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

Theoremlautcnv 30901 The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)

Theoremlautlt 30902 Less-than property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautcvr 30903 Covering property of a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremlautj 30904 Meet property of a lattice automorphism. (Contributed by NM, 25-May-2012.)

Theoremlautm 30905 Meet property of a lattice automorphism. (Contributed by NM, 19-May-2012.)

Theoremlauteq 30906* A lattice automorphism argument is equal to its value if all atoms are equal to their values. (Contributed by NM, 24-May-2012.)

Theoremidlaut 30907 The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.)

Theoremlautco 30908 The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)

TheorempautsetN 30909* The set of projective automorphisms. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

TheoremispautN 30910* The predictate "is a projective automorphism." (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

Syntaxcldil 30911 Extend class notation with set of all lattice dilations.

Syntaxcltrn 30912 Extend class notation with set of all lattice translations.

SyntaxcdilN 30913 Extend class notation with set of all dilations.

SyntaxctrnN 30914 Extend class notation with set of all translations.

Definitiondf-ldil 30915* Define set of all lattice dilations. Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Definitiondf-ltrn 30916* Define set of all lattice translations. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Definitiondf-dilN 30917* Define set of all dilations. Definition of dilation in [Crawley] p. 111. (Contributed by NM, 30-Jan-2012.)

Definitiondf-trnN 30918* Define set of all translations. Definition of translation in [Crawley] p. 111. (Contributed by NM, 4-Feb-2012.)

Theoremldilfset 30919* The mapping from fiducial co-atom to its set of lattice dilations. (Contributed by NM, 11-May-2012.)

Theoremldilset 30920* The set of lattice dilations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)

Theoremisldil 30921* The predicate "is a lattice dilation". Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Theoremldillaut 30922 A lattice dilation is an automorphism. (Contributed by NM, 20-May-2012.)

Theoremldil1o 30923 A lattice dilation is a one-to-one onto function. (Contributed by NM, 19-Apr-2013.)

Theoremldilval 30924 Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.)

Theoremidldil 30925 The identity function is a lattice dilation. (Contributed by NM, 18-May-2012.)

Theoremldilcnv 30926 The converse of a lattice dilation is a lattice dilation. (Contributed by NM, 10-May-2013.)

Theoremldilco 30927 The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)

Theoremltrnfset 30928* The set of all lattice translations for a lattice . (Contributed by NM, 11-May-2012.)

Theoremltrnset 30929* The set of lattice translations for a fiducial co-atom . (Contributed by NM, 11-May-2012.)

Theoremisltrn 30930* The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Theoremisltrn2N 30931* The predicate "is a lattice translation". Version of isltrn 30930 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 30932 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 30933 A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.)

Theoremltrnlaut 30934 A lattice translation is a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremltrn1o 30935 A lattice translation is a one-to-one onto function. (Contributed by NM, 20-May-2012.)

Theoremltrncl 30936 Closure of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremltrn11 30937 One-to-one property of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremltrncnvnid 30938 If a translation is different from the identity, so is its converse. (Contributed by NM, 17-Jun-2013.)

TheoremltrncoidN 30939 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 30940 Less-than or equal property of a lattice translation. (Contributed by NM, 20-May-2012.)

TheoremltrncnvleN 30941 Less-than or equal property of lattice translation converse. (Contributed by NM, 10-May-2013.) (New usage is discouraged.)

Theoremltrnm 30942 Lattice translation of a meet. (Contributed by NM, 20-May-2012.)

Theoremltrnj 30943 Lattice translation of a meet. TODO: change antecedent to (Contributed by NM, 25-May-2012.)

Theoremltrncvr 30944 Covering property of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremltrnval1 30945 Value of a lattice translation under its co-atom. (Contributed by NM, 20-May-2012.)

Theoremltrnid 30946* 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 30947* 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 30948 The lattice translation of an atom is an atom. (Contributed by NM, 20-May-2012.)

Theoremltrncnvatb 30949 The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.)

Theoremltrnel 30950 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 30951 The lattice translation of an atom is also an atom. TODO: See if this can shorten some ltrnel 30950 uses. (Contributed by NM, 25-May-2012.)

Theoremltrncnvat 30952 The converse of the lattice translation of an atom is an atom. (Contributed by NM, 9-May-2013.)

Theoremltrncnvel 30953 The converse of the lattice translation of an atom not under the fiducial co-atom. (Contributed by NM, 10-May-2013.)

TheoremltrncoelN 30954 Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 30950 uses. (Contributed by NM, 1-May-2013.) (New usage is discouraged.)

Theoremltrncoat 30955 Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 30950, ltrnat 30951 uses. (Contributed by NM, 1-May-2013.)

Theoremltrncoval 30956 Two ways to express value of translation composition. (Contributed by NM, 31-May-2013.)

Theoremltrncnv 30957 The converse of a lattice translation is a lattice translation. (Contributed by NM, 10-May-2013.)

Theoremltrn11at 30958 Frequently used one-to-one property of lattice translation atoms. (Contributed by NM, 5-May-2013.)

Theoremltrneq2 30959* The equality of two translations is determined by their equality at atoms. (Contributed by NM, 2-Mar-2014.)

Theoremltrneq 30960* The equality of two translations is determined by their equality at atoms not under co-atom . (Contributed by NM, 20-Jun-2013.)

Theoremidltrn 30961 The identity function is a lattice translation. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 18-May-2012.)

Theoremltrnmw 30962 Property of lattice translation value. Remark below Lemma B in [Crawley] p. 112. TODO: Can this be used in more places? (Contributed by NM, 20-May-2012.)

TheoremdilfsetN 30963* The mapping from fiducial atom to set of dilations. (Contributed by NM, 30-Jan-2012.) (New usage is discouraged.)

TheoremdilsetN 30964* The set of dilations for a fiducial atom . (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

TheoremisdilN 30965* The predicate "is a dilation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

TheoremtrnfsetN 30966* The mapping from fiducial atom to set of translations. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

TheoremtrnsetN 30967* The set of translations for a fiducial atom . (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

TheoremistrnN 30968* The predicate "is a translation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

Syntaxctrl 30969 Extend class notation with set of all traces of lattice translations.

Definitiondf-trl 30970* Define trace of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremtrlfset 30971* The set of all traces of lattice translations for a lattice . (Contributed by NM, 20-May-2012.)

Theoremtrlset 30972* The set of traces of lattice translations for a fiducial co-atom . (Contributed by NM, 20-May-2012.)

Theoremtrlval 30973* The value of the trace of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremtrlval2 30974 The value of the trace of a lattice translation, given any atom not under the fiducial co-atom . Note: this requires only the weaker assumption ; we use for convenience. (Contributed by NM, 20-May-2012.)

Theoremtrlcl 30975 Closure of the trace of a lattice translation. (Contributed by NM, 22-May-2012.)

Theoremtrlcnv 30976 The trace of the converse of a lattice translation. (Contributed by NM, 10-May-2013.)

Theoremtrljat1 30977 The value of a translation of an atom not under the fiducial co-atom , joined with trace. Equation above Lemma C in [Crawley] p. 112. Todo: shorten with atmod3i1 30675? (Contributed by NM, 22-May-2012.)

Theoremtrljat2 30978 The value of a translation of an atom not under the fiducial co-atom , joined with trace. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)

Theoremtrljat3 30979 The value of a translation of an atom not under the fiducial co-atom , joined with trace. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 22-May-2012.)

Theoremtrlat 30980 If an atom differs from its translation, the trace is an atom. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)

Theoremtrl0 30981 If an atom not under the fiducial co-atom equals its lattice translation, the trace of the translation is zero. (Contributed by NM, 24-May-2012.)

Theoremtrlator0 30982 The trace of a lattice translation is an atom or zero. (Contributed by NM, 5-May-2013.)

Theoremtrlatn0 30983 The trace of a lattice translation is an atom iff it is nonzero. (Contributed by NM, 14-Jun-2013.)

Theoremtrlnidat 30984 The trace of a lattice translation other than the identity is an atom. Remark above Lemma C in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)

Theoremltrnnidn 30985 If a lattice translation is not the identity, then the translation of any atom not under the fiducial co-atom is different from the atom. Remark above Lemma C in [Crawley] p. 112. (Contributed by NM, 24-May-2012.)

Theoremltrnideq 30986 Property of the identity lattice translation. (Contributed by NM, 27-May-2012.)

Theoremtrlid0 30987 The trace of the identity translation is zero. (Contributed by NM, 11-Jun-2013.)

Theoremtrlnidatb 30988 A lattice translation is not the identity iff its trace is an atom. TODO: Can proofs be reorganized so this goes with trlnidat 30984? Why do both this and ltrnideq 30986 need trlnidat 30984? (Contributed by NM, 4-Jun-2013.)

Theoremtrlid0b 30989 A lattice translation is the identity iff its trace is zero. (Contributed by NM, 14-Jun-2013.)

Theoremtrlnid 30990 Different translations with the same trace cannot be the identity. (Contributed by NM, 26-Jul-2013.)

Theoremltrn2ateq 30991 Property of the equality of a lattice translation with its value. (Contributed by NM, 27-May-2012.)

Theoremltrnateq 30992 If any atom (under ) is not equal to its translation, so is any other atom. (Contributed by NM, 6-May-2013.)

Theoremltrnatneq 30993 If any atom (under ) is not equal to its translation, so is any other atom. TODO: isn't needed to prove this. Will removing it shorten (and not lengthen) proofs using it? (Contributed by NM, 6-May-2013.)

Theoremltrnatlw 30994 If the value of an atom equals the atom in a non-identity translation, the atom is under the fiducial hyperplane. (Contributed by NM, 15-May-2013.)

Theoremtrlle 30995 The trace of a lattice translation is less than the fiducial co-atom . (Contributed by NM, 25-May-2012.)

Theoremtrlne 30996 The trace of a lattice translation is not equal to any atom not under the fiducial co-atom . Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)

Theoremtrlnle 30997 The atom not under the fiducial co-atom is not less than the trace of a lattice translation. Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 26-May-2012.)

Theoremtrlval3 30998 The value of the trace of a lattice translation in terms of 2 atoms. TODO: Try to shorten proof. (Contributed by NM, 3-May-2013.)

Theoremtrlval4 30999 The value of the trace of a lattice translation in terms of 2 atoms. (Contributed by NM, 3-May-2013.)

Theoremtrlval5 31000 The value of the trace of a lattice translation in terms of itself. (Contributed by NM, 19-Jul-2013.)

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