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Theorem List for Metamath Proof Explorer - 22801-22900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempjococi 22801 Proof of orthocomplement theorem using projections. Compare ococ 22770. (Contributed by NM, 5-Nov-1999.) (New usage is discouraged.)

Theorempjoc2i 22802 Projection of a vector in the orthocomplement of the projection subspace. Lemma 4.4(iii) of [Beran] p. 111. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)

Theorempjoc2 22803 Projection of a vector in the orthocomplement of the projection subspace. Lemma 4.4(iii) of [Beran] p. 111. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

18.5.3  Hilbert lattice operations

Theoremsh0le 22804 The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremch0le 22805 The zero subspace is the smallest member of . (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)

Theoremshle0 22806 No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)

Theoremchle0 22807 No Hilbert lattice element is smaller than zero. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)

Theoremchnlen0 22808 A Hilbert lattice element that is not a subset of another is nonzero. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)

Theoremch0pss 22809 The zero subspace is a proper subset of nonzero Hilbert lattice elements. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)

Theoremorthin 22810 The intersection of orthogonal subspaces is the zero subspace. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)

Theoremssjo 22811 The lattice join of a subset with its orthocomplement is the whole space. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremshne0i 22812* A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)

Theoremshs0i 22813 Hilbert subspace sum with the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)

Theoremshs00i 22814 Two subspaces are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)

Theoremch0lei 22815 The closed subspace zero is the smallest member of . (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchle0i 22816 No Hilbert closed subspace is smaller than zero. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)

Theoremchne0i 22817* A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)

Theoremchocini 22818 Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)

Theoremchj0i 22819 Join with lattice zero in . (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchm1i 22820 Meet with lattice one in . (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremchjcli 22821 Closure of join. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)

Theoremchsleji 22822 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)

Theoremchseli 22823* Membership in subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremchincli 22824 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchsscon3i 22825 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchsscon1i 22826 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchsscon2i 22827 Hilbert lattice contraposition law. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchcon2i 22828 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)

Theoremchcon1i 22829 Hilbert lattice contraposition law. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)

Theoremchcon3i 22830 Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.)

Theoremchunssji 22831 Union is smaller than join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchjcomi 22832 Commutative law for join in . (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

Theoremchub1i 22833 join is an upper bound of two elements. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchub2i 22834 join is an upper bound of two elements. (Contributed by NM, 5-Nov-2000.) (New usage is discouraged.)

Theoremchlubi 22835 Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 11-Jun-2004.) (New usage is discouraged.)

Theoremchlubii 22836 Hilbert lattice join is the least upper bound of two elements (one direction of chlubi 22835). (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchlej1i 22837 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremchlej2i 22838 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremchlej12i 22839 Add join to both sides of a Hilbert lattice ordering. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremchlejb1i 22840 Hilbert lattice ordering in terms of join. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchdmm1i 22841 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm2i 22842 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm3i 22843 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm4i 22844 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj1i 22845 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj2i 22846 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj3i 22847 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj4i 22848 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchnlei 22849 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)

Theoremchjassi 22850 Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)

Theoremchj00i 22851 Two Hilbert lattice elements are zero iff their join is zero. (Contributed by NM, 7-Aug-2004.) (New usage is discouraged.)

Theoremchjoi 22852 The join of a closed subspace and its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremchj1i 22853 Join with Hilbert lattice unit. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)

Theoremchm0i 22854 Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)

Theoremchm0 22855 Meet with Hilbert lattice zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)

Theoremshjshsi 22856 Hilbert lattice join equals the double orthocomplement of subspace sum. (Contributed by NM, 27-Nov-2004.) (New usage is discouraged.)

Theoremshjshseli 22857 A closed subspace sum equals Hilbert lattice join. Part of Lemma 31.1.5 of [MaedaMaeda] p. 136. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)

Theoremchne0 22858* A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)

Theoremchocin 22859 Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)

Theoremchssoc 22860 A closed subspace less than its orthocomplement is zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)

Theoremchj0 22861 Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremchslej 22862 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)

Theoremchincl 22863 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)

Theoremchsscon3 22864 Hilbert lattice contraposition law. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchsscon1 22865 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchsscon2 22866 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchpsscon3 22867 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchpsscon1 22868 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchpsscon2 22869 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchjcom 22870 Commutative law for Hilbert lattice join. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchub1 22871 Hilbert lattice join is greater than or equal to its first argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchub2 22872 Hilbert lattice join is greater than or equal to its second argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchlub 22873 Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchlej1 22874 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremchlej2 22875 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremchlejb1 22876 Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)

Theoremchlejb2 22877 Hilbert lattice ordering in terms of join. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)

Theoremchnle 22878 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)

Theoremchjo 22879 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 22880 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)

Theoremchabs2 22881 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)

Theoremchabs1i 22882 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)

Theoremchabs2i 22883 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)

Theoremchjidm 22884 Idempotent law for Hilbert lattice join. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Theoremchjidmi 22885 Idempotent law for Hilbert lattice join. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)

Theoremchj12i 22886 A rearrangement of Hilbert lattice join. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremchj4i 22887 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremchjjdiri 22888 Hilbert lattice join distributes over itself. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremchdmm1 22889 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm2 22890 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm3 22891 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm4 22892 De Morgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj1 22893 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj2 22894 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj3 22895 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj4 22896 De Morgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchjass 22897 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 22898 A rearrangement of Hilbert lattice join. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)

Theoremchj4 22899 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)

Theoremledii 22900 An ortholattice is distributive in one ordering direction. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)

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