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Theorem List for Metamath Proof Explorer - 33601-33700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremosumcllem10N 33601 Lemma for osumclN 33603. Contradict osumcllem9N 33600. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

Theoremosumcllem11N 33602 Lemma for osumclN 33603. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

TheoremosumclN 33603 Closure of orthogonal sum. If and are orthogonal closed projective subspaces, then their sum is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

TheorempmapojoinN 33604 For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 33488 where only one direction holds. (Contributed by NM, 11-Apr-2012.) (New usage is discouraged.)

TheorempexmidN 33605 Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 33589. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 33603. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

Theorempexmidlem1N 33606 Lemma for pexmidN 33605. Holland's proof implicitly requires , which we prove here. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)

Theorempexmidlem2N 33607 Lemma for pexmidN 33605. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)

Theorempexmidlem3N 33608 Lemma for pexmidN 33605. Use atom exchange hlatexch1 33031 to swap and . (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)

Theorempexmidlem4N 33609* Lemma for pexmidN 33605. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)

Theorempexmidlem5N 33610 Lemma for pexmidN 33605. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)

Theorempexmidlem6N 33611 Lemma for pexmidN 33605. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)

Theorempexmidlem7N 33612 Lemma for pexmidN 33605. Contradict pexmidlem6N 33611. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)

Theorempexmidlem8N 33613 Lemma for pexmidN 33605. The contradiction of pexmidlem6N 33611 and pexmidlem7N 33612 shows that there can be no atom that is not in , which is therefore the whole atom space. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)

TheorempexmidALTN 33614 Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 33589. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables , , , , in place of Hollands' l, m, P, Q, L respectively. TODO: should we make this obsolete? (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)

Theorempl42lem1N 33615 Lemma for pl42N 33619. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

Theorempl42lem2N 33616 Lemma for pl42N 33619. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

Theorempl42lem3N 33617 Lemma for pl42N 33619. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

Theorempl42lem4N 33618 Lemma for pl42N 33619. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

Theorempl42N 33619 Law holding in a Hilbert lattice that fails in orthomodular lattice L42 (Figure 7 in [MegPav2000] p. 2366). (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

Syntaxclh 33620 Extend class notation with set of all co-atoms (lattice hyperplanes).

Syntaxclaut 33621 Extend class notation with set of all lattice automorphisms.

SyntaxcwpointsN 33622 Extend class notation with W points.

SyntaxcpautN 33623 Extend class notation with set of all projective automorphisms.

Definitiondf-lhyp 33624* Define the set of lattice hyperplanes, which are all lattice elements covered by 1 (i.e. all co-atoms). We call them "hyperplanes" instead of "co-atoms" in analogy with projective geometry hyperplanes. (Contributed by NM, 11-May-2012.)

Definitiondf-laut 33625* Define set of lattice autoisomorphisms. (Contributed by NM, 11-May-2012.)

Definitiondf-watsN 33626* Define W-atoms corresponding to an arbitrary "fiducial (i.e. reference) atom" . These are all atoms not in the polarity of , which is the hyperplane determined by . Definition of set W in [Crawley] p. 111. (Contributed by NM, 26-Jan-2012.)

Definitiondf-pautN 33627* Define set of all projective automorphisms. This is the intended definition of automorphism in [Crawley] p. 112. (Contributed by NM, 26-Jan-2012.)

TheoremwatfvalN 33628* The W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

TheoremwatvalN 33629 Value of the W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

TheoremiswatN 33630 The predicate "is a W atom" (corresponding to fiducial atom ). (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

Theoremlhpset 33631* The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012.)

Theoremislhp 33632 The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 11-May-2012.)

Theoremislhp2 33633 The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 18-May-2012.)

Theoremlhpbase 33634 A co-atom is a member of the lattice base set (i.e. a lattice element). (Contributed by NM, 18-May-2012.)

Theoremlhp1cvr 33635 The lattice unit covers a co-atom (lattice hyperplane). (Contributed by NM, 18-May-2012.)

Theoremlhplt 33636 An atom under a co-atom is strictly less than it. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)

Theoremlhp2lt 33637 The join of two atoms under a co-atom is strictly less than it. (Contributed by NM, 8-Jul-2013.)

Theoremlhpexlt 33638* There exists an atom less than a co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)

Theoremlhp0lt 33639 A co-atom is greater than zero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)

Theoremlhpn0 33640 A co-atom is nonzero. TODO: is this needed? (Contributed by NM, 26-Apr-2013.)

Theoremlhpexle 33641* There exists an atom under a co-atom. (Contributed by NM, 26-Apr-2013.)

Theoremlhpexnle 33642* There exists an atom not under a co-atom. (Contributed by NM, 12-Apr-2013.)

Theoremlhpexle1lem 33643* Lemma for lhpexle1 33644 and others that eliminates restrictions on . (Contributed by NM, 24-Jul-2013.)

Theoremlhpexle1 33644* There exists an atom under a co-atom different from any given element. (Contributed by NM, 24-Jul-2013.)

Theoremlhpexle2lem 33645* Lemma for lhpexle2 33646. (Contributed by NM, 19-Jun-2013.)

Theoremlhpexle2 33646* There exists atom under a co-atom different from any two other elements. (Contributed by NM, 24-Jul-2013.)

Theoremlhpexle3lem 33647* 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 33648* There exists atom under a co-atom different from any three other elements. (Contributed by NM, 24-Jul-2013.)

Theoremlhpex2leN 33649* 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 33650 The orthocomplement of a co-atom (lattice hyperplane) is an atom. (Contributed by NM, 18-May-2012.)

Theoremlhpoc2N 33651 The orthocomplement of an atom is a co-atom (lattice hyperplane). (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)

Theoremlhpocnle 33652 The orthocomplement of a co-atom is not under it. (Contributed by NM, 22-May-2012.)

Theoremlhpocat 33653 The orthocomplement of a co-atom is an atom. (Contributed by NM, 9-Feb-2013.)

Theoremlhpocnel 33654 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 33655 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 33656 The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 18-May-2012.)

Theoremlhpjat2 33657 The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 4-Jun-2012.)

Theoremlhpj1 33658 The join of a co-atom (hyperplane) and an element not under it is the lattice unit. (Contributed by NM, 7-Dec-2012.)

Theoremlhpmcvr 33659 The meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 7-Dec-2012.)

Theoremlhpmcvr2 33660* 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 33661 Specialization of lhpmcvr2 33660. TODO: Use this to simplify many uses of to become . (Contributed by NM, 6-Apr-2014.)

Theoremlhpmcvr4N 33662 Specialization of lhpmcvr2 33660. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremlhpmcvr5N 33663* Specialization of lhpmcvr2 33660. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremlhpmcvr6N 33664* Specialization of lhpmcvr2 33660. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Theoremlhpm0atN 33665 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 33666 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 33667 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 33668 Join and meet with different atoms under co-atom . (Contributed by NM, 15-Jun-2013.)

Theoremlhp2atnle 33669 Inequality for 2 different atoms under co-atom . (Contributed by NM, 17-Jun-2013.)

Theoremlhp2atne 33670 Inequality for joins with 2 different atoms under co-atom . (Contributed by NM, 22-Jul-2013.)

Theoremlhp2at0nle 33671 Inequality for 2 different atoms (or an atom and zero) under co-atom . (Contributed by NM, 28-Jul-2013.)

Theoremlhp2at0ne 33672 Inequality for joins with 2 different atoms (or an atom and zero) under co-atom . (Contributed by NM, 28-Jul-2013.)

Theoremlhpelim 33673 Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 33666 to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013.)

Theoremlhpmod2i2 33674 Modular law for hyperplanes analogous to atmod2i2 33498 for atoms. (Contributed by NM, 9-Feb-2013.)

Theoremlhpmod6i1 33675 Modular law for hyperplanes analogous to complement of atmod2i1 33497 for atoms. (Contributed by NM, 1-Jun-2013.)

Theoremlhprelat3N 33676* The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 33048. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)

Theoremcdlemb2 33677* 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 33678 Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014.)

Theoremlhpat 33679 Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)

Theoremlhpat4N 33680 Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)

Theoremlhpat2 33681 Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 21-Nov-2012.)

Theoremlhpat3 33682 There is only one atom under both and co-atom . (Contributed by NM, 21-Nov-2012.)

Theorem4atexlemk 33683 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemw 33684 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlempw 33685 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemp 33686 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemq 33687 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlems 33688 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemt 33689 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemutvt 33690 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlempnq 33691 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemnslpq 33692 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemkl 33693 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemkc 33694 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemwb 33695 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlempsb 33696 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemqtb 33697 Lemma for 4atexlem7 33711. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlempns 33698 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemswapqr 33699 Lemma for 4atexlem7 33711. 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 33700 Lemma for 4atexlem7 33711. (Contributed by NM, 23-Nov-2012.)

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