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Type | Label | Description |
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Statement | ||
Theorem | atleneN 33001 | Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.) |
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Theorem | atltcvr 33002 | An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.) |
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Theorem | atle 33003* | Any non-zero element has an atom under it. (Contributed by NM, 28-Jun-2012.) |
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Theorem | atlt 33004 | Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012.) |
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Theorem | atlelt 33005 | Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.) |
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Theorem | 2atlt 33006* | Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.) |
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Theorem | atexchcvrN 33007 | Atom exchange property. Version of hlatexch2 32963 with covers relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.) |
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Theorem | atexchltN 33008 | Atom exchange property. Version of hlatexch2 32963 with less-than ordering. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.) |
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Theorem | cvrat3 33009 | A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 28061 analog.) (Contributed by NM, 30-Nov-2011.) |
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Theorem | cvrat4 33010* | A condition implying existence of an atom with the properties shown. Lemma 3.2.20 in [PtakPulmannova] p. 68. Also Lemma 9.2(delta) in [MaedaMaeda] p. 41. (atcvat4i 28062 analog.) (Contributed by NM, 30-Nov-2011.) |
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Theorem | cvrat42 33011* | Commuted version of cvrat4 33010. (Contributed by NM, 28-Jan-2012.) |
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Theorem | 2atjm 33012 | The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.) |
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Theorem | atbtwn 33013 |
Property of a 3rd atom ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | atbtwnexOLDN 33014* |
There exists a 3rd atom ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | atbtwnex 33015* |
Given atoms ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | 3noncolr2 33016 | Two ways to express 3 non-colinear atoms (rotated right 2 places). (Contributed by NM, 12-Jul-2012.) |
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Theorem | 3noncolr1N 33017 | Two ways to express 3 non-colinear atoms (rotated right 1 place). (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.) |
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Theorem | hlatcon3 33018 | Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.) |
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Theorem | hlatcon2 33019 | Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.) |
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Theorem | 4noncolr3 33020 | A way to express 4 non-colinear atoms (rotated right 3 places). (Contributed by NM, 11-Jul-2012.) |
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Theorem | 4noncolr2 33021 | A way to express 4 non-colinear atoms (rotated right 2 places). (Contributed by NM, 11-Jul-2012.) |
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Theorem | 4noncolr1 33022 | A way to express 4 non-colinear atoms (rotated right 1 places). (Contributed by NM, 11-Jul-2012.) |
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Theorem | athgt 33023* | A Hilbert lattice, whose height is at least 4, has a chain of 4 successively covering atom joins. (Contributed by NM, 3-May-2012.) |
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Theorem | 3dim0 33024* | There exists a 3-dimensional (height-4) element i.e. a volume. (Contributed by NM, 25-Jul-2012.) |
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Theorem | 3dimlem1 33025 | Lemma for 3dim1 33034. (Contributed by NM, 25-Jul-2012.) |
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Theorem | 3dimlem2 33026 | Lemma for 3dim1 33034. (Contributed by NM, 25-Jul-2012.) |
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Theorem | 3dimlem3a 33027 | Lemma for 3dim3 33036. (Contributed by NM, 27-Jul-2012.) |
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Theorem | 3dimlem3 33028 | Lemma for 3dim1 33034. (Contributed by NM, 25-Jul-2012.) |
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Theorem | 3dimlem3OLDN 33029 | Lemma for 3dim1 33034. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | 3dimlem4a 33030 | Lemma for 3dim3 33036. (Contributed by NM, 27-Jul-2012.) |
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Theorem | 3dimlem4 33031 | Lemma for 3dim1 33034. (Contributed by NM, 25-Jul-2012.) |
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Theorem | 3dimlem4OLDN 33032 | Lemma for 3dim1 33034. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | 3dim1lem5 33033* | Lemma for 3dim1 33034. (Contributed by NM, 26-Jul-2012.) |
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Theorem | 3dim1 33034* |
Construct a 3-dimensional volume (height-4 element) on top of a given
atom ![]() |
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Theorem | 3dim2 33035* | Construct 2 new layers on top of 2 given atoms. (Contributed by NM, 27-Jul-2012.) |
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Theorem | 3dim3 33036* | Construct a new layer on top of 3 given atoms. (Contributed by NM, 27-Jul-2012.) |
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Theorem | 2dim 33037* | Generate a height-3 element (2-dimensional plane) from an atom. (Contributed by NM, 3-May-2012.) |
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Theorem | 1dimN 33038* | An atom is covered by a height-2 element (1-dimensional line). (Contributed by NM, 3-May-2012.) (New usage is discouraged.) |
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Theorem | 1cvrco 33039 | The orthocomplement of an element covered by 1 is an atom. (Contributed by NM, 7-May-2012.) |
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Theorem | 1cvratex 33040* | There exists an atom less than an element covered by 1. (Contributed by NM, 7-May-2012.) (Revised by Mario Carneiro, 13-Jun-2014.) |
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Theorem | 1cvratlt 33041 | An atom less than or equal to an element covered by 1 is less than the element. (Contributed by NM, 7-May-2012.) |
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Theorem | 1cvrjat 33042 | An element covered by the lattice unit, when joined with an atom not under it, equals the lattice unit. (Contributed by NM, 30-Apr-2012.) |
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Theorem | 1cvrat 33043 | Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 30-Apr-2012.) |
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Theorem | ps-1 33044 |
The join of two atoms ![]() ![]() ![]() |
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Theorem | ps-2 33045* | Lattice analogue for the projective geometry axiom, "if a line intersects two sides of a triangle at different points then it also intersects the third side." Projective space condition PS2 in [MaedaMaeda] p. 68 and part of Theorem 16.4 in [MaedaMaeda] p. 69. (Contributed by NM, 1-Dec-2011.) |
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Theorem | 2atjlej 33046 | Two atoms are different if their join majorizes the join of two different atoms. (Contributed by NM, 4-Jun-2013.) |
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Theorem | hlatexch3N 33047 | Rearrange join of atoms in an equality. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.) |
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Theorem | hlatexch4 33048 | Exchange 2 atoms. (Contributed by NM, 13-May-2013.) |
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Theorem | ps-2b 33049 | Variation of projective geometry axiom ps-2 33045. (Contributed by NM, 3-Jul-2012.) |
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Theorem | 3atlem1 33050 | Lemma for 3at 33057. (Contributed by NM, 22-Jun-2012.) |
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Theorem | 3atlem2 33051 | Lemma for 3at 33057. (Contributed by NM, 22-Jun-2012.) |
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Theorem | 3atlem3 33052 | Lemma for 3at 33057. (Contributed by NM, 23-Jun-2012.) |
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Theorem | 3atlem4 33053 | Lemma for 3at 33057. (Contributed by NM, 23-Jun-2012.) |
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Theorem | 3atlem5 33054 | Lemma for 3at 33057. (Contributed by NM, 23-Jun-2012.) |
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Theorem | 3atlem6 33055 | Lemma for 3at 33057. (Contributed by NM, 23-Jun-2012.) |
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Theorem | 3atlem7 33056 | Lemma for 3at 33057. (Contributed by NM, 23-Jun-2012.) |
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Theorem | 3at 33057 | Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analogue of ps-1 33044 for lines and 4at 33180 for volumes. I could not find this proof in the literature on projective geometry (where it is either given as an axiom or stated as an unproved fact), but it is similar to Theorem 15 of Veblen, "The Foundations of Geometry" (1911), p. 18, which uses different axioms. This proof was written before I became aware of Veblen's, and it is possible that a shorter proof could be obtained by using Veblen's proof for hints. (Contributed by NM, 23-Jun-2012.) |
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Syntax | clln 33058 | Extend class notation with set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice. |
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Syntax | clpl 33059 | Extend class notation with set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice. |
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Syntax | clvol 33060 | Extend class notation with set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice. |
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Syntax | clines 33061 | Extend class notation with set of all projective lines for a Hilbert lattice. |
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Syntax | cpointsN 33062 | Extend class notation with set of all projective points. |
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Syntax | cpsubsp 33063 | Extend class notation with set of all projective subspaces. |
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Syntax | cpmap 33064 | Extend class notation with projective map. |
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Definition | df-llines 33065* |
Define the set of all "lattice lines" (lattice elements which cover
an
atom) in a Hilbert lattice ![]() |
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Definition | df-lplanes 33066* |
Define the set of all "lattice planes" (lattice elements which cover
a
line) in a Hilbert lattice ![]() |
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Definition | df-lvols 33067* |
Define the set of all 3-dimensional "lattice volumes" (lattice
elements
which cover a plane) in a Hilbert lattice ![]() |
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Definition | df-lines 33068* | Define set of all projective lines for a Hilbert lattice (actually in any set at all, for simplicity). The join of two distinct atoms equals a line. Definition of lines in item 1 of [Holland95] p. 222. (Contributed by NM, 19-Sep-2011.) |
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Definition | df-pointsN 33069* | Define set of all projective points in a Hilbert lattice (actually in any set at all, for simplicity). A projective point is the singleton of a lattice atom. Definition 15.1 of [MaedaMaeda] p. 61. Note that item 1 in [Holland95] p. 222 defines a point as the atom itself, but this leads to a complicated subspace ordering that may be either membership or inclusion based on its arguments. (Contributed by NM, 2-Oct-2011.) |
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Definition | df-psubsp 33070* | Define set of all projective subspaces. Based on definition of subspace in [Holland95] p. 212. (Contributed by NM, 2-Oct-2011.) |
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Definition | df-pmap 33071* |
Define projective map for ![]() ![]() |
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Theorem | llnset 33072* | The set of lattice lines in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.) |
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Theorem | islln 33073* | The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.) |
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Theorem | islln4 33074* | The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.) |
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Theorem | llni 33075 | Condition implying a lattice line. (Contributed by NM, 17-Jun-2012.) |
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Theorem | llnbase 33076 | A lattice line is a lattice element. (Contributed by NM, 16-Jun-2012.) |
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Theorem | islln3 33077* | The predicate "is a lattice line". (Contributed by NM, 17-Jun-2012.) |
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Theorem | islln2 33078* | The predicate "is a lattice line". (Contributed by NM, 23-Jun-2012.) |
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Theorem | llni2 33079 | The join of two different atoms is a lattice line. (Contributed by NM, 26-Jun-2012.) |
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Theorem | llnnleat 33080 | An atom cannot majorize a lattice line. (Contributed by NM, 8-Jul-2012.) |
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Theorem | llnneat 33081 | A lattice line is not an atom. (Contributed by NM, 19-Jun-2012.) |
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Theorem | 2atneat 33082 | The join of two distinct atoms is not an atom. (Contributed by NM, 12-Oct-2012.) |
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Theorem | llnn0 33083 | A lattice line is non-zero. (Contributed by NM, 15-Jul-2012.) |
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Theorem | islln2a 33084 | The predicate "is a lattice line" in terms of atoms. (Contributed by NM, 15-Jul-2012.) |
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Theorem | llnle 33085* | Any element greater than 0 and not an atom majorizes a lattice line. (Contributed by NM, 28-Jun-2012.) |
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Theorem | atcvrlln2 33086 | An atom under a line is covered by it. (Contributed by NM, 2-Jul-2012.) |
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Theorem | atcvrlln 33087 | An element covering an atom is a lattice line and vice-versa. (Contributed by NM, 18-Jun-2012.) |
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Theorem | llnexatN 33088* | Given an atom on a line, there is another atom whose join equals the line. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.) |
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Theorem | llncmp 33089 | If two lattice lines are comparable, they are equal. (Contributed by NM, 19-Jun-2012.) |
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Theorem | llnnlt 33090 | Two lattice lines cannot satisfy the less than relation. (Contributed by NM, 26-Jun-2012.) |
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Theorem | 2llnmat 33091 | Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.) |
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Theorem | 2at0mat0 33092 | Special case of 2atmat0 33093 where one atom could be zero. (Contributed by NM, 30-May-2013.) |
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Theorem | 2atmat0 33093 | The meet of two unequal lines (expressed as joins of atoms) is an atom or zero. (Contributed by NM, 2-Dec-2012.) |
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Theorem | 2atm 33094 | An atom majorized by two different atom joins (which could be atoms or lines) is equal to their intersection. (Contributed by NM, 30-Jun-2013.) |
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Theorem | ps-2c 33095 | Variation of projective geometry axiom ps-2 33045. (Contributed by NM, 3-Jul-2012.) |
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Theorem | lplnset 33096* | The set of lattice planes in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.) |
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Theorem | islpln 33097* | The predicate "is a lattice plane". (Contributed by NM, 16-Jun-2012.) |
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Theorem | islpln4 33098* | The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.) |
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Theorem | lplni 33099 | Condition implying a lattice plane. (Contributed by NM, 20-Jun-2012.) |
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Theorem | islpln3 33100* | The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.) |
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