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Theorem List for Metamath Proof Explorer - 23001-23100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremchub1 23001 Hilbert lattice join is greater than or equal to its first argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchub2 23002 Hilbert lattice join is greater than or equal to its second argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchlub 23003 Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchlej1 23004 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremchlej2 23005 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremchlejb1 23006 Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)

Theoremchlejb2 23007 Hilbert lattice ordering in terms of join. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)

Theoremchnle 23008 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)

Theoremchjo 23009 The join of a closed subspace and its orthocomplement is all of Hilbert space. (Contributed by NM, 31-Oct-2005.) (New usage is discouraged.)

Theoremchabs1 23010 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)

Theoremchabs2 23011 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)

Theoremchabs1i 23012 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)

Theoremchabs2i 23013 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)

Theoremchjidm 23014 Idempotent law for Hilbert lattice join. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Theoremchjidmi 23015 Idempotent law for Hilbert lattice join. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)

Theoremchj12i 23016 A rearrangement of Hilbert lattice join. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremchj4i 23017 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremchjjdiri 23018 Hilbert lattice join distributes over itself. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremchdmm1 23019 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm2 23020 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm3 23021 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm4 23022 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj1 23023 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj2 23024 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj3 23025 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj4 23026 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchjass 23027 Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)

Theoremchj12 23028 A rearrangement of Hilbert lattice join. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)

Theoremchj4 23029 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)

Theoremledii 23030 An ortholattice is distributive in one ordering direction. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)

Theoremlediri 23031 An ortholattice is distributive in one ordering direction. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremlejdii 23032 An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremlejdiri 23033 An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremledi 23034 An ortholattice is distributive in one ordering direction. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)

18.5.4  Span (cont.) and one-dimensional subspaces

Theoremspansn0 23035 The span of the singleton of the zero vector is the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)

Theoremspan0 23036 The span of the empty set is the zero subspace. Remark 11.6.e of [Schechter] p. 276. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremelspani 23037* Membership in the span of a subset of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremspanuni 23038 The span of a union is the subspace sum of spans. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremspanun 23039 The span of a union is the subspace sum of spans. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)

Theoremsshhococi 23040 The join of two Hilbert space subsets (not necessarily closed subspaces) equals the join of their closures (double orthocomplements). (Contributed by NM, 1-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremhne0 23041 Hilbert space has a nonzero vector iff it is not trivial. (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)

Theoremchsup0 23042 The supremum of the empty set. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)

Theoremh1deoi 23043 Membership in orthocomplement of 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (New usage is discouraged.)

Theoremh1dei 23044* Membership in 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremh1did 23045 A generating vector belongs to the 1-dimensional subspace it generates. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)

Theoremh1dn0 23046 A nonzero vector generates a (nonzero) 1-dimensional subspace. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)

Theoremh1de2i 23047 Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 17-Jul-2001.) (New usage is discouraged.)

Theoremh1de2bi 23048 Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremh1de2ctlem 23049* Lemma for h1de2ci 23050. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremh1de2ci 23050* Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 21-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremspansni 23051 The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremelspansni 23052* Membership in the span of a singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremspansn 23053 The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)

Theoremspansnch 23054 The span of a Hilbert space singleton belongs to the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)

Theoremspansnsh 23055 The span of a Hilbert space singleton is a subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)

Theoremspansnchi 23056 The span of a singleton in Hilbert space is a closed subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremspansnid 23057 A vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremspansnmul 23058 A scalar product with a vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremelspansncl 23059 A member of a span of a singleton is a vector. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)

Theoremelspansn 23060* Membership in the span of a singleton. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)

Theoremelspansn2 23061 Membership in the span of a singleton. All members are collinear with the generating vector. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)

Theoremspansncol 23062 The singletons of collinear vectors have the same span. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)

Theoremspansneleqi 23063 Membership relation implied by equality of spans. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)

Theoremspansneleq 23064 Membership relation that implies equality of spans. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)

Theoremspansnss 23065 The span of the singleton of an element of a subspace is included in the subspace. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)

Theoremelspansn3 23066 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (Contributed by NM, 16-Dec-2004.) (New usage is discouraged.)

Theoremelspansn4 23067 A span membership condition implying two vectors belong to the same subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)

Theoremelspansn5 23068 A vector belonging to both a subspace and the span of the singleton of a vector not in it must be zero. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)

Theoremspansnss2 23069 The span of the singleton of an element of a subspace is included in the subspace. (Contributed by NM, 16-Dec-2004.) (New usage is discouraged.)

Theoremnormcan 23070 Cancellation-type law that "extracts" a vector from its inner product with a proportional vector . (Contributed by NM, 18-Mar-2006.) (New usage is discouraged.)

Theorempjspansn 23071 A projection on the span of a singleton. (The proof ws shortened by Mario Carneiro, 15-Dec-2013.) (Contributed by NM, 28-May-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremspansnpji 23072 A subset of Hilbert space is orthogonal to the span of the singleton of a projection onto its orthocomplement. (Contributed by NM, 4-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremspanunsni 23073 The span of the union of a closed subspace with a singleton equals the span of its union with an orthogonal singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremspanpr 23074 The span of a pair of vectors. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)

Theoremh1datomi 23075 A 1-dimensional subspace is an atom. (Contributed by NM, 20-Jul-2001.) (New usage is discouraged.)

Theoremh1datom 23076 A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)

18.5.5  Commutes relation for Hilbert lattice elements

Definitiondf-cm 23077* Define the commutes relation (on the Hilbert lattice). Definition of commutes in [Kalmbach] p. 20, who uses the notation xCy for "x commutes with y." See cmbri 23084 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)

Theoremcmbr 23078 Binary relation expressing commutes with . Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)

Theorempjoml2i 23079 Variation of orthomodular law. Definition in [Kalmbach] p. 22. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)

Theorempjoml3i 23080 Variation of orthomodular law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)

Theorempjoml4i 23081 Variation of orthomodular law. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)

Theorempjoml5i 23082 The orthomodular law. Remark in [Kalmbach] p. 22. (Contributed by NM, 12-Jun-2006.) (New usage is discouraged.)

Theorempjoml6i 23083* An equivalent of the orthomodular law. Theorem 29.13(e) of [MaedaMaeda] p. 132. (Contributed by NM, 30-May-2004.) (New usage is discouraged.)

Theoremcmbri 23084 Binary relation expressing the commutes relation. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)

Theoremcmcmlem 23085 Commutation is symmetric. Theorem 3.4 of [Beran] p. 45. (Contributed by NM, 3-Nov-2000.) (New usage is discouraged.)

Theoremcmcmi 23086 Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.)

Theoremcmcm2i 23087 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.)

Theoremcmcm3i 23088 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 4-Nov-2000.) (New usage is discouraged.)

Theoremcmcm4i 23089 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)

Theoremcmbr2i 23090 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)

Theoremcmcmii 23091 Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)

Theoremcmcm2ii 23092 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)

Theoremcmcm3ii 23093 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)

Theoremcmbr3i 23094 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)

Theoremcmbr4i 23095 Alternate definition for the commutes relation. (Contributed by NM, 6-Dec-2000.) (New usage is discouraged.)

Theoremlecmi 23096 Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 16-Jan-2005.) (New usage is discouraged.)

Theoremlecmii 23097 Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)

Theoremcmj1i 23098 A Hilbert lattice element commutes with its join. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)

Theoremcmj2i 23099 A Hilbert lattice element commutes with its join. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)

Theoremcmm1i 23100 A Hilbert lattice element commutes with its meet. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)

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