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Theorem List for Metamath Proof Explorer - 30601-30700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisltrn 30601* The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)

Theoremisltrn2N 30602* The predicate "is a lattice translation". Version of isltrn 30601 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 30603 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 30604 A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.)

Theoremltrnlaut 30605 A lattice translation is a lattice automorphism. (Contributed by NM, 20-May-2012.)

Theoremltrn1o 30606 A lattice translation is a one-to-one onto function. (Contributed by NM, 20-May-2012.)

Theoremltrncl 30607 Closure of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremltrn11 30608 One-to-one property of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremltrncnvnid 30609 If a translation is different from the identity, so is its converse. (Contributed by NM, 17-Jun-2013.)

TheoremltrncoidN 30610 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 30611 Less-than or equal property of a lattice translation. (Contributed by NM, 20-May-2012.)

TheoremltrncnvleN 30612 Less-than or equal property of lattice translation converse. (Contributed by NM, 10-May-2013.) (New usage is discouraged.)

Theoremltrnm 30613 Lattice translation of a meet. (Contributed by NM, 20-May-2012.)

Theoremltrnj 30614 Lattice translation of a meet. TODO: change antecedent to (Contributed by NM, 25-May-2012.)

Theoremltrncvr 30615 Covering property of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremltrnval1 30616 Value of a lattice translation under its co-atom. (Contributed by NM, 20-May-2012.)

Theoremltrnid 30617* 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 30618* 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 30619 The lattice translation of an atom is an atom. (Contributed by NM, 20-May-2012.)

Theoremltrncnvatb 30620 The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.)

Theoremltrnel 30621 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 30622 The lattice translation of an atom is also an atom. TODO: See if this can shorten some ltrnel 30621 uses. (Contributed by NM, 25-May-2012.)

Theoremltrncnvat 30623 The converse of the lattice translation of an atom is an atom. (Contributed by NM, 9-May-2013.)

Theoremltrncnvel 30624 The converse of the lattice translation of an atom not under the fiducial co-atom. (Contributed by NM, 10-May-2013.)

TheoremltrncoelN 30625 Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 30621 uses. (Contributed by NM, 1-May-2013.) (New usage is discouraged.)

Theoremltrncoat 30626 Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 30621, ltrnat 30622 uses. (Contributed by NM, 1-May-2013.)

Theoremltrncoval 30627 Two ways to express value of translation composition. (Contributed by NM, 31-May-2013.)

Theoremltrncnv 30628 The converse of a lattice translation is a lattice translation. (Contributed by NM, 10-May-2013.)

Theoremltrn11at 30629 Frequently used one-to-one property of lattice translation atoms. (Contributed by NM, 5-May-2013.)

Theoremltrneq2 30630* The equality of two translations is determined by their equality at atoms. (Contributed by NM, 2-Mar-2014.)

Theoremltrneq 30631* The equality of two translations is determined by their equality at atoms not under co-atom . (Contributed by NM, 20-Jun-2013.)

Theoremidltrn 30632 The identity function is a lattice translation. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 18-May-2012.)

Theoremltrnmw 30633 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 30634* The mapping from fiducial atom to set of dilations. (Contributed by NM, 30-Jan-2012.) (New usage is discouraged.)

TheoremdilsetN 30635* The set of dilations for a fiducial atom . (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

TheoremisdilN 30636* The predicate "is a dilation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

TheoremtrnfsetN 30637* The mapping from fiducial atom to set of translations. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

TheoremtrnsetN 30638* The set of translations for a fiducial atom . (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

TheoremistrnN 30639* The predicate "is a translation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

Syntaxctrl 30640 Extend class notation with set of all traces of lattice translations.

Definitiondf-trl 30641* Define trace of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremtrlfset 30642* The set of all traces of lattice translations for a lattice . (Contributed by NM, 20-May-2012.)

Theoremtrlset 30643* The set of traces of lattice translations for a fiducial co-atom . (Contributed by NM, 20-May-2012.)

Theoremtrlval 30644* The value of the trace of a lattice translation. (Contributed by NM, 20-May-2012.)

Theoremtrlval2 30645 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 30646 Closure of the trace of a lattice translation. (Contributed by NM, 22-May-2012.)

Theoremtrlcnv 30647 The trace of the converse of a lattice translation. (Contributed by NM, 10-May-2013.)

Theoremtrljat1 30648 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 30346? (Contributed by NM, 22-May-2012.)

Theoremtrljat2 30649 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 30650 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 30651 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 30652 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 30653 The trace of a lattice translation is an atom or zero. (Contributed by NM, 5-May-2013.)

Theoremtrlatn0 30654 The trace of a lattice translation is an atom iff it is nonzero. (Contributed by NM, 14-Jun-2013.)

Theoremtrlnidat 30655 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 30656 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 30657 Property of the identity lattice translation. (Contributed by NM, 27-May-2012.)

Theoremtrlid0 30658 The trace of the identity translation is zero. (Contributed by NM, 11-Jun-2013.)

Theoremtrlnidatb 30659 A lattice translation is not the identity iff its trace is an atom. TODO: Can proofs be reorganized so this goes with trlnidat 30655? Why do both this and ltrnideq 30657 need trlnidat 30655? (Contributed by NM, 4-Jun-2013.)

Theoremtrlid0b 30660 A lattice translation is the identity iff its trace is zero. (Contributed by NM, 14-Jun-2013.)

Theoremtrlnid 30661 Different translations with the same trace cannot be the identity. (Contributed by NM, 26-Jul-2013.)

Theoremltrn2ateq 30662 Property of the equality of a lattice translation with its value. (Contributed by NM, 27-May-2012.)

Theoremltrnateq 30663 If any atom (under ) is not equal to its translation, so is any other atom. (Contributed by NM, 6-May-2013.)

Theoremltrnatneq 30664 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 30665 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 30666 The trace of a lattice translation is less than the fiducial co-atom . (Contributed by NM, 25-May-2012.)

Theoremtrlne 30667 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 30668 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 30669 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 30670 The value of the trace of a lattice translation in terms of 2 atoms. (Contributed by NM, 3-May-2013.)

Theoremtrlval5 30671 The value of the trace of a lattice translation in terms of itself. (Contributed by NM, 19-Jul-2013.)

Theoremarglem1N 30672 Lemma for Desargues' law. Theorem 13.3 of [Crawley] p. 110, 3rd and 4th lines from bottom. In these lemmas, , , , , , , , , , , and represent Crawley's a0, a1, a2, b0, b1, b2, c, z0, z1, z2, and p respectively. (Contributed by NM, 28-Jun-2012.) (New usage is discouraged.)

Theoremcdlemc1 30673 Part of proof of Lemma C in [Crawley] p. 112. TODO: shorten with atmod3i1 30346? (Contributed by NM, 29-May-2012.)

Theoremcdlemc2 30674 Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)

Theoremcdlemc3 30675 Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)

Theoremcdlemc4 30676 Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)

Theoremcdlemc5 30677 Lemma for cdlemc 30679. (Contributed by NM, 26-May-2012.)

Theoremcdlemc6 30678 Lemma for cdlemc 30679. (Contributed by NM, 26-May-2012.)

Theoremcdlemc 30679 Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)

Theoremcdlemd1 30680 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)

Theoremcdlemd2 30681 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)

Theoremcdlemd3 30682 Part of proof of Lemma D in [Crawley] p. 113. The requirement is not mentioned in their proof. (Contributed by NM, 29-May-2012.)

Theoremcdlemd4 30683 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)

Theoremcdlemd5 30684 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)

Theoremcdlemd6 30685 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-May-2012.)

Theoremcdlemd7 30686 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)

Theoremcdlemd8 30687 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)

Theoremcdlemd9 30688 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)

Theoremcdlemd 30689 If two translations agree at any atom not under the fiducial co-atom , then they are equal. Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)

Theoremltrneq3 30690 Two translations agree at any atom not under the fiducial co-atom iff they are equal. (Contributed by NM, 25-Jul-2013.)

Theoremcdleme00a 30691 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme0aa 30692 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)

Theoremcdleme0a 30693 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)

Theoremcdleme0b 30694 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme0c 30695 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)

Theoremcdleme0cp 30696 Part of proof of Lemma E in [Crawley] p. 113. TODO: Reformat as in cdlemg3a 31079- swap consequent equality; make antecedent use df-3an 938. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme0cq 30697 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Apr-2013.)

Theoremcdleme0dN 30698 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)

Theoremcdleme0e 30699 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)

Theoremcdleme0fN 30700 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)

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