  S  =  (Scalar `  W )   &    |-  R  =  ( Base `  S )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LVec
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( ( K `  G )  C_  ( K `  H )  <->  E. r  e.  R  H  =  ( r  .x.  G ) ) )
 
TheoremlkreqN 28049 Proportional functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
 |-  S  =  (Scalar `  W )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  A  e.  ( R  \  {  .0.  } ) )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  G  =  ( A  .x.  H )
 )   =>    |-  ( ph  ->  ( K `  G )  =  ( K `  H ) )
 
TheoremlkrlspeqN 28050 Condition for colinear functionals to have equal kernels. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
 |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  N  =  ( LSpan `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  G  e.  (
 ( N `  { H } )  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( L `  G )  =  ( L `  H ) )
 
16.22.6  Ortholattices and orthomodular lattices
 
Syntaxcops 28051 Extend class notation with orthoposets.
 class  OP
 
SyntaxccmtN 28052 Extend class notation with the commutes relation.
 class  cm
 
Syntaxcol 28053 Extend class notation with orthlattices.
 class  OL
 
Syntaxcoml 28054 Extend class notation with orthomodular lattices.
 class  OML
 
Definitiondf-oposet 28055* Define the class of orthoposets. (Contributed by NM, 20-Oct-2011.)
 |-  OP  =  { p  e.  Poset  |  ( ( ( 0. `  p )  e.  ( Base `  p )  /\  ( 1. `  p )  e.  ( Base `  p ) )  /\  E. o
 ( o  =  ( oc `  p ) 
 /\  A. a  e.  ( Base `  p ) A. b  e.  ( Base `  p ) ( ( ( o `  a
 )  e.  ( Base `  p )  /\  (
 o `  ( o `  a ) )  =  a  /\  ( a ( le `  p ) b  ->  ( o `
  b ) ( le `  p ) ( o `  a
 ) ) )  /\  ( a ( join `  p ) ( o `
  a ) )  =  ( 1. `  p )  /\  ( a (
 meet `  p ) ( o `  a ) )  =  ( 0. `  p ) ) ) ) }
 
Definitiondf-cmtN 28056* Define the commutes relation for orthoposets. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Nov-2011.)
 |-  cm  =  ( p  e.  _V  |->  {
 <. x ,  y >.  |  ( x  e.  ( Base `  p )  /\  y  e.  ( Base `  p )  /\  x  =  ( ( x (
 meet `  p ) y ) ( join `  p ) ( x (
 meet `  p ) ( ( oc `  p ) `  y ) ) ) ) } )
 
Definitiondf-ol 28057 Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
 |-  OL  =  ( Lat  i^i  OP )
 
Definitiondf-oml 28058* Define the class of orthomodular lattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
 |-  OML  =  { l  e.  OL  |  A. a  e.  ( Base `  l ) A. b  e.  ( Base `  l ) ( a ( le `  l
 ) b  ->  b  =  ( a ( join `  l ) ( b ( meet `  l )
 ( ( oc `  l ) `  a
 ) ) ) ) }
 
Theoremisopos 28059* The predicate "is an orthoposet." (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( K  e.  OP  <->  (
 ( K  e.  Poset  /\ 
 .0.  e.  B  /\  .1.  e.  B )  /\  A. x  e.  B  A. y  e.  B  (
 ( (  ._|_  `  x )  e.  B  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x  /\  ( x  .<_  y  ->  (  ._|_  `  y )  .<_  (  ._|_  `  x ) ) )  /\  ( x  .\/  (  ._|_  `  x ) )  =  .1.  /\  ( x  ./\  (  ._|_  `  x ) )  =  .0.  ) ) )
 
TheoremisopiN 28060* Properties that determine an orthoposet (constructed structure version). (Contributed by NM, 13-Sep-2011.) (New usage is discouraged.)
 |-  K  e.  Poset   &    |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  .0.  e.  B   &    |-  .1.  e.  B   &    |-  ( x  e.  B  ->  (  ._|_  `  x )  e.  B )   &    |-  ( x  e.  B  ->  (  ._|_  `  (  ._|_  `  x ) )  =  x )   &    |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .<_  y  ->  (  ._|_  `  y )  .<_  ( 
 ._|_  `  x ) ) )   &    |-  ( x  e.  B  ->  ( x  .\/  (  ._|_  `  x ) )  =  .1.  )   &    |-  ( x  e.  B  ->  ( x  ./\  (  ._|_  `  x ) )  =  .0.  )   =>    |-  K  e.  OP
 
Theoremopposet 28061 Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)
 |-  ( K  e.  OP  ->  K  e.  Poset )
 
Theoremoposlem 28062 Lemma for orthoposet properties. (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( ( (  ._|_  `  X )  e.  B  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  ( X  .<_  Y  ->  (  ._|_  `  Y )  .<_  ( 
 ._|_  `  X ) ) )  /\  ( X 
 .\/  (  ._|_  `  X ) )  =  .1.  /\  ( X  ./\  (  ._|_  `  X ) )  =  .0.  ) )
 
Theoremop0cl 28063 An orthoposet has a zero element. (h0elch 21664 analog.) (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( K  e.  OP  ->  .0.  e.  B )
 
Theoremop1cl 28064 An orthoposet has a unit element. (helch 21653 analog.) (Contributed by NM, 22-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .1.  =  ( 1. `  K )   =>    |-  ( K  e.  OP  ->  .1.  e.  B )
 
Theoremop0le 28065 Orthoposet zero is less than or equal to any element. (ch0le 21850 analog.) (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  .<_  X )
 
Theoremople0 28066 An element less than or equal to zero equals zero. (chle0 21852 analog.) (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .<_  .0.  <->  X  =  .0.  ) )
 
Theoremopnlen0 28067 An element not less than another is nonzero. TODO: Look for uses of necon3bd 2449 and op0le 28065 to see if this is useful elsewhere. (Contributed by NM, 5-May-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  /\  -.  X  .<_  Y )  ->  X  =/=  .0.  )
 
Theoremlub0N 28068 The least upper bound of the empty set is the zero element. (Contributed by NM, 15-Sep-2013.) (New usage is discouraged.)
 |-  .1.  =  ( lub `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( K  e.  OP  ->  (  .1.  `  (/) )  =  .0.  )
 
Theoremopltn0 28069 A lattice element greater than zero is nonzero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  .<  X  <->  X  =/=  .0.  ) )
 
Theoremople1 28070 Any element is less than the orthoposet unit. (chss 21639 analog.) (Contributed by NM, 23-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  .<_  .1.  )
 
Theoremop1le 28071 If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 21852 analog.) (Contributed by NM, 5-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .1.  .<_  X  <->  X  =  .1.  )
 )
 
Theoremglb0N 28072 The greatest lower bound of the empty set is the unit element. (Contributed by NM, 5-Dec-2011.) (New usage is discouraged.)
 |-  G  =  ( glb `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( K  e.  OP  ->  ( G `  (/) )  =  .1.  )
 
Theoremopoccl 28073 Closure of orthocomplement operation. (choccl 21715 analog.) (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  X )  e.  B )
 
Theoremopococ 28074 Double negative law for orthoposets. (ococ 21815 analog.) (Contributed by NM, 13-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
 
Theoremopcon3b 28075 Contraposition law for orthoposets. (chcon3i 21875 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  =  Y  <->  ( 
 ._|_  `  Y )  =  (  ._|_  `  X ) ) )
 
Theoremopcon2b 28076 Orthocomplement contraposition law. (negcon2 8980 analog.) (Contributed by NM, 16-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  =  ( 
 ._|_  `  Y )  <->  Y  =  (  ._|_  `  X ) ) )
 
Theoremopcon1b 28077 Orthocomplement contraposition law. (negcon1 8979 analog.) (Contributed by NM, 24-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( (  ._|_  `  X )  =  Y  <->  (  ._|_  `  Y )  =  X )
 )
 
Theoremoplecon3 28078 Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .<_  Y  ->  ( 
 ._|_  `  Y )  .<_  ( 
 ._|_  `  X ) ) )
 
Theoremoplecon3b 28079 Contraposition law for orthoposets. (chsscon3 21909 analog.) (Contributed by NM, 4-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .<_  Y  <->  (  ._|_  `  Y )  .<_  (  ._|_  `  X ) ) )
 
Theoremoplecon1b 28080 Contraposition law for strict ordering in orthoposets. (chsscon1 21910 analog.) (Contributed by NM, 6-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( (  ._|_  `  X )  .<_  Y  <->  (  ._|_  `  Y )  .<_  X ) )
 
Theoremopoc1 28081 Orthocomplement of orthoposet unit. (Contributed by NM, 24-Jan-2012.)
 |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( K  e.  OP  ->  (  ._|_  `  .1.  )  =  .0.  )
 
Theoremopoc0 28082 Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)
 |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( K  e.  OP  ->  (  ._|_  `  .0.  )  =  .1.  )
 
Theoremopltcon3b 28083 Contraposition law for strict ordering in orthoposets. (chpsscon3 21912 analog.) (Contributed by NM, 4-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .<  Y  <->  (  ._|_  `  Y )  .<  (  ._|_  `  X ) ) )
 
Theoremopltcon1b 28084 Contraposition law for strict ordering in orthoposets. (chpsscon1 21913 analog.) (Contributed by NM, 5-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( (  ._|_  `  X )  .<  Y  <->  (  ._|_  `  Y )  .<  X ) )
 
Theoremopltcon2b 28085 Contraposition law for strict ordering in orthoposets. (chsscon2 21911 analog.) (Contributed by NM, 5-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .<  (  ._|_  `  Y )  <->  Y  .<  (  ._|_  `  X ) ) )
 
Theoremopexmid 28086 Law of excluded middle for orthoposets. (chjo 21924 analog.) (Contributed by NM, 13-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  .\/ 
 =  ( join `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .\/  (  ._|_  `  X )
 )  =  .1.  )
 
Theoremopnoncon 28087 Law of contradiction for orthoposets. (chocin 21904 analog.) (Contributed by NM, 13-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  ./\ 
 =  ( meet `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  ./\  (  ._|_  `  X )
 )  =  .0.  )
 
TheoremriotaocN 28088* The orthocomplement of the unique poset element such that  ps. (riotaneg 9609 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  ( x  =  (  ._|_  `  y )  ->  ( ph  <->  ps ) )   =>    |-  ( ( K  e.  OP  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  B ph )  =  (  ._|_  `  ( iota_ y  e.  B ps ) ) )
 
TheoremcmtfvalN 28089* Value of commutes relation. (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( K  e.  A  ->  C  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
 
TheoremcmtvalN 28090 Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 22011 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X C Y  <->  X  =  ( ( X 
 ./\  Y )  .\/  ( X  ./\  (  ._|_  `  Y ) ) ) ) )
 
Theoremisolat 28091 The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OL  <->  ( K  e.  Lat  /\  K  e.  OP )
 )
 
Theoremollat 28092 An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OL  ->  K  e.  Lat )
 
Theoremolop 28093 An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OL  ->  K  e.  OP )
 
TheoremolposN 28094 An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
 |-  ( K  e.  OL  ->  K  e.  Poset )
 
TheoremisolatiN 28095 Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
 |-  K  e.  Lat   &    |-  K  e.  OP   =>    |-  K  e.  OL
 
Theoremoldmm1 28096 DeMorgan's law for meet in an ortholattice. (chdmm1 21934 analog.) (Contributed by NM, 6-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  ( X  ./\  Y ) )  =  ( (  ._|_  `  X )  .\/  (  ._|_  `  Y ) ) )
 
Theoremoldmm2 28097 DeMorgan's law for meet in an ortholattice. (chdmm2 21935 analog.) (Contributed by NM, 6-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  (
 (  ._|_  `  X )  ./\ 
 Y ) )  =  ( X  .\/  (  ._|_  `  Y ) ) )
 
Theoremoldmm3N 28098 DeMorgan's law for meet in an ortholattice. (chdmm3 21936 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  ( X  ./\  (  ._|_  `  Y ) ) )  =  ( (  ._|_  `  X )  .\/  Y ) )
 
Theoremoldmm4 28099 DeMorgan's law for meet in an ortholattice. (chdmm4 21937 analog.) (Contributed by NM, 6-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  (
 (  ._|_  `  X )  ./\  (  ._|_  `  Y ) ) )  =  ( X  .\/  Y )
 )
 
Theoremoldmj1 28100 DeMorgan's law for join in an ortholattice. (chdmj1 21938 analog.) (Contributed by NM, 6-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  ( X  .\/  Y ) )  =  ( (  ._|_  `  X )  ./\  (  ._|_  `  Y ) ) )
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