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Theorem List for Metamath Proof Explorer - 30001-30100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremldualvsass 30001 Associative law for scalar product operation. (Contributed by NM, 20-Oct-2014.)
LFnl       Scalar                     LDual

Theoremldualvsass2 30002 Associative law for scalar product operation, using operations from the dual space. (Contributed by NM, 20-Oct-2014.)
LFnl       Scalar              LDual       Scalar

Theoremldualvsdi1 30003 Distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
LFnl       Scalar              LDual

Theoremldualvsdi2 30004 Reverse distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.)
LFnl       Scalar                     LDual

Theoremldualgrplem 30005 Lemma for ldualgrp 30006. (Contributed by NM, 22-Oct-2014.)
LDual                            LFnl       Scalar                     oppr

Theoremldualgrp 30006 The dual of a vector space is a group. (Contributed by NM, 21-Oct-2014.)
LDual

Theoremldual0 30007 The zero scalar of the dual of a vector space. (Contributed by NM, 28-Dec-2014.)
Scalar              LDual       Scalar

Theoremldual1 30008 The unit scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015.)
Scalar              LDual       Scalar

Theoremldualneg 30009 The negative of a scalar of the dual of a vector space. (Contributed by NM, 26-Feb-2015.)
Scalar              LDual       Scalar

Theoremldual0v 30010 The zero vector of the dual of a vector space. (Contributed by NM, 24-Oct-2014.)
Scalar              LDual

Theoremldual0vcl 30011 The dual zero vector is a functional. (Contributed by NM, 5-Mar-2015.)
LFnl       LDual

Theoremlduallmodlem 30012 Lemma for lduallmod 30013. (Contributed by NM, 22-Oct-2014.)
LDual                            LFnl       Scalar                     oppr

Theoremlduallmod 30013 The dual of a left module is also a left module. (Contributed by NM, 22-Oct-2014.)
LDual

Theoremlduallvec 30014 The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 15730; otherwise, the dual would be a right vector space as is sometimes the case in the literature. (Contributed by NM, 22-Oct-2014.)
LDual

Theoremldualvsub 30015 The value of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
Scalar                     LFnl       LDual

Theoremldualvsubcl 30016 Closure of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.)
LFnl       LDual

Theoremldualvsubval 30017 The value of the value of vector subtraction in the dual of a vector space. TODO: shorten with ldualvsub 30015? (Requires to oppr conversion.) (Contributed by NM, 26-Feb-2015.)
Scalar              LFnl       LDual

Theoremldualssvscl 30018 Closure of scalar product in a dual subspace.) (Contributed by NM, 5-Feb-2015.)
Scalar              LDual

Theoremldualssvsubcl 30019 Closure of vector subtraction in a dual subspace.) (Contributed by NM, 9-Mar-2015.)
LDual

Theoremldual0vs 30020 Scalar zero times a functional is the zero functional. (Contributed by NM, 17-Feb-2015.)
LFnl       Scalar              LDual

Theoremlkr0f2 30021 The kernel of the zero functional is the set of all vectors. (Contributed by NM, 4-Feb-2015.)
LFnl       LKer       LDual

Theoremlduallkr3 30022 The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 22-Feb-2015.)
LSHyp       LFnl       LKer       LDual

TheoremlkrpssN 30023 Proper subset relation between kernels. (Contributed by NM, 16-Feb-2015.) (New usage is discouraged.)
LFnl       LKer       LDual

Theoremlkrin 30024 Intersection of the kernels of 2 functionals is included in the kernel of their sum. (Contributed by NM, 7-Jan-2015.)
LFnl       LKer       LDual

Theoremeqlkr4 30025* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 4-Feb-2015.)
Scalar              LFnl       LKer       LDual

Theoremldual1dim 30026* Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)
LFnl       LKer       LDual

Theoremldualkrsc 30027 The kernel of a non-zero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 28-Dec-2014.)
Scalar                     LFnl       LKer       LDual

Theoremlkrss 30028 The kernel of a scalar product of a functional includes the kernel of the functional. (Contributed by NM, 27-Jan-2015.)
Scalar              LFnl       LKer       LDual

Theoremlkrss2N 30029* Two functionals with kernels in a subset relationship. (Contributed by NM, 17-Feb-2015.) (New usage is discouraged.)
Scalar              LFnl       LKer       LDual

TheoremlkreqN 30030 Proportional functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
Scalar                     LFnl       LKer       LDual

TheoremlkrlspeqN 30031 Condition for colinear functionals to have equal kernels. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
LFnl       LKer       LDual

19.26.6  Ortholattices and orthomodular lattices

Syntaxcops 30032 Extend class notation with orthoposets.

SyntaxccmtN 30033 Extend class notation with the commutes relation.

Syntaxcol 30034 Extend class notation with orthlattices.

Syntaxcoml 30035 Extend class notation with orthomodular lattices.

Definitiondf-oposet 30036* Define the class of orthoposets. (Contributed by NM, 20-Oct-2011.)

Definitiondf-cmtN 30037* Define the commutes relation for orthoposets. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Nov-2011.)

Definitiondf-ol 30038 Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)

Definitiondf-oml 30039* Define the class of orthomodular lattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)

Theoremisopos 30040* The predicate "is an orthoposet." (Contributed by NM, 20-Oct-2011.)

TheoremisopiN 30041* Properties that determine an orthoposet (constructed structure version). (Contributed by NM, 13-Sep-2011.) (New usage is discouraged.)

Theoremopposet 30042 Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)

Theoremoposlem 30043 Lemma for orthoposet properties. (Contributed by NM, 20-Oct-2011.)

Theoremop0cl 30044 An orthoposet has a zero element. (h0elch 22759 analog.) (Contributed by NM, 12-Oct-2011.)

Theoremop1cl 30045 An orthoposet has a unit element. (helch 22748 analog.) (Contributed by NM, 22-Oct-2011.)

Theoremop0le 30046 Orthoposet zero is less than or equal to any element. (ch0le 22945 analog.) (Contributed by NM, 12-Oct-2011.)

Theoremople0 30047 An element less than or equal to zero equals zero. (chle0 22947 analog.) (Contributed by NM, 21-Oct-2011.)

Theoremopnlen0 30048 An element not less than another is nonzero. TODO: Look for uses of necon3bd 2640 and op0le 30046 to see if this is useful elsewhere. (Contributed by NM, 5-May-2013.)

Theoremlub0N 30049 The least upper bound of the empty set is the zero element. (Contributed by NM, 15-Sep-2013.) (New usage is discouraged.)

Theoremopltn0 30050 A lattice element greater than zero is nonzero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)

Theoremople1 30051 Any element is less than the orthoposet unit. (chss 22734 analog.) (Contributed by NM, 23-Oct-2011.)

Theoremop1le 30052 If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 22947 analog.) (Contributed by NM, 5-Dec-2011.)

Theoremglb0N 30053 The greatest lower bound of the empty set is the unit element. (Contributed by NM, 5-Dec-2011.) (New usage is discouraged.)

Theoremopoccl 30054 Closure of orthocomplement operation. (choccl 22810 analog.) (Contributed by NM, 20-Oct-2011.)

Theoremopococ 30055 Double negative law for orthoposets. (ococ 22910 analog.) (Contributed by NM, 13-Sep-2011.)

Theoremopcon3b 30056 Contraposition law for orthoposets. (chcon3i 22970 analog.) (Contributed by NM, 8-Nov-2011.)

Theoremopcon2b 30057 Orthocomplement contraposition law. (negcon2 9356 analog.) (Contributed by NM, 16-Jan-2012.)

Theoremopcon1b 30058 Orthocomplement contraposition law. (negcon1 9355 analog.) (Contributed by NM, 24-Jan-2012.)

Theoremoplecon3 30059 Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)

Theoremoplecon3b 30060 Contraposition law for orthoposets. (chsscon3 23004 analog.) (Contributed by NM, 4-Nov-2011.)

Theoremoplecon1b 30061 Contraposition law for strict ordering in orthoposets. (chsscon1 23005 analog.) (Contributed by NM, 6-Nov-2011.)

Theoremopoc1 30062 Orthocomplement of orthoposet unit. (Contributed by NM, 24-Jan-2012.)

Theoremopoc0 30063 Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)

Theoremopltcon3b 30064 Contraposition law for strict ordering in orthoposets. (chpsscon3 23007 analog.) (Contributed by NM, 4-Nov-2011.)

Theoremopltcon1b 30065 Contraposition law for strict ordering in orthoposets. (chpsscon1 23008 analog.) (Contributed by NM, 5-Nov-2011.)

Theoremopltcon2b 30066 Contraposition law for strict ordering in orthoposets. (chsscon2 23006 analog.) (Contributed by NM, 5-Nov-2011.)

Theoremopexmid 30067 Law of excluded middle for orthoposets. (chjo 23019 analog.) (Contributed by NM, 13-Sep-2011.)

Theoremopnoncon 30068 Law of contradiction for orthoposets. (chocin 22999 analog.) (Contributed by NM, 13-Sep-2011.)

TheoremriotaocN 30069* The orthocomplement of the unique poset element such that . (riotaneg 9985 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.)

TheoremcmtfvalN 30070* Value of commutes relation. (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)

TheoremcmtvalN 30071 Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 23088 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)

Theoremisolat 30072 The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)

Theoremollat 30073 An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)

Theoremolop 30074 An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)

TheoremolposN 30075 An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)

TheoremisolatiN 30076 Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)

Theoremoldmm1 30077 De Morgan's law for meet in an ortholattice. (chdmm1 23029 analog.) (Contributed by NM, 6-Nov-2011.)

Theoremoldmm2 30078 De Morgan's law for meet in an ortholattice. (chdmm2 23030 analog.) (Contributed by NM, 6-Nov-2011.)

Theoremoldmm3N 30079 De Morgan's law for meet in an ortholattice. (chdmm3 23031 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremoldmm4 30080 De Morgan's law for meet in an ortholattice. (chdmm4 23032 analog.) (Contributed by NM, 6-Nov-2011.)

Theoremoldmj1 30081 De Morgan's law for join in an ortholattice. (chdmj1 23033 analog.) (Contributed by NM, 6-Nov-2011.)

Theoremoldmj2 30082 De Morgan's law for join in an ortholattice. (chdmj2 23034 analog.) (Contributed by NM, 7-Nov-2011.)

Theoremoldmj3 30083 De Morgan's law for join in an ortholattice. (chdmj3 23035 analog.) (Contributed by NM, 7-Nov-2011.)

Theoremoldmj4 30084 De Morgan's law for join in an ortholattice. (chdmj4 23036 analog.) (Contributed by NM, 7-Nov-2011.)

Theoremolj01 30085 An ortholattice element joined with zero equals itself. (chj0 23001 analog.) (Contributed by NM, 19-Oct-2011.)

Theoremolj02 30086 An ortholattice element joined with zero equals itself. (Contributed by NM, 28-Jan-2012.)

Theoremolm11 30087 The meet of an ortholattice element with one equals itself. (chm1i 22960 analog.) (Contributed by NM, 22-May-2012.)

Theoremolm12 30088 The meet of an ortholattice element with one equals itself. (Contributed by NM, 22-May-2012.)

TheoremlatmassOLD 30089 Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3553 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremlatm12 30090 A rearrangement of lattice meet. (in12 3554 analog.) (Contributed by NM, 8-Nov-2011.)

Theoremlatm32 30091 A rearrangement of lattice meet. (in12 3554 analog.) (Contributed by NM, 13-Nov-2012.)

Theoremlatmrot 30092 Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012.)

Theoremlatm4 30093 Rearrangement of lattice meet of 4 classes. (in4 3559 analog.) (Contributed by NM, 8-Nov-2011.)

TheoremlatmmdiN 30094 Lattice meet distributes over itself. (inindi 3560 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremlatmmdir 30095 Lattice meet distributes over itself. (inindir 3561 analog.) (Contributed by NM, 6-Jun-2012.)

Theoremolm01 30096 Meet with lattice zero is zero. (chm0 22995 analog.) (Contributed by NM, 8-Nov-2011.)

Theoremolm02 30097 Meet with lattice zero is zero. (Contributed by NM, 9-Oct-2012.)

Theoremisoml 30098* The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)

TheoremisomliN 30099* Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)

Theoremomlol 30100 An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)

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