HomeHome Metamath Proof Explorer
Theorem List (p. 286 of 314)
< 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
(1-21458)
  Hilbert Space Explorer  Hilbert Space Explorer
(21459-22981)
  Users' Mathboxes  Users' Mathboxes
(22982-31321)
 

Theorem List for Metamath Proof Explorer - 28501-28600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremldualvsdi2 28501 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 28502 Lemma for ldualgrp 28503. (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 28503 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 28504 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 28505 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 28506 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 28507 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 28508 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 28509 Lemma for lduallmod 28510. (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 28510 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 28511 The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 15367; 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 28512 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 28513 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 28514 The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 28512? (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 28515 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 28516 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 28517 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 28518 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 28519 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 28520 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 28521 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 28522* 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 28523* 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 28524 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 28525 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 28526* 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 28527 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 28528 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 ) )
 
18.25.10  Ortholattices and orthomodular lattices
 
Syntaxcops 28529 Extend class notation with orthoposets.
 class  OP
 
SyntaxccmtN 28530 Extend class notation with the commutes relation.
 class  cm
 
Syntaxcol 28531 Extend class notation with orthlattices.
 class  OL
 
Syntaxcoml 28532 Extend class notation with orthomodular lattices.
 class  OML
 
Definitiondf-oposet 28533* 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 28534* 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 28535 Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
 |-  OL  =  ( Lat  i^i  OP )
 
Definitiondf-oml 28536* 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 28537* 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 28538* 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 28539 Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)
 |-  ( K  e.  OP  ->  K  e.  Poset )
 
Theoremoposlem 28540 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 28541 An orthoposet has a zero element. (h0elch 21794 analog.) (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( K  e.  OP  ->  .0.  e.  B )
 
Theoremop1cl 28542 An orthoposet has a unit element. (helch 21783 analog.) (Contributed by NM, 22-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .1.  =  ( 1. `  K )   =>    |-  ( K  e.  OP  ->  .1.  e.  B )
 
Theoremop0le 28543 Orthoposet zero is less than or equal to any element. (ch0le 21980 analog.) (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  .<_  X )
 
Theoremople0 28544 An element less than or equal to zero equals zero. (chle0 21982 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 28545 An element not less than another is nonzero. TODO: Look for uses of necon3bd 2458 and op0le 28543 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 28546 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 28547 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 28548 Any element is less than the orthoposet unit. (chss 21769 analog.) (Contributed by NM, 23-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  .<_  .1.  )
 
Theoremop1le 28549 If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 21982 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 28550 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 28551 Closure of orthocomplement operation. (choccl 21845 analog.) (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  X )  e.  B )
 
Theoremopococ 28552 Double negative law for orthoposets. (ococ 21945 analog.) (Contributed by NM, 13-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
 
Theoremopcon3b 28553 Contraposition law for orthoposets. (chcon3i 22005 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 28554 Orthocomplement contraposition law. (negcon2 9068 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 28555 Orthocomplement contraposition law. (negcon1 9067 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 28556 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 28557 Contraposition law for orthoposets. (chsscon3 22039 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 28558 Contraposition law for strict ordering in orthoposets. (chsscon1 22040 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 28559 Orthocomplement of orthoposet unit. (Contributed by NM, 24-Jan-2012.)
 |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( K  e.  OP  ->  (  ._|_  `  .1.  )  =  .0.  )
 
Theoremopoc0 28560 Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)
 |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( K  e.  OP  ->  (  ._|_  `  .0.  )  =  .1.  )
 
Theoremopltcon3b 28561 Contraposition law for strict ordering in orthoposets. (chpsscon3 22042 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 28562 Contraposition law for strict ordering in orthoposets. (chpsscon1 22043 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 28563 Contraposition law for strict ordering in orthoposets. (chsscon2 22041 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 28564 Law of excluded middle for orthoposets. (chjo 22054 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 28565 Law of contradiction for orthoposets. (chocin 22034 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 28566* The orthocomplement of the unique poset element such that  ps. (riotaneg 9697 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 28567* 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 28568 Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 22123 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 28569 The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OL  <->  ( K  e.  Lat  /\  K  e.  OP )
 )
 
Theoremollat 28570 An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OL  ->  K  e.  Lat )
 
Theoremolop 28571 An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OL  ->  K  e.  OP )
 
TheoremolposN 28572 An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
 |-  ( K  e.  OL  ->  K  e.  Poset )
 
TheoremisolatiN 28573 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 28574 DeMorgan's law for meet in an ortholattice. (chdmm1 22064 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 28575 DeMorgan's law for meet in an ortholattice. (chdmm2 22065 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 28576 DeMorgan's law for meet in an ortholattice. (chdmm3 22066 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 28577 DeMorgan's law for meet in an ortholattice. (chdmm4 22067 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 28578 DeMorgan's law for join in an ortholattice. (chdmj1 22068 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 ) ) )
 
Theoremoldmj2 28579 DeMorgan's law for join in an ortholattice. (chdmj2 22069 analog.) (Contributed by NM, 7-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  (
 (  ._|_  `  X )  .\/  Y ) )  =  ( X  ./\  (  ._|_  `  Y ) ) )
 
Theoremoldmj3 28580 DeMorgan's law for join in an ortholattice. (chdmj3 22070 analog.) (Contributed by NM, 7-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  ( X  .\/  (  ._|_  `  Y ) ) )  =  ( (  ._|_  `  X )  ./\  Y ) )
 
Theoremoldmj4 28581 DeMorgan's law for join in an ortholattice. (chdmj4 22071 analog.) (Contributed by NM, 7-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  (
 (  ._|_  `  X )  .\/  (  ._|_  `  Y ) ) )  =  ( X  ./\  Y )
 )
 
Theoremolj01 28582 An ortholattice element joined with zero equals itself. (chj0 22036 analog.) (Contributed by NM, 19-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  .\/  .0.  )  =  X )
 
Theoremolj02 28583 An ortholattice element joined with zero equals itself. (Contributed by NM, 28-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B )  ->  (  .0.  .\/  X )  =  X )
 
Theoremolm11 28584 The meet of an ortholattice element with one equals itself. (chm1i 21995 analog.) (Contributed by NM, 22-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .1.  )  =  X )
 
Theoremolm12 28585 The meet of an ortholattice element with one equals itself. (Contributed by NM, 22-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B )  ->  (  .1.  ./\  X )  =  X )
 
TheoremlatmassOLD 28586 Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3354 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  ./\  Y )  ./\  Z )  =  ( X 
 ./\  ( Y  ./\  Z ) ) )
 
Theoremlatm12 28587 A rearrangement of lattice meet. (in12 3355 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  ./\  ( Y  ./\  Z ) )  =  ( Y 
 ./\  ( X  ./\  Z ) ) )
 
Theoremlatm32 28588 A rearrangement of lattice meet. (in12 3355 analog.) (Contributed by NM, 13-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  ./\  Y )  ./\  Z )  =  ( ( X  ./\  Z )  ./\ 
 Y ) )
 
Theoremlatmrot 28589 Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  ./\  Y )  ./\  Z )  =  ( ( Z  ./\  X )  ./\ 
 Y ) )
 
Theoremlatm4 28590 Rearrangement of lattice meet of 4 classes. (in4 3360 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B ) 
 /\  ( Z  e.  B  /\  W  e.  B ) )  ->  ( ( X  ./\  Y )  ./\  ( Z  ./\  W ) )  =  ( ( X  ./\  Z )  ./\  ( Y  ./\  W ) ) )
 
TheoremlatmmdiN 28591 Lattice meet distributes over itself. (inindi 3361 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  ./\  ( Y  ./\  Z ) )  =  ( ( X  ./\  Y )  ./\  ( X  ./\  Z ) ) )
 
Theoremlatmmdir 28592 Lattice meet distributes over itself. (inindir 3362 analog.) (Contributed by NM, 6-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  ./\  Y )  ./\  Z )  =  ( ( X  ./\  Z )  ./\  ( Y  ./\  Z ) ) )
 
Theoremolm01 28593 Meet with lattice zero is zero. (chm0 22030 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .0.  )  =  .0.  )
 
Theoremolm02 28594 Meet with lattice zero is zero. (Contributed by NM, 9-Oct-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B )  ->  (  .0.  ./\  X )  =  .0.  )
 
Theoremisoml 28595* The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( K  e.  OML  <->  ( K  e.  OL  /\  A. x  e.  B  A. y  e.  B  ( x  .<_  y 
 ->  y  =  ( x  .\/  ( y  ./\  (  ._|_  `  x )
 ) ) ) ) )
 
TheoremisomliN 28596* Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
 |-  K  e.  OL   &    |-  B  =  (
 Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .<_  y  ->  y  =  ( x  .\/  (
 y  ./\  (  ._|_  `  x ) ) ) ) )   =>    |-  K  e.  OML
 
Theoremomlol 28597 An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OML  ->  K  e.  OL )
 
Theoremomlop 28598 An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)
 |-  ( K  e.  OML  ->  K  e.  OP )
 
Theoremomllat 28599 An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)
 |-  ( K  e.  OML  ->  K  e.  Lat )
 
Theoremomllaw 28600 The orthomodular law. (Contributed by NM, 18-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  Y  =  ( X  .\/  ( Y  ./\  (  ._|_  `  X ) ) ) ) )
    < 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 31301-31321
  Copyright terms: Public domain < Previous  Next >