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Theorem List for Metamath Proof Explorer - 30901-31000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlhple 30901 Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014.)

Theoremlhpat 30902 Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)

Theoremlhpat4N 30903 Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)

Theoremlhpat2 30904 Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 21-Nov-2012.)

Theoremlhpat3 30905 There is only one atom under both and co-atom . (Contributed by NM, 21-Nov-2012.)

Theorem4atexlemk 30906 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemw 30907 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlempw 30908 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemp 30909 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemq 30910 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlems 30911 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemt 30912 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemutvt 30913 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlempnq 30914 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemnslpq 30915 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemkl 30916 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemkc 30917 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemwb 30918 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlempsb 30919 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemqtb 30920 Lemma for 4atexlem7 30934. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlempns 30921 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemswapqr 30922 Lemma for 4atexlem7 30934. 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 30923 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemv 30924 Lemma for 4atexlem7 30934. (Contributed by NM, 23-Nov-2012.)

Theorem4atexlemunv 30925 Lemma for 4atexlem7 30934. (Contributed by NM, 21-Nov-2012.)

Theorem4atexlemtlw 30926 Lemma for 4atexlem7 30934. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemntlpq 30927 Lemma for 4atexlem7 30934. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemc 30928 Lemma for 4atexlem7 30934. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemnclw 30929 Lemma for 4atexlem7 30934. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemex2 30930* Lemma for 4atexlem7 30934. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 25-Nov-2012.)

Theorem4atexlemcnd 30931 Lemma for 4atexlem7 30934. (Contributed by NM, 24-Nov-2012.)

Theorem4atexlemex4 30932* Lemma for 4atexlem7 30934. Show that when , satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012.)

Theorem4atexlemex6 30933* Lemma for 4atexlem7 30934. (Contributed by NM, 25-Nov-2012.)

Theorem4atexlem7 30934* 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 30203, our is a shorter way to express . With a longer proof, the condition could be eliminated (see 4atex 30935), although for some purposes this more restricted lemma may be adequate. (Contributed by NM, 25-Nov-2012.)

Theorem4atex 30935* 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 30203, our is a shorter way to express . (Contributed by NM, 27-May-2013.)

Theorem4atex2 30936* More general version of 4atex 30935 for a line not necessarily connected to . (Contributed by NM, 27-May-2013.)

Theorem4atex2-0aOLDN 30937* Same as 4atex2 30936 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Theorem4atex2-0bOLDN 30938* Same as 4atex2 30936 except that is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Theorem4atex2-0cOLDN 30939* Same as 4atex2 30936 except that and are zero. TODO: do we need this one or 4atex2-0aOLDN 30937 or 4atex2-0bOLDN 30938? (Contributed by NM, 27-May-2013.) (New usage is discouraged.)

Theorem4atex3 30940* More general version of 4atex 30935 for a line not necessarily connected to . (Contributed by NM, 29-May-2013.)

Theoremlautset 30941* The set of lattice automorphisms. (Contributed by NM, 11-May-2012.)

Theoremislaut 30942* The predictate "is a lattice automorphism." (Contributed by NM, 11-May-2012.)

Theoremlautle 30943 Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.)

Theoremlaut1o 30944 A lattice automorphism is one-to-one and onto. (Contributed by NM, 19-May-2012.)

Theoremlaut11 30945 One-to-one property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautcl 30946 A lattice automorphism value belongs to the base set. (Contributed by NM, 20-May-2012.)

TheoremlautcnvclN 30947 Reverse closure of a lattice automorphism. (Contributed by NM, 25-May-2012.) (New usage is discouraged.)

Theoremlautcnvle 30948 Less-than or equal property of lattice automorphism converse. (Contributed by NM, 19-May-2012.)

Theoremlautcnv 30949 The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)

Theoremlautlt 30950 Less-than property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautcvr 30951 Covering property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautj 30952 Meet property of a lattice automorphism. (Contributed by NM, 25-May-2012.)

Theoremlautm 30953 Meet property of a lattice automorphism. (Contributed by NM, 19-May-2012.)

Theoremlauteq 30954* A lattice automorphism argument is equal to its value if all atoms are equal to their values. (Contributed by NM, 24-May-2012.)

Theoremidlaut 30955 The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.)

Theoremlautco 30956 The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)

TheorempautsetN 30957* The set of projective automorphisms. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

TheoremispautN 30958* The predictate "is a projective automorphism." (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

Syntaxcldil 30959 Extend class notation with set of all lattice dilations.

Syntaxcltrn 30960 Extend class notation with set of all lattice translations.

SyntaxcdilN 30961 Extend class notation with set of all dilations.

SyntaxctrnN 30962 Extend class notation with set of all translations.

Definitiondf-ldil 30963* Define set of all lattice dilations. Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Definitiondf-ltrn 30964* Define set of all lattice translations. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Definitiondf-dilN 30965* Define set of all dilations. Definition of dilation in [Crawley] p. 111. (Contributed by NM, 30-Jan-2012.)

Definitiondf-trnN 30966* Define set of all translations. Definition of translation in [Crawley] p. 111. (Contributed by NM, 4-Feb-2012.)

Theoremldilfset 30967* The mapping from fiducial co-atom to its set of lattice dilations. (Contributed by NM, 11-May-2012.)

Theoremldilset 30968* The set of lattice dilations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)

Theoremisldil 30969* The predicate "is a lattice dilation". Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Theoremldillaut 30970 A lattice dilation is an automorphism. (Contributed by NM, 20-May-2012.)

Theoremldil1o 30971 A lattice dilation is a one-to-one onto function. (Contributed by NM, 19-Apr-2013.)

Theoremldilval 30972 Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.)

Theoremidldil 30973 The identity function is a lattice dilation. (Contributed by NM, 18-May-2012.)

Theoremldilcnv 30974 The converse of a lattice dilation is a lattice dilation. (Contributed by NM, 10-May-2013.)

Theoremldilco 30975 The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)

Theoremltrnfset 30976* The set of all lattice translations for a lattice . (Contributed by NM, 11-May-2012.)

Theoremltrnset 30977* The set of lattice translations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)

Theoremisltrn 30978* The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Theoremisltrn2N 30979* The predicate "is a lattice translation". Version of isltrn 30978 that considers only different and . TODO: Can this eliminate some separate proofs for the case? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)

Theoremltrnu 30980 Uniqueness property of a lattice translation value for atoms not under the fiducial co-atom . Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 20-May-2012.)

Theoremltrnldil 30981 A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.)

Theoremltrnlaut 30982 A lattice translation is a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremltrn1o 30983 A lattice translation is a one-to-one onto function. (Contributed by NM, 20-May-2012.)

Theoremltrncl 30984 Closure of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremltrn11 30985 One-to-one property of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremltrncnvnid 30986 If a translation is different from the identity, so is its converse. (Contributed by NM, 17-Jun-2013.)

TheoremltrncoidN 30987 Two translations are equal if the composition of one with the converse of the other is the zero translation. This is an analog of vector subtraction. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremltrnle 30988 Less-than or equal property of a lattice translation. (Contributed by NM, 20-May-2012.)

TheoremltrncnvleN 30989 Less-than or equal property of lattice translation converse. (Contributed by NM, 10-May-2013.) (New usage is discouraged.)

Theoremltrnm 30990 Lattice translation of a meet. (Contributed by NM, 20-May-2012.)

Theoremltrnj 30991 Lattice translation of a meet. TODO: change antecedent to (Contributed by NM, 25-May-2012.)

Theoremltrncvr 30992 Covering property of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremltrnval1 30993 Value of a lattice translation under its co-atom. (Contributed by NM, 20-May-2012.)

Theoremltrnid 30994* A lattice translation is the identity function iff all atoms not under the fiducial co-atom are equal to their values. (Contributed by NM, 24-May-2012.)

Theoremltrnnid 30995* If a lattice translation is not the identity, then there is an atom not under the fiducial co-atom and not equal to its translation. (Contributed by NM, 24-May-2012.)

Theoremltrnatb 30996 The lattice translation of an atom is an atom. (Contributed by NM, 20-May-2012.)

Theoremltrncnvatb 30997 The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.)

Theoremltrnel 30998 The lattice translation of an atom not under the fiducial co-atom is also an atom not under the fiducial co-atom. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 22-May-2012.)

Theoremltrnat 30999 The lattice translation of an atom is also an atom. TODO: See if this can shorten some ltrnel 30998 uses. (Contributed by NM, 25-May-2012.)

Theoremltrncnvat 31000 The converse of the lattice translation of an atom is an atom. (Contributed by NM, 9-May-2013.)

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