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Theorem List for Metamath Proof Explorer - 30501-30600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlhpocnel2 30501 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 30502 The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 18-May-2012.)

Theoremlhpjat2 30503 The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 4-Jun-2012.)

Theoremlhpj1 30504 The join of a co-atom (hyperplane) and an element not under it is the lattice unit. (Contributed by NM, 7-Dec-2012.)

Theoremlhpmcvr 30505 The meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 7-Dec-2012.)

Theoremlhpmcvr2 30506* 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 30507 Specialization of lhpmcvr2 30506. TODO: Use this to simplify many uses of to become . (Contributed by NM, 6-Apr-2014.)

Theoremlhpmcvr4N 30508 Specialization of lhpmcvr2 30506. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremlhpmcvr5N 30509* Specialization of lhpmcvr2 30506. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremlhpmcvr6N 30510* Specialization of lhpmcvr2 30506. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremlhpm0atN 30511 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 30512 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 30513 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 30514 Join and meet with different atoms under co-atom . (Contributed by NM, 15-Jun-2013.)

Theoremlhp2atnle 30515 Inequality for 2 different atoms under co-atom . (Contributed by NM, 17-Jun-2013.)

Theoremlhp2atne 30516 Inequality for joins with 2 different atoms under co-atom . (Contributed by NM, 22-Jul-2013.)

Theoremlhp2at0nle 30517 Inequality for 2 different atoms (or an atom and zero) under co-atom . (Contributed by NM, 28-Jul-2013.)

Theoremlhp2at0ne 30518 Inequality for joins with 2 different atoms (or an atom and zero) under co-atom . (Contributed by NM, 28-Jul-2013.)

Theoremlhpelim 30519 Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 30512 to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013.)

Theoremlhpmod2i2 30520 Modular law for hyperplanes analogous to atmod2i2 30344 for atoms. (Contributed by NM, 9-Feb-2013.)

Theoremlhpmod6i1 30521 Modular law for hyperplanes analogous to complement of atmod2i1 30343 for atoms. (Contributed by NM, 1-Jun-2013.)

Theoremlhprelat3N 30522* The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 29894. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)

Theoremcdlemb2 30523* 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 30524 Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014.)

Theoremlhpat 30525 Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)

Theoremlhpat4N 30526 Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)

Theoremlhpat2 30527 Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 21-Nov-2012.)

Theoremlhpat3 30528 There is only one atom under both and co-atom . (Contributed by NM, 21-Nov-2012.)

Theorem4atexlemk 30529 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemw 30530 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlempw 30531 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemp 30532 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemq 30533 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlems 30534 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemt 30535 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemutvt 30536 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlempnq 30537 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemnslpq 30538 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemkl 30539 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemkc 30540 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemwb 30541 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlempsb 30542 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemqtb 30543 Lemma for 4atexlem7 30557. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlempns 30544 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemswapqr 30545 Lemma for 4atexlem7 30557. 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 30546 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemv 30547 Lemma for 4atexlem7 30557. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemunv 30548 Lemma for 4atexlem7 30557. (Contributed by NM, 21-Nov-2012.)

Theorem4atexlemtlw 30549 Lemma for 4atexlem7 30557. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemntlpq 30550 Lemma for 4atexlem7 30557. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemc 30551 Lemma for 4atexlem7 30557. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemnclw 30552 Lemma for 4atexlem7 30557. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemex2 30553* Lemma for 4atexlem7 30557. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 25-Nov-2012.)

Theorem4atexlemcnd 30554 Lemma for 4atexlem7 30557. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemex4 30555* Lemma for 4atexlem7 30557. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012.)

Theorem4atexlemex6 30556* Lemma for 4atexlem7 30557. (Contributed by NM, 25-Nov-2012.)

Theorem4atexlem7 30557* 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 29826, our is a shorter way to express . With a longer proof, the condition could be eliminated (see 4atex 30558), although for some purposes this more restricted lemma may be adequate. (Contributed by NM, 25-Nov-2012.)

Theorem4atex 30558* 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 29826, our is a shorter way to express . (Contributed by NM, 27-May-2013.)

Theorem4atex2 30559* More general version of 4atex 30558 for a line not necessarily connected to . (Contributed by NM, 27-May-2013.)

Theorem4atex2-0aOLDN 30560* Same as 4atex2 30559 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Theorem4atex2-0bOLDN 30561* Same as 4atex2 30559 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Theorem4atex2-0cOLDN 30562* Same as 4atex2 30559 except that and are zero. TODO: do we need this one or 4atex2-0aOLDN 30560 or 4atex2-0bOLDN 30561? (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Theorem4atex3 30563* More general version of 4atex 30558 for a line not necessarily connected to . (Contributed by NM, 29-May-2013.)

Theoremlautset 30564* The set of lattice automorphisms. (Contributed by NM, 11-May-2012.)

Theoremislaut 30565* The predictate "is a lattice automorphism." (Contributed by NM, 11-May-2012.)

Theoremlautle 30566 Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.)

Theoremlaut1o 30567 A lattice automorphism is one-to-one and onto. (Contributed by NM, 19-May-2012.)

Theoremlaut11 30568 One-to-one property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautcl 30569 A lattice automorphism value belongs to the base set. (Contributed by NM, 20-May-2012.)

TheoremlautcnvclN 30570 Reverse closure of a lattice automorphism. (Contributed by NM, 25-May-2012.) (New usage is discouraged.)

Theoremlautcnvle 30571 Less-than or equal property of lattice automorphism converse. (Contributed by NM, 19-May-2012.)

Theoremlautcnv 30572 The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)

Theoremlautlt 30573 Less-than property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautcvr 30574 Covering property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautj 30575 Meet property of a lattice automorphism. (Contributed by NM, 25-May-2012.)

Theoremlautm 30576 Meet property of a lattice automorphism. (Contributed by NM, 19-May-2012.)

Theoremlauteq 30577* A lattice automorphism argument is equal to its value if all atoms are equal to their values. (Contributed by NM, 24-May-2012.)

Theoremidlaut 30578 The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.)

Theoremlautco 30579 The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)

TheorempautsetN 30580* The set of projective automorphisms. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

TheoremispautN 30581* The predictate "is a projective automorphism." (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

Syntaxcldil 30582 Extend class notation with set of all lattice dilations.

Syntaxcltrn 30583 Extend class notation with set of all lattice translations.

SyntaxcdilN 30584 Extend class notation with set of all dilations.

SyntaxctrnN 30585 Extend class notation with set of all translations.

Definitiondf-ldil 30586* Define set of all lattice dilations. Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Definitiondf-ltrn 30587* Define set of all lattice translations. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Definitiondf-dilN 30588* Define set of all dilations. Definition of dilation in [Crawley] p. 111. (Contributed by NM, 30-Jan-2012.)

Definitiondf-trnN 30589* Define set of all translations. Definition of translation in [Crawley] p. 111. (Contributed by NM, 4-Feb-2012.)

Theoremldilfset 30590* The mapping from fiducial co-atom to its set of lattice dilations. (Contributed by NM, 11-May-2012.)

Theoremldilset 30591* The set of lattice dilations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)

Theoremisldil 30592* The predicate "is a lattice dilation". Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Theoremldillaut 30593 A lattice dilation is an automorphism. (Contributed by NM, 20-May-2012.)

Theoremldil1o 30594 A lattice dilation is a one-to-one onto function. (Contributed by NM, 19-Apr-2013.)

Theoremldilval 30595 Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.)

Theoremidldil 30596 The identity function is a lattice dilation. (Contributed by NM, 18-May-2012.)

Theoremldilcnv 30597 The converse of a lattice dilation is a lattice dilation. (Contributed by NM, 10-May-2013.)

Theoremldilco 30598 The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)

Theoremltrnfset 30599* The set of all lattice translations for a lattice . (Contributed by NM, 11-May-2012.)

Theoremltrnset 30600* The set of lattice translations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)

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