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Type | Label | Description |
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Statement | ||
Theorem | dochkrshp2 35001 | Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.) |
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Theorem | dochkrshp3 35002 | Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | dochkrshp4 35003 | Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | dochdmj1 35004 | De Morgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | dochnoncon 35005 | Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | dochnel2 35006 | A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | dochnel 35007 | A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Syntax | cdjh 35008 |
Extend class notation with subspace join for ![]() |
![]() | ||
Definition | df-djh 35009* |
Define (closed) subspace join for ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhffval 35010* |
Subspace join for ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhfval 35011* |
Subspace join for ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhval 35012 |
Subspace join for ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhval2 35013 |
Value of subspace join for ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhcl 35014 |
Closure of subspace join for ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhlj 35015 |
Transfer lattice join to ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhljjN 35016 |
Lattice join in terms of ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhjlj 35017 |
![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhj 35018 |
![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhcom 35019 | Subspace join commutes. (Contributed by NM, 8-Aug-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhspss 35020 | Subspace span of union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhsumss 35021 | Subspace sum is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | dihsumssj 35022 | The subspace sum of two isomorphisms of lattice elements is less than the isomorphism of their lattice join. (Contributed by NM, 23-Sep-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhunssN 35023 | Subspace union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) (New usage is discouraged.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | dochdmm1 35024 | De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhexmid 35025 |
Excluded middle property of ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djh01 35026 | Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djh02 35027 | Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhlsmcl 35028 | A closed subspace sum equals subspace join. (shjshseli 27202 analog.) (Contributed by NM, 13-Aug-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | djhcvat42 35029* | A covering property. (cvrat42 33055 analog.) (Contributed by NM, 17-Aug-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | dihjatb 35030 | Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-Sep-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | dihjatc 35031 | Isomorphism H of lattice join of an element under the fiducial hyperplane with atom not under it. (Contributed by NM, 26-Aug-2014.) |
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Theorem | dihjatcclem1 35032 | Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.) |
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Theorem | dihjatcclem2 35033 | Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.) |
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Theorem | dihjatcclem3 35034* | Lemma for dihjatcc 35036. (Contributed by NM, 28-Sep-2014.) |
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Theorem | dihjatcclem4 35035* | Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.) |
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Theorem | dihjatcc 35036 | Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.) |
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Theorem | dihjat 35037 | Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.) |
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Theorem | dihprrnlem1N 35038 | Lemma for dihprrn 35040, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.) |
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Theorem | dihprrnlem2 35039 | Lemma for dihprrn 35040. (Contributed by NM, 29-Sep-2014.) |
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Theorem | dihprrn 35040 | The span of a vector pair belongs to the range of isomorphism H i.e. is a closed subspace. (Contributed by NM, 29-Sep-2014.) |
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Theorem | djhlsmat 35041 | The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 35040; should we directly use dihjat 35037? (Contributed by NM, 13-Aug-2014.) |
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Theorem | dihjat1lem 35042 | Subspace sum of a closed subspace and an atom. (pmapjat1 33464 analog.) TODO: merge into dihjat1 35043? (Contributed by NM, 18-Aug-2014.) |
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Theorem | dihjat1 35043 | Subspace sum of a closed subspace and an atom. (pmapjat1 33464 analog.) (Contributed by NM, 1-Oct-2014.) |
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Theorem | dihsmsprn 35044 | Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015.) |
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Theorem | dihjat2 35045 | The subspace sum of a closed subspace and an atom is the same as their subspace join. (Contributed by NM, 1-Oct-2014.) |
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Theorem | dihjat3 35046 | Isomorphism H of lattice join with an atom. (Contributed by NM, 25-Apr-2015.) |
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Theorem | dihjat4 35047 | Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.) |
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Theorem | dihjat6 35048 | Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.) |
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Theorem | dihsmsnrn 35049 | The subspace sum of two singleton spans is closed. (Contributed by NM, 27-Feb-2015.) |
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Theorem | dihsmatrn 35050 | The subspace sum of a closed subspace and an atom is closed. TODO: see if proof at http://math.stackexchange.com/a/1233211/50776 and Mon, 13 Apr 2015 20:44:07 -0400 email could be used instead of this and dihjat2 35045. (Contributed by NM, 15-Jan-2015.) |
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Theorem | dihjat5N 35051 | Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.) |
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Theorem | dvh4dimat 35052* | There is an atom that is outside the subspace sum of 3 others. (Contributed by NM, 25-Apr-2015.) |
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Theorem | dvh3dimatN 35053* | There is an atom that is outside the subspace sum of 2 others. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.) |
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Theorem | dvh2dimatN 35054* | Given an atom, there exists another. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.) |
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Theorem | dvh1dimat 35055* | There exists an atom. (Contributed by NM, 25-Apr-2015.) |
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Theorem | dvh1dim 35056* | There exists a nonzero vector. (Contributed by NM, 26-Apr-2015.) |
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Theorem | dvh4dimlem 35057* | Lemma for dvh4dimN 35061. (Contributed by NM, 22-May-2015.) |
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Theorem | dvhdimlem 35058* | Lemma for dvh2dim 35059 and dvh3dim 35060. TODO: make this obsolete and use dvh4dimlem 35057 directly? (Contributed by NM, 24-Apr-2015.) |
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Theorem | dvh2dim 35059* | There is a vector that is outside the span of another. (Contributed by NM, 25-Apr-2015.) |
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Theorem | dvh3dim 35060* | There is a vector that is outside the span of 2 others. (Contributed by NM, 24-Apr-2015.) |
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Theorem | dvh4dimN 35061* | There is a vector that is outside the span of 3 others. (Contributed by NM, 22-May-2015.) (New usage is discouraged.) |
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Theorem | dvh3dim2 35062* | There is a vector that is outside of 2 spans with a common vector. (Contributed by NM, 13-May-2015.) |
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Theorem | dvh3dim3N 35063* | There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 35062 everywhere. If this one is needed, make dvh3dim2 35062 into a lemma. (Contributed by NM, 21-May-2015.) (New usage is discouraged.) |
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Theorem | dochsnnz 35064 | The orthocomplement of a singleton is nonzero. (Contributed by NM, 13-Jun-2015.) |
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Theorem | dochsatshp 35065 | The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 27-Jul-2014.) (Revised by Mario Carneiro, 1-Oct-2014.) |
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Theorem | dochsatshpb 35066 | The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 29-Oct-2014.) |
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Theorem | dochsnshp 35067 | The orthocomplement of a nonzero singleton is a hyperplane. (Contributed by NM, 3-Jan-2015.) |
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Theorem | dochshpsat 35068 | A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.) |
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Theorem | dochkrsat 35069 | The orthocomplement of a kernel is an atom iff it is nonzero. (Contributed by NM, 1-Nov-2014.) |
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Theorem | dochkrsat2 35070 | The orthocomplement of a kernel is an atom iff the double orthocomplement is not the vector space. (Contributed by NM, 1-Jan-2015.) |
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Theorem | dochsat0 35071 | The orthocomplement of a kernel is either an atom or zero. (Contributed by NM, 29-Jan-2015.) |
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Theorem | dochkrsm 35072 | The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 35028 can be used to convert sum to join.) (Contributed by NM, 29-Jan-2015.) |
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Theorem | dochexmidat 35073 | Special case of excluded middle for the singleton of a vector. (Contributed by NM, 27-Oct-2014.) |
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Theorem | dochexmidlem1 35074 |
Lemma for dochexmid 35082. Holland's proof implicitly requires
![]() ![]() ![]() |
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Theorem | dochexmidlem2 35075 | Lemma for dochexmid 35082. (Contributed by NM, 14-Jan-2015.) |
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Theorem | dochexmidlem3 35076 |
Lemma for dochexmid 35082. Use atom exchange lsatexch1 32658 to swap ![]() ![]() |
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Theorem | dochexmidlem4 35077 | Lemma for dochexmid 35082. (Contributed by NM, 15-Jan-2015.) |
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Theorem | dochexmidlem5 35078 | Lemma for dochexmid 35082. (Contributed by NM, 15-Jan-2015.) |
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Theorem | dochexmidlem6 35079 | Lemma for dochexmid 35082. (Contributed by NM, 15-Jan-2015.) |
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Theorem | dochexmidlem7 35080 | Lemma for dochexmid 35082. Contradict dochexmidlem6 35079. (Contributed by NM, 15-Jan-2015.) |
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Theorem | dochexmidlem8 35081 |
Lemma for dochexmid 35082. The contradiction of dochexmidlem6 35079 and
dochexmidlem7 35080 shows that there can be no atom ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dochexmid 35082 |
Excluded middle law for closed subspaces, which is equivalent to (and
derived from) the orthomodular law dihoml4 34991. Lemma 3.3(2) in
[Holland95] p. 215. In our proof, we
use the variables ![]() ![]() ![]() ![]() ![]() |
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Theorem | dochsnkrlem1 35083 | Lemma for dochsnkr 35086. (Contributed by NM, 1-Jan-2015.) |
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Theorem | dochsnkrlem2 35084 | Lemma for dochsnkr 35086. (Contributed by NM, 1-Jan-2015.) |
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Theorem | dochsnkrlem3 35085 | Lemma for dochsnkr 35086. (Contributed by NM, 2-Jan-2015.) |
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Theorem | dochsnkr 35086 | A (closed) kernel expressed in terms of a nonzero vector in its orthocomplement. TODO: consolidate lemmas unless they're needed for something else (in which case break out as theorems). (Contributed by NM, 2-Jan-2015.) |
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Theorem | dochsnkr2 35087* |
Kernel of the explicit functional ![]() ![]() |
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Theorem | dochsnkr2cl 35088* |
The ![]() ![]() |
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Theorem | dochflcl 35089* |
Closure of the explicit functional ![]() ![]() |
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Theorem | dochfl1 35090* |
The value of the explicit functional ![]() ![]() |
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Theorem | dochfln0 35091 | The value of a functional is nonzero at a nonzero vector in the orthocomplement of its kernel. (Contributed by NM, 2-Jan-2015.) |
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Theorem | dochkr1 35092* | A non-zero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 32682. (Contributed by NM, 2-Jan-2015.) |
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Theorem | dochkr1OLDN 35093* | A non-zero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 32682. (Contributed by NM, 2-Jan-2015.) (New usage is discouraged.) |
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Syntax | clpoN 35094 | Extend class notation with all polarities of a left module or left vector space. |
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Definition | df-lpolN 35095* | Define the set of all polarities of a left module or left vector space. A polarity is a kind of complementation operation on a subspace. The double polarity of a subspace is a closure operation. Based on Definition 3.2 of [Holland95] p. 214 for projective geometry polarities. For convenience, we open up the domain to include all vector subsets and not just subspaces, but any more restricted polarity can be converted to this one by taking the span of its argument. (Contributed by NM, 24-Nov-2014.) |
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Theorem | lpolsetN 35096* | The set of polarities of a left module or left vector space. (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.) |
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Theorem | islpolN 35097* | The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.) |
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Theorem | islpoldN 35098* | Properties that determine a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
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Theorem | lpolfN 35099 | Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
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Theorem | lpolvN 35100 | The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
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