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Theorem List for Metamath Proof Explorer - 30801-30900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlhpn0 30801 A co-atom is nonzero. TODO: is this needed? (Contributed by NM, 26-Apr-2013.)

Theoremlhpexle 30802* There exists an atom under a co-atom. (Contributed by NM, 26-Apr-2013.)

Theoremlhpexnle 30803* There exists an atom not under a co-atom. (Contributed by NM, 12-Apr-2013.)

Theoremlhpexle1lem 30804* Lemma for lhpexle1 30805 and others that eliminates restrictions on . (Contributed by NM, 24-Jul-2013.)

Theoremlhpexle1 30805* There exists an atom under a co-atom different from any given element. (Contributed by NM, 24-Jul-2013.)

Theoremlhpexle2lem 30806* Lemma for lhpexle2 30807. (Contributed by NM, 19-Jun-2013.)

Theoremlhpexle2 30807* There exists atom under a co-atom different from any two other elements. (Contributed by NM, 24-Jul-2013.)

Theoremlhpexle3lem 30808* There exists atom under a co-atom different from any 3 other atoms. TODO: study if adant*,simp* usage can be improved. (Contributed by NM, 9-Jul-2013.)

Theoremlhpexle3 30809* There exists atom under a co-atom different from any three other elements. (Contributed by NM, 24-Jul-2013.)

Theoremlhpex2leN 30810* There exist at least two different atoms under a co-atom. This allows us to create a line under the co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) (New usage is discouraged.)

Theoremlhpoc 30811 The orthocomplement of a co-atom (lattice hyperplane) is an atom. (Contributed by NM, 18-May-2012.)

Theoremlhpoc2N 30812 The orthocomplement of an atom is a co-atom (lattice hyperplane). (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)

Theoremlhpocnle 30813 The orthocomplement of a co-atom is not under it. (Contributed by NM, 22-May-2012.)

Theoremlhpocat 30814 The orthocomplement of a co-atom is an atom. (Contributed by NM, 9-Feb-2013.)

Theoremlhpocnel 30815 The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 25-May-2012.)

Theoremlhpocnel2 30816 The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 20-Feb-2014.)

Theoremlhpjat1 30817 The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 18-May-2012.)

Theoremlhpjat2 30818 The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 4-Jun-2012.)

Theoremlhpj1 30819 The join of a co-atom (hyperplane) and an element not under it is the lattice unit. (Contributed by NM, 7-Dec-2012.)

Theoremlhpmcvr 30820 The meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 7-Dec-2012.)

Theoremlhpmcvr2 30821* Alternate way to express that the meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 9-Apr-2013.)

Theoremlhpmcvr3 30822 Specialization of lhpmcvr2 30821. TODO: Use this to simplify many uses of to become . (Contributed by NM, 6-Apr-2014.)

Theoremlhpmcvr4N 30823 Specialization of lhpmcvr2 30821. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremlhpmcvr5N 30824* Specialization of lhpmcvr2 30821. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremlhpmcvr6N 30825* Specialization of lhpmcvr2 30821. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremlhpm0atN 30826 If the meet of a lattice hyperplane with a nonzero element is zero, the element is an atom. (Contributed by NM, 28-Apr-2014.) (New usage is discouraged.)

Theoremlhpmat 30827 An element covered by the lattice unit, when conjoined with an atom not under it, equals the lattice zero. (Contributed by NM, 6-Jun-2012.)

Theoremlhpmatb 30828 An element covered by the lattice unit, when conjoined with an atom, equals zero iff the atom is not under it. (Contributed by NM, 15-Jun-2013.)

Theoremlhp2at0 30829 Join and meet with different atoms under co-atom . (Contributed by NM, 15-Jun-2013.)

Theoremlhp2atnle 30830 Inequality for 2 different atoms under co-atom . (Contributed by NM, 17-Jun-2013.)

Theoremlhp2atne 30831 Inequality for joins with 2 different atoms under co-atom . (Contributed by NM, 22-Jul-2013.)

Theoremlhp2at0nle 30832 Inequality for 2 different atoms (or an atom and zero) under co-atom . (Contributed by NM, 28-Jul-2013.)

Theoremlhp2at0ne 30833 Inequality for joins with 2 different atoms (or an atom and zero) under co-atom . (Contributed by NM, 28-Jul-2013.)

Theoremlhpelim 30834 Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 30827 to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013.)

Theoremlhpmod2i2 30835 Modular law for hyperplanes analogous to atmod2i2 30659 for atoms. (Contributed by NM, 9-Feb-2013.)

Theoremlhpmod6i1 30836 Modular law for hyperplanes analogous to complement of atmod2i1 30658 for atoms. (Contributed by NM, 1-Jun-2013.)

Theoremlhprelat3N 30837* The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 30209. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)

Theoremcdlemb2 30838* Given two atoms not under the fiducial (reference) co-atom , there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 30-May-2012.)

Theoremlhple 30839 Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014.)

Theoremlhpat 30840 Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)

Theoremlhpat4N 30841 Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)

Theoremlhpat2 30842 Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 21-Nov-2012.)

Theoremlhpat3 30843 There is only one atom under both and co-atom . (Contributed by NM, 21-Nov-2012.)

Theorem4atexlemk 30844 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemw 30845 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlempw 30846 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemp 30847 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemq 30848 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlems 30849 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemt 30850 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemutvt 30851 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlempnq 30852 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemnslpq 30853 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemkl 30854 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemkc 30855 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemwb 30856 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlempsb 30857 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemqtb 30858 Lemma for 4atexlem7 30872. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlempns 30859 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemswapqr 30860 Lemma for 4atexlem7 30872. Swap and , so that theorems involving can be reused for . Note that must be expanded because it involves . (Contributed by NM, 25-Nov-2012.)

Theorem4atexlemu 30861 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemv 30862 Lemma for 4atexlem7 30872. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemunv 30863 Lemma for 4atexlem7 30872. (Contributed by NM, 21-Nov-2012.)

Theorem4atexlemtlw 30864 Lemma for 4atexlem7 30872. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemntlpq 30865 Lemma for 4atexlem7 30872. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemc 30866 Lemma for 4atexlem7 30872. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemnclw 30867 Lemma for 4atexlem7 30872. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemex2 30868* Lemma for 4atexlem7 30872. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 25-Nov-2012.)

Theorem4atexlemcnd 30869 Lemma for 4atexlem7 30872. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemex4 30870* Lemma for 4atexlem7 30872. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012.)

Theorem4atexlemex6 30871* Lemma for 4atexlem7 30872. (Contributed by NM, 25-Nov-2012.)

Theorem4atexlem7 30872* Whenever there are at least 4 atoms under (specifically, , , , and ), there are also at least 4 atoms under . This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p q/0 and hence p s/0 contains at least four atoms..." Note that by cvlsupr2 30141, our is a shorter way to express . With a longer proof, the condition could be eliminated (see 4atex 30873), although for some purposes this more restricted lemma may be adequate. (Contributed by NM, 25-Nov-2012.)

Theorem4atex 30873* Whenever there are at least 4 atoms under (specifically, , , , and ), there are also at least 4 atoms under . This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p q/0 and hence p s/0 contains at least four atoms..." Note that by cvlsupr2 30141, our is a shorter way to express . (Contributed by NM, 27-May-2013.)

Theorem4atex2 30874* More general version of 4atex 30873 for a line not necessarily connected to . (Contributed by NM, 27-May-2013.)

Theorem4atex2-0aOLDN 30875* Same as 4atex2 30874 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Theorem4atex2-0bOLDN 30876* Same as 4atex2 30874 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Theorem4atex2-0cOLDN 30877* Same as 4atex2 30874 except that and are zero. TODO: do we need this one or 4atex2-0aOLDN 30875 or 4atex2-0bOLDN 30876? (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Theorem4atex3 30878* More general version of 4atex 30873 for a line not necessarily connected to . (Contributed by NM, 29-May-2013.)

Theoremlautset 30879* The set of lattice automorphisms. (Contributed by NM, 11-May-2012.)

Theoremislaut 30880* The predictate "is a lattice automorphism." (Contributed by NM, 11-May-2012.)

Theoremlautle 30881 Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.)

Theoremlaut1o 30882 A lattice automorphism is one-to-one and onto. (Contributed by NM, 19-May-2012.)

Theoremlaut11 30883 One-to-one property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautcl 30884 A lattice automorphism value belongs to the base set. (Contributed by NM, 20-May-2012.)

TheoremlautcnvclN 30885 Reverse closure of a lattice automorphism. (Contributed by NM, 25-May-2012.) (New usage is discouraged.)

Theoremlautcnvle 30886 Less-than or equal property of lattice automorphism converse. (Contributed by NM, 19-May-2012.)

Theoremlautcnv 30887 The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)

Theoremlautlt 30888 Less-than property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautcvr 30889 Covering property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautj 30890 Meet property of a lattice automorphism. (Contributed by NM, 25-May-2012.)

Theoremlautm 30891 Meet property of a lattice automorphism. (Contributed by NM, 19-May-2012.)

Theoremlauteq 30892* A lattice automorphism argument is equal to its value if all atoms are equal to their values. (Contributed by NM, 24-May-2012.)

Theoremidlaut 30893 The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.)

Theoremlautco 30894 The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)

TheorempautsetN 30895* The set of projective automorphisms. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

TheoremispautN 30896* The predictate "is a projective automorphism." (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

Syntaxcldil 30897 Extend class notation with set of all lattice dilations.

Syntaxcltrn 30898 Extend class notation with set of all lattice translations.

SyntaxcdilN 30899 Extend class notation with set of all dilations.

SyntaxctrnN 30900 Extend class notation with set of all translations.

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