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Theorem List for Metamath Proof Explorer - 29701-29800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremllni2 29701 The join of two different atoms is a lattice line. (Contributed by NM, 26-Jun-2012.)

Theoremllnnleat 29702 An atom cannot majorize a lattice line. (Contributed by NM, 8-Jul-2012.)

Theoremllnneat 29703 A lattice line is not an atom. (Contributed by NM, 19-Jun-2012.)

Theorem2atneat 29704 The join of two distinct atoms is not an atom. (Contributed by NM, 12-Oct-2012.)

Theoremllnn0 29705 A lattice line is non-zero. (Contributed by NM, 15-Jul-2012.)

Theoremislln2a 29706 The predicate "is a lattice line" in terms of atoms. (Contributed by NM, 15-Jul-2012.)

Theoremllnle 29707* Any element greater than 0 and not an atom majorizes a lattice line. (Contributed by NM, 28-Jun-2012.)

Theorematcvrlln2 29708 An atom under a line is covered by it. (Contributed by NM, 2-Jul-2012.)

Theorematcvrlln 29709 An element covering an atom is a lattice line and vice-versa. (Contributed by NM, 18-Jun-2012.)

TheoremllnexatN 29710* 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.)

Theoremllncmp 29711 If two lattice lines are comparable, they are equal. (Contributed by NM, 19-Jun-2012.)

Theoremllnnlt 29712 Two lattice lines cannot satisfy the less than relation. (Contributed by NM, 26-Jun-2012.)

Theorem2llnmat 29713 Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)

Theorem2at0mat0 29714 Special case of 2atmat0 29715 where one atom could be zero. (Contributed by NM, 30-May-2013.)

Theorem2atmat0 29715 The meet of two unequal lines (expressed as joins of atoms) is an atom or zero. (Contributed by NM, 2-Dec-2012.)

Theorem2atm 29716 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.)

Theoremps-2c 29717 Variation of projective geometry axiom ps-2 29667. (Contributed by NM, 3-Jul-2012.)

Theoremlplnset 29718* The set of lattice planes in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)

Theoremislpln 29719* The predicate "is a lattice plane". (Contributed by NM, 16-Jun-2012.)

Theoremislpln4 29720* The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.)

Theoremlplni 29721 Condition implying a lattice plane. (Contributed by NM, 20-Jun-2012.)

Theoremislpln3 29722* The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.)

Theoremlplnbase 29723 A lattice plane is a lattice element. (Contributed by NM, 17-Jun-2012.)

Theoremislpln5 29724* The predicate "is a lattice plane" in terms of atoms. (Contributed by NM, 24-Jun-2012.)

Theoremislpln2 29725* The predicate "is a lattice plane" in terms of atoms. (Contributed by NM, 25-Jun-2012.)

Theoremlplni2 29726 The join of 3 different atoms is a lattice plane. (Contributed by NM, 4-Jul-2012.)

Theoremlvolex3N 29727* There is an atom outside of a lattice plane i.e. a 3-dimensional lattice volume exists. (Contributed by NM, 28-Jul-2012.) (New usage is discouraged.)

TheoremllnmlplnN 29728 The intersection of a line with a plane not containing it is an atom. (Contributed by NM, 29-Jun-2012.) (New usage is discouraged.)

Theoremlplnle 29729* Any element greater than 0 and not an atom and not a lattice line majorizes a lattice plane. (Contributed by NM, 28-Jun-2012.)

Theoremlplnnle2at 29730 A lattice lattice line (or atom) cannot majorize a lattice plane. (Contributed by NM, 8-Jul-2012.)

Theoremlplnnleat 29731 A lattice plane cannot majorize an atom. (Contributed by NM, 14-Jul-2012.)

Theoremlplnnlelln 29732 A lattice plane is not less than or equal to a lattice line. (Contributed by NM, 14-Jul-2012.)

Theorem2atnelpln 29733 The join of two atoms is not a lattice plane. (Contributed by NM, 16-Jul-2012.)

Theoremlplnneat 29734 No lattice plane is an atom. (Contributed by NM, 15-Jul-2012.)

Theoremlplnnelln 29735 No lattice plane is a lattice line. (Contributed by NM, 19-Jun-2012.)

Theoremlplnn0N 29736 A lattice plane is non-zero. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)

Theoremislpln2a 29737 The predicate "is a lattice plane" for join of atoms. (Contributed by NM, 16-Jul-2012.)

Theoremislpln2ah 29738 The predicate "is a lattice plane" for join of atoms. Version of islpln2a 29737 expressed with an abbreviation hypothesis. (Contributed by NM, 30-Jul-2012.)

TheoremlplnriaN 29739 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)

TheoremlplnribN 29740 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)

Theoremlplnric 29741 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.)

Theoremlplnri1 29742 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.)

Theoremlplnri2N 29743 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)

Theoremlplnri3N 29744 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)

TheoremlplnllnneN 29745 Two lattice lines defined by atoms defining a lattice plane are not equal. (Contributed by NM, 9-Oct-2012.) (New usage is discouraged.)

Theoremllncvrlpln2 29746 A lattice line under a lattice plane is covered by it. (Contributed by NM, 24-Jun-2012.)

Theoremllncvrlpln 29747 An element covering a lattice line is a lattice plane and vice-versa. (Contributed by NM, 26-Jun-2012.)

Theorem2lplnmN 29748 If the join of two lattice planes covers one of them, their meet is a lattice line. (Contributed by NM, 30-Jun-2012.) (New usage is discouraged.)

Theorem2llnmj 29749 The meet of two lattice lines is an atom iff their join is a lattice plane. (Contributed by NM, 27-Jun-2012.)

Theorem2atmat 29750 The meet of two intersecting lines (expressed as joins of atoms) is an atom. (Contributed by NM, 21-Nov-2012.)

Theoremlplncmp 29751 If two lattice planes are comparable, they are equal. (Contributed by NM, 24-Jun-2012.)

TheoremlplnexatN 29752* Given a lattice line on a lattice plane, there is an atom whose join with the line equals the plane. (Contributed by NM, 29-Jun-2012.) (New usage is discouraged.)

TheoremlplnexllnN 29753* Given an atom on a lattice plane, there is a lattice line whose join with the atom equals the plane. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)

Theoremlplnnlt 29754 Two lattice planes cannot satisfy the less than relation. (Contributed by NM, 7-Jul-2012.)

Theorem2llnjaN 29755 The join of two different lattice lines in a lattice plane equals the plane (version of 2llnjN 29756 in terms of atoms). (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)

Theorem2llnjN 29756 The join of two different lattice lines in a lattice plane equals the plane. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)

Theorem2llnm2N 29757 The meet of two different lattice lines in a lattice plane is an atom. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)

Theorem2llnm3N 29758 Two lattice lines in a lattice plane always meet. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)

Theorem2llnm4 29759 Two lattice lines that majorize the same atom always meet. (Contributed by NM, 20-Jul-2012.)

Theorem2llnmeqat 29760 An atom equals the intersection of two majorizing lines. (Contributed by NM, 3-Apr-2013.)

Theoremlvolset 29761* The set of 3-dim lattice volumes in a Hilbert lattice. (Contributed by NM, 1-Jul-2012.)

Theoremislvol 29762* The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)

Theoremislvol4 29763* The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)

Theoremlvoli 29764 Condition implying a 3-dim lattice volume. (Contributed by NM, 1-Jul-2012.)

Theoremislvol3 29765* The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)

Theoremlvoli3 29766 Condition implying a 3-dim lattice volume. (Contributed by NM, 2-Aug-2012.)

Theoremlvolbase 29767 A 3-dim lattice volume is a lattice element. (Contributed by NM, 1-Jul-2012.)

Theoremislvol5 29768* The predicate "is a 3-dim lattice volume" in terms of atoms. (Contributed by NM, 1-Jul-2012.)

Theoremislvol2 29769* The predicate "is a 3-dim lattice volume" in terms of atoms. (Contributed by NM, 1-Jul-2012.)

Theoremlvoli2 29770 The join of 4 different atoms is a lattice volume. (Contributed by NM, 8-Jul-2012.)

Theoremlvolnle3at 29771 A lattice plane (or lattice line or atom) cannot majorize a lattice volume. (Contributed by NM, 8-Jul-2012.)

Theoremlvolnleat 29772 An atom cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)

Theoremlvolnlelln 29773 A lattice line cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)

Theoremlvolnlelpln 29774 A lattice plane cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)

Theorem3atnelvolN 29775 The join of 3 atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)

Theorem2atnelvolN 29776 The join of two atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)

TheoremlvolneatN 29777 No lattice volume is an atom. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)

Theoremlvolnelln 29778 No lattice volume is a lattice line. (Contributed by NM, 15-Jul-2012.)

Theoremlvolnelpln 29779 No lattice volume is a lattice plane. (Contributed by NM, 19-Jun-2012.)

Theoremlvoln0N 29780 A lattice volume is non-zero. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)

Theoremislvol2aN 29781 The predicate "is a lattice volume". (Contributed by NM, 16-Jul-2012.) (New usage is discouraged.)

Theorem4atlem0a 29782 Lemma for 4at 29802. (Contributed by NM, 10-Jul-2012.)

Theorem4atlem0ae 29783 Lemma for 4at 29802. (Contributed by NM, 10-Jul-2012.)

Theorem4atlem0be 29784 Lemma for 4at 29802. (Contributed by NM, 10-Jul-2012.)

Theorem4atlem3 29785 Lemma for 4at 29802. Break inequality into 4 cases. (Contributed by NM, 8-Jul-2012.)

Theorem4atlem3a 29786 Lemma for 4at 29802. Break inequality into 3 cases. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem3b 29787 Lemma for 4at 29802. Break inequality into 2 cases. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem4a 29788 Lemma for 4at 29802. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem4b 29789 Lemma for 4at 29802. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem4c 29790 Lemma for 4at 29802. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem4d 29791 Lemma for 4at 29802. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem9 29792 Lemma for 4at 29802. Substitute for . (Contributed by NM, 9-Jul-2012.)

Theorem4atlem10a 29793 Lemma for 4at 29802. Substitute for . (Contributed by NM, 9-Jul-2012.)

Theorem4atlem10b 29794 Lemma for 4at 29802. Substitute for (cont.). (Contributed by NM, 10-Jul-2012.)

Theorem4atlem10 29795 Lemma for 4at 29802. Combine both possible cases. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem11a 29796 Lemma for 4at 29802. Substitute for . (Contributed by NM, 9-Jul-2012.)

Theorem4atlem11b 29797 Lemma for 4at 29802. Substitute for (cont.). (Contributed by NM, 10-Jul-2012.)

Theorem4atlem11 29798 Lemma for 4at 29802. Combine all three possible cases. (Contributed by NM, 10-Jul-2012.)

Theorem4atlem12a 29799 Lemma for 4at 29802. Substitute for . (Contributed by NM, 9-Jul-2012.)

Theorem4atlem12b 29800 Lemma for 4at 29802. Substitute for (cont.). (Contributed by NM, 11-Jul-2012.)

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