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Type | Label | Description |
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Statement | ||
Theorem | lhpmcvr4N 39201 | Specialization of lhpmcvr2 39199. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ ((π β π΅ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΅ β§ (π β§ π) β€ π β§ π β€ π)) β Β¬ π β€ π) | ||
Theorem | lhpmcvr5N 39202* | Specialization of lhpmcvr2 39199. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π) β§ (π β π΅ β§ (π β§ π) β€ π)) β βπ β π΄ (Β¬ π β€ π β§ Β¬ π β€ π β§ (π β¨ (π β§ π)) = π)) | ||
Theorem | lhpmcvr6N 39203* | Specialization of lhpmcvr2 39199. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ Β¬ π β€ π) β§ (π β π΅ β§ (π β§ π) β€ π)) β βπ β π΄ (Β¬ π β€ π β§ Β¬ π β€ π β§ π β€ π)) | ||
Theorem | lhpm0atN 39204 | 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.) |
β’ π΅ = (BaseβπΎ) & β’ β§ = (meetβπΎ) & β’ 0 = (0.βπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β 0 β§ (π β§ π) = 0 )) β π β π΄) | ||
Theorem | lhpmat 39205 | An element covered by the lattice unity, when conjoined with an atom not under it, equals the lattice zero. (Contributed by NM, 6-Jun-2012.) |
β’ β€ = (leβπΎ) & β’ β§ = (meetβπΎ) & β’ 0 = (0.βπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β (π β§ π) = 0 ) | ||
Theorem | lhpmatb 39206 | An element covered by the lattice unity, when conjoined with an atom, equals zero iff the atom is not under it. (Contributed by NM, 15-Jun-2013.) |
β’ β€ = (leβπΎ) & β’ β§ = (meetβπΎ) & β’ 0 = (0.βπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ π β π΄) β (Β¬ π β€ π β (π β§ π) = 0 )) | ||
Theorem | lhp2at0 39207 | Join and meet with different atoms under co-atom π. (Contributed by NM, 15-Jun-2013.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ 0 = (0.βπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ π β π) β§ (π β π΄ β§ π β€ π) β§ (π β π΄ β§ π β€ π)) β ((π β¨ π) β§ π) = 0 ) | ||
Theorem | lhp2atnle 39208 | Inequality for 2 different atoms under co-atom π. (Contributed by NM, 17-Jun-2013.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ π β π) β§ (π β π΄ β§ π β€ π) β§ (π β π΄ β§ π β€ π)) β Β¬ π β€ (π β¨ π)) | ||
Theorem | lhp2atne 39209 | Inequality for joins with 2 different atoms under co-atom π. (Contributed by NM, 22-Jul-2013.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ π β π΄) β§ ((π β π΄ β§ π β€ π) β§ (π β π΄ β§ π β€ π)) β§ π β π) β (π β¨ π) β (π β¨ π)) | ||
Theorem | lhp2at0nle 39210 | Inequality for 2 different atoms (or an atom and zero) under co-atom π. (Contributed by NM, 28-Jul-2013.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ 0 = (0.βπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ π β π) β§ ((π β π΄ β¨ π = 0 ) β§ π β€ π) β§ (π β π΄ β§ π β€ π)) β Β¬ π β€ (π β¨ π)) | ||
Theorem | lhp2at0ne 39211 | Inequality for joins with 2 different atoms (or an atom and zero) under co-atom π. (Contributed by NM, 28-Jul-2013.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ 0 = (0.βπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ π β π΄) β§ (((π β π΄ β¨ π = 0 ) β§ π β€ π) β§ (π β π΄ β§ π β€ π)) β§ π β π) β (π β¨ π) β (π β¨ π)) | ||
Theorem | lhpelim 39212 | Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 39205 to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ π β π΅) β ((π β¨ (π β§ π)) β§ π) = (π β§ π)) | ||
Theorem | lhpmod2i2 39213 | Modular law for hyperplanes analogous to atmod2i2 39037 for atoms. (Contributed by NM, 9-Feb-2013.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ π β€ π) β ((π β§ π) β¨ π) = (π β§ (π β¨ π))) | ||
Theorem | lhpmod6i1 39214 | Modular law for hyperplanes analogous to complement of atmod2i1 39036 for atoms. (Contributed by NM, 1-Jun-2013.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ π β€ π) β (π β¨ (π β§ π)) = ((π β¨ π) β§ π)) | ||
Theorem | lhprelat3N 39215* | The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 38587. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ < = (ltβπΎ) & β’ β§ = (meetβπΎ) & β’ πΆ = ( β βπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π΅ β§ π β π΅) β§ π < π) β βπ€ β π» (π β€ (π β§ π€) β§ (π β§ π€)πΆπ)) | ||
Theorem | cdlemb2 39216* | Given two atoms not under the fiducial (reference) co-atom π, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 30-May-2012.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ π β π) β βπ β π΄ (Β¬ π β€ π β§ Β¬ π β€ (π β¨ π))) | ||
Theorem | lhple 39217 | Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΅ β§ π β€ π)) β ((π β¨ π) β§ π) = π) | ||
Theorem | lhpat 39218 | Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 23-May-2012.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ π β π)) β ((π β¨ π) β§ π) β π΄) | ||
Theorem | lhpat4N 39219 | Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ π β€ π)) β ((π β¨ π) β§ π) = π) | ||
Theorem | lhpat2 39220 | Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 21-Nov-2012.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((π β¨ π) β§ π) β β’ (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ π β π)) β π β π΄) | ||
Theorem | lhpat3 39221 | There is only one atom under both π β¨ π and co-atom π. (Contributed by NM, 21-Nov-2012.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((π β¨ π) β§ π) β β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ π β π΄) β§ (π β π β§ π β€ (π β¨ π))) β (Β¬ π β€ π β π β π )) | ||
Theorem | 4atexlemk 39222 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) β β’ (π β πΎ β HL) | ||
Theorem | 4atexlemw 39223 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) β β’ (π β π β π») | ||
Theorem | 4atexlempw 39224 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) β β’ (π β (π β π΄ β§ Β¬ π β€ π)) | ||
Theorem | 4atexlemp 39225 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) β β’ (π β π β π΄) | ||
Theorem | 4atexlemq 39226 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) β β’ (π β π β π΄) | ||
Theorem | 4atexlems 39227 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) β β’ (π β π β π΄) | ||
Theorem | 4atexlemt 39228 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) β β’ (π β π β π΄) | ||
Theorem | 4atexlemutvt 39229 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) β β’ (π β (π β¨ π) = (π β¨ π)) | ||
Theorem | 4atexlempnq 39230 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) β β’ (π β π β π) | ||
Theorem | 4atexlemnslpq 39231 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) β β’ (π β Β¬ π β€ (π β¨ π)) | ||
Theorem | 4atexlemkl 39232 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) β β’ (π β πΎ β Lat) | ||
Theorem | 4atexlemkc 39233 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) β β’ (π β πΎ β CvLat) | ||
Theorem | 4atexlemwb 39234 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) & β’ π» = (LHypβπΎ) β β’ (π β π β (BaseβπΎ)) | ||
Theorem | 4atexlempsb 39235 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) β β’ (π β (π β¨ π) β (BaseβπΎ)) | ||
Theorem | 4atexlemqtb 39236 | Lemma for 4atexlem7 39250. (Contributed by NM, 24-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) β β’ (π β (π β¨ π) β (BaseβπΎ)) | ||
Theorem | 4atexlempns 39237 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) β β’ (π β π β π) | ||
Theorem | 4atexlemswapqr 39238 | Lemma for 4atexlem7 39250. Swap π and π , so that theorems involving πΆ can be reused for π·. Note that π must be expanded because it involves π. (Contributed by NM, 25-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π = ((π β¨ π) β§ π) β β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π) = (π β¨ π)) β§ (π β π΄ β§ (((π β¨ π ) β§ π) β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π )))) | ||
Theorem | 4atexlemu 39239 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((π β¨ π) β§ π) β β’ (π β π β π΄) | ||
Theorem | 4atexlemv 39240 | Lemma for 4atexlem7 39250. (Contributed by NM, 23-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((π β¨ π) β§ π) & β’ π = ((π β¨ π) β§ π) β β’ (π β π β π΄) | ||
Theorem | 4atexlemunv 39241 | Lemma for 4atexlem7 39250. (Contributed by NM, 21-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((π β¨ π) β§ π) & β’ π = ((π β¨ π) β§ π) β β’ (π β π β π) | ||
Theorem | 4atexlemtlw 39242 | Lemma for 4atexlem7 39250. (Contributed by NM, 24-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((π β¨ π) β§ π) & β’ π = ((π β¨ π) β§ π) β β’ (π β π β€ π) | ||
Theorem | 4atexlemntlpq 39243 | Lemma for 4atexlem7 39250. (Contributed by NM, 24-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((π β¨ π) β§ π) & β’ π = ((π β¨ π) β§ π) β β’ (π β Β¬ π β€ (π β¨ π)) | ||
Theorem | 4atexlemc 39244 | Lemma for 4atexlem7 39250. (Contributed by NM, 24-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((π β¨ π) β§ π) & β’ π = ((π β¨ π) β§ π) & β’ πΆ = ((π β¨ π) β§ (π β¨ π)) β β’ (π β πΆ β π΄) | ||
Theorem | 4atexlemnclw 39245 | Lemma for 4atexlem7 39250. (Contributed by NM, 24-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((π β¨ π) β§ π) & β’ π = ((π β¨ π) β§ π) & β’ πΆ = ((π β¨ π) β§ (π β¨ π)) β β’ (π β Β¬ πΆ β€ π) | ||
Theorem | 4atexlemex2 39246* | Lemma for 4atexlem7 39250. Show that when πΆ β π, πΆ satisfies the existence condition of the consequent. (Contributed by NM, 25-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((π β¨ π) β§ π) & β’ π = ((π β¨ π) β§ π) & β’ πΆ = ((π β¨ π) β§ (π β¨ π)) β β’ ((π β§ πΆ β π) β βπ§ β π΄ (Β¬ π§ β€ π β§ (π β¨ π§) = (π β¨ π§))) | ||
Theorem | 4atexlemcnd 39247 | Lemma for 4atexlem7 39250. (Contributed by NM, 24-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((π β¨ π) β§ π) & β’ π = ((π β¨ π) β§ π) & β’ πΆ = ((π β¨ π) β§ (π β¨ π)) & β’ π· = ((π β¨ π) β§ (π β¨ π)) β β’ (π β πΆ β π·) | ||
Theorem | 4atexlemex4 39248* | Lemma for 4atexlem7 39250. Show that when πΆ = π, π· satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012.) |
β’ (π β (((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π΄ β§ (π β π΄ β§ Β¬ π β€ π β§ (π β¨ π ) = (π β¨ π )) β§ (π β π΄ β§ (π β¨ π) = (π β¨ π))) β§ (π β π β§ Β¬ π β€ (π β¨ π)))) & β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((π β¨ π) β§ π) & β’ π = ((π β¨ π) β§ π) & β’ πΆ = ((π β¨ π) β§ (π β¨ π)) & β’ π· = ((π β¨ π) β§ (π β¨ π)) β β’ ((π β§ πΆ = π) β βπ§ β π΄ (Β¬ π§ β€ π β§ (π β¨ π§) = (π β¨ π§))) | ||
Theorem | 4atexlemex6 39249* | Lemma for 4atexlem7 39250. (Contributed by NM, 25-Nov-2012.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ ((((πΎ β HL β§ π β π») β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ ((π β π΄ β§ Β¬ π β€ π) β§ π β π΄) β§ ((π β¨ π ) = (π β¨ π ) β§ π β π β§ Β¬ π β€ (π β¨ π))) β βπ§ β π΄ (Β¬ π§ β€ π β§ (π β¨ π§) = (π β¨ π§))) | ||
Theorem | 4atexlem7 39250* | Whenever there are at least 4 atoms under π β¨ π (specifically, π, π, π, and (π β¨ π) β§ π), there are also at least 4 atoms under π β¨ π. This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p β¨ q/0 and hence p β¨ s/0 contains at least four atoms..." Note that by cvlsupr2 38517, our (π β¨ π) = (π β¨ π) is a shorter way to express π β π β§ π β π β§ π β€ (π β¨ π). With a longer proof, the condition Β¬ π β€ (π β¨ π) could be eliminated (see 4atex 39251), although for some purposes this more restricted lemma may be adequate. (Contributed by NM, 25-Nov-2012.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π) β§ π β π΄) β§ (π β π β§ Β¬ π β€ (π β¨ π) β§ βπ β π΄ (Β¬ π β€ π β§ (π β¨ π) = (π β¨ π)))) β βπ§ β π΄ (Β¬ π§ β€ π β§ (π β¨ π§) = (π β¨ π§))) | ||
Theorem | 4atex 39251* | Whenever there are at least 4 atoms under π β¨ π (specifically, π, π, π, and (π β¨ π) β§ π), there are also at least 4 atoms under π β¨ π. This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p β¨ q/0 and hence p β¨ s/0 contains at least four atoms..." Note that by cvlsupr2 38517, our (π β¨ π) = (π β¨ π) is a shorter way to express π β π β§ π β π β§ π β€ (π β¨ π). (Contributed by NM, 27-May-2013.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π) β§ π β π΄) β§ (π β π β§ βπ β π΄ (Β¬ π β€ π β§ (π β¨ π) = (π β¨ π)))) β βπ§ β π΄ (Β¬ π§ β€ π β§ (π β¨ π§) = (π β¨ π§))) | ||
Theorem | 4atex2 39252* | More general version of 4atex 39251 for a line π β¨ π not necessarily connected to π β¨ π. (Contributed by NM, 27-May-2013.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π β§ π β π΄ β§ βπ β π΄ (Β¬ π β€ π β§ (π β¨ π) = (π β¨ π)))) β βπ§ β π΄ (Β¬ π§ β€ π β§ (π β¨ π§) = (π β¨ π§))) | ||
Theorem | 4atex2-0aOLDN 39253* | Same as 4atex2 39252 except that π is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π) β§ π = (0.βπΎ)) β§ (π β π β§ (π β π΄ β§ Β¬ π β€ π) β§ βπ β π΄ (Β¬ π β€ π β§ (π β¨ π) = (π β¨ π)))) β βπ§ β π΄ (Β¬ π§ β€ π β§ (π β¨ π§) = (π β¨ π§))) | ||
Theorem | 4atex2-0bOLDN 39254* | Same as 4atex2 39252 except that π is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π β§ π = (0.βπΎ) β§ βπ β π΄ (Β¬ π β€ π β§ (π β¨ π) = (π β¨ π)))) β βπ§ β π΄ (Β¬ π§ β€ π β§ (π β¨ π§) = (π β¨ π§))) | ||
Theorem | 4atex2-0cOLDN 39255* | Same as 4atex2 39252 except that π and π are zero. TODO: do we need this one or 4atex2-0aOLDN 39253 or 4atex2-0bOLDN 39254? (Contributed by NM, 27-May-2013.) (New usage is discouraged.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π) β§ π = (0.βπΎ)) β§ (π β π β§ π = (0.βπΎ) β§ βπ β π΄ (Β¬ π β€ π β§ (π β¨ π) = (π β¨ π)))) β βπ§ β π΄ (Β¬ π§ β€ π β§ (π β¨ π§) = (π β¨ π§))) | ||
Theorem | 4atex3 39256* | More general version of 4atex 39251 for a line π β¨ π not necessarily connected to π β¨ π. (Contributed by NM, 29-May-2013.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (((πΎ β HL β§ π β π») β§ ((π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β§ (π β π β§ (π β π΄ β§ π β π) β§ βπ β π΄ (Β¬ π β€ π β§ (π β¨ π) = (π β¨ π)))) β βπ§ β π΄ (Β¬ π§ β€ π β§ (π§ β π β§ π§ β π β§ π§ β€ (π β¨ π)))) | ||
Theorem | lautset 39257* | The set of lattice automorphisms. (Contributed by NM, 11-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ πΌ = (LAutβπΎ) β β’ (πΎ β π΄ β πΌ = {π β£ (π:π΅β1-1-ontoβπ΅ β§ βπ₯ β π΅ βπ¦ β π΅ (π₯ β€ π¦ β (πβπ₯) β€ (πβπ¦)))}) | ||
Theorem | islaut 39258* | The predicate "is a lattice automorphism". (Contributed by NM, 11-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ πΌ = (LAutβπΎ) β β’ (πΎ β π΄ β (πΉ β πΌ β (πΉ:π΅β1-1-ontoβπ΅ β§ βπ₯ β π΅ βπ¦ β π΅ (π₯ β€ π¦ β (πΉβπ₯) β€ (πΉβπ¦))))) | ||
Theorem | lautle 39259 | Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ πΌ = (LAutβπΎ) β β’ (((πΎ β π β§ πΉ β πΌ) β§ (π β π΅ β§ π β π΅)) β (π β€ π β (πΉβπ) β€ (πΉβπ))) | ||
Theorem | laut1o 39260 | A lattice automorphism is one-to-one and onto. (Contributed by NM, 19-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ πΌ = (LAutβπΎ) β β’ ((πΎ β π΄ β§ πΉ β πΌ) β πΉ:π΅β1-1-ontoβπ΅) | ||
Theorem | laut11 39261 | One-to-one property of a lattice automorphism. (Contributed by NM, 20-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ πΌ = (LAutβπΎ) β β’ (((πΎ β π β§ πΉ β πΌ) β§ (π β π΅ β§ π β π΅)) β ((πΉβπ) = (πΉβπ) β π = π)) | ||
Theorem | lautcl 39262 | A lattice automorphism value belongs to the base set. (Contributed by NM, 20-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ πΌ = (LAutβπΎ) β β’ (((πΎ β π β§ πΉ β πΌ) β§ π β π΅) β (πΉβπ) β π΅) | ||
Theorem | lautcnvclN 39263 | Reverse closure of a lattice automorphism. (Contributed by NM, 25-May-2012.) (New usage is discouraged.) |
β’ π΅ = (BaseβπΎ) & β’ πΌ = (LAutβπΎ) β β’ (((πΎ β π β§ πΉ β πΌ) β§ π β π΅) β (β‘πΉβπ) β π΅) | ||
Theorem | lautcnvle 39264 | Less-than or equal property of lattice automorphism converse. (Contributed by NM, 19-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ πΌ = (LAutβπΎ) β β’ (((πΎ β π β§ πΉ β πΌ) β§ (π β π΅ β§ π β π΅)) β (π β€ π β (β‘πΉβπ) β€ (β‘πΉβπ))) | ||
Theorem | lautcnv 39265 | The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.) |
β’ πΌ = (LAutβπΎ) β β’ ((πΎ β π β§ πΉ β πΌ) β β‘πΉ β πΌ) | ||
Theorem | lautlt 39266 | Less-than property of a lattice automorphism. (Contributed by NM, 20-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ < = (ltβπΎ) & β’ πΌ = (LAutβπΎ) β β’ ((πΎ β π΄ β§ (πΉ β πΌ β§ π β π΅ β§ π β π΅)) β (π < π β (πΉβπ) < (πΉβπ))) | ||
Theorem | lautcvr 39267 | Covering property of a lattice automorphism. (Contributed by NM, 20-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ πΆ = ( β βπΎ) & β’ πΌ = (LAutβπΎ) β β’ ((πΎ β π΄ β§ (πΉ β πΌ β§ π β π΅ β§ π β π΅)) β (ππΆπ β (πΉβπ)πΆ(πΉβπ))) | ||
Theorem | lautj 39268 | Meet property of a lattice automorphism. (Contributed by NM, 25-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ β¨ = (joinβπΎ) & β’ πΌ = (LAutβπΎ) β β’ ((πΎ β Lat β§ (πΉ β πΌ β§ π β π΅ β§ π β π΅)) β (πΉβ(π β¨ π)) = ((πΉβπ) β¨ (πΉβπ))) | ||
Theorem | lautm 39269 | Meet property of a lattice automorphism. (Contributed by NM, 19-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ β§ = (meetβπΎ) & β’ πΌ = (LAutβπΎ) β β’ ((πΎ β Lat β§ (πΉ β πΌ β§ π β π΅ β§ π β π΅)) β (πΉβ(π β§ π)) = ((πΉβπ) β§ (πΉβπ))) | ||
Theorem | lauteq 39270* | A lattice automorphism argument is equal to its value if all atoms are equal to their values. (Contributed by NM, 24-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ πΌ = (LAutβπΎ) β β’ (((πΎ β HL β§ πΉ β πΌ β§ π β π΅) β§ βπ β π΄ (πΉβπ) = π) β (πΉβπ) = π) | ||
Theorem | idlaut 39271 | The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ πΌ = (LAutβπΎ) β β’ (πΎ β π΄ β ( I βΎ π΅) β πΌ) | ||
Theorem | lautco 39272 | The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.) |
β’ πΌ = (LAutβπΎ) β β’ ((πΎ β π β§ πΉ β πΌ β§ πΊ β πΌ) β (πΉ β πΊ) β πΌ) | ||
Theorem | pautsetN 39273* | The set of projective automorphisms. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.) |
β’ π = (PSubSpβπΎ) & β’ π = (PAutβπΎ) β β’ (πΎ β π΅ β π = {π β£ (π:πβ1-1-ontoβπ β§ βπ₯ β π βπ¦ β π (π₯ β π¦ β (πβπ₯) β (πβπ¦)))}) | ||
Theorem | ispautN 39274* | The predicate "is a projective automorphism". (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.) |
β’ π = (PSubSpβπΎ) & β’ π = (PAutβπΎ) β β’ (πΎ β π΅ β (πΉ β π β (πΉ:πβ1-1-ontoβπ β§ βπ₯ β π βπ¦ β π (π₯ β π¦ β (πΉβπ₯) β (πΉβπ¦))))) | ||
Syntax | cldil 39275 | Extend class notation with set of all lattice dilations. |
class LDil | ||
Syntax | cltrn 39276 | Extend class notation with set of all lattice translations. |
class LTrn | ||
Syntax | cdilN 39277 | Extend class notation with set of all dilations. |
class Dil | ||
Syntax | ctrnN 39278 | Extend class notation with set of all translations. |
class Trn | ||
Definition | df-ldil 39279* | Define set of all lattice dilations. Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.) |
β’ LDil = (π β V β¦ (π€ β (LHypβπ) β¦ {π β (LAutβπ) β£ βπ₯ β (Baseβπ)(π₯(leβπ)π€ β (πβπ₯) = π₯)})) | ||
Definition | df-ltrn 39280* | Define set of all lattice translations. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.) |
β’ LTrn = (π β V β¦ (π€ β (LHypβπ) β¦ {π β ((LDilβπ)βπ€) β£ βπ β (Atomsβπ)βπ β (Atomsβπ)((Β¬ π(leβπ)π€ β§ Β¬ π(leβπ)π€) β ((π(joinβπ)(πβπ))(meetβπ)π€) = ((π(joinβπ)(πβπ))(meetβπ)π€))})) | ||
Definition | df-dilN 39281* | Define set of all dilations. Definition of dilation in [Crawley] p. 111. (Contributed by NM, 30-Jan-2012.) |
β’ Dil = (π β V β¦ (π β (Atomsβπ) β¦ {π β (PAutβπ) β£ βπ₯ β (PSubSpβπ)(π₯ β ((WAtomsβπ)βπ) β (πβπ₯) = π₯)})) | ||
Definition | df-trnN 39282* | Define set of all translations. Definition of translation in [Crawley] p. 111. (Contributed by NM, 4-Feb-2012.) |
β’ Trn = (π β V β¦ (π β (Atomsβπ) β¦ {π β ((Dilβπ)βπ) β£ βπ β ((WAtomsβπ)βπ)βπ β ((WAtomsβπ)βπ)((π(+πβπ)(πβπ)) β© ((β₯πβπ)β{π})) = ((π(+πβπ)(πβπ)) β© ((β₯πβπ)β{π}))})) | ||
Theorem | ldilfset 39283* | The mapping from fiducial co-atom π€ to its set of lattice dilations. (Contributed by NM, 11-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ π» = (LHypβπΎ) & β’ πΌ = (LAutβπΎ) β β’ (πΎ β πΆ β (LDilβπΎ) = (π€ β π» β¦ {π β πΌ β£ βπ₯ β π΅ (π₯ β€ π€ β (πβπ₯) = π₯)})) | ||
Theorem | ldilset 39284* | The set of lattice dilations for a fiducial co-atom π. (Contributed by NM, 11-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ π» = (LHypβπΎ) & β’ πΌ = (LAutβπΎ) & β’ π· = ((LDilβπΎ)βπ) β β’ ((πΎ β πΆ β§ π β π») β π· = {π β πΌ β£ βπ₯ β π΅ (π₯ β€ π β (πβπ₯) = π₯)}) | ||
Theorem | isldil 39285* | The predicate "is a lattice dilation". Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ π» = (LHypβπΎ) & β’ πΌ = (LAutβπΎ) & β’ π· = ((LDilβπΎ)βπ) β β’ ((πΎ β πΆ β§ π β π») β (πΉ β π· β (πΉ β πΌ β§ βπ₯ β π΅ (π₯ β€ π β (πΉβπ₯) = π₯)))) | ||
Theorem | ldillaut 39286 | A lattice dilation is an automorphism. (Contributed by NM, 20-May-2012.) |
β’ π» = (LHypβπΎ) & β’ πΌ = (LAutβπΎ) & β’ π· = ((LDilβπΎ)βπ) β β’ (((πΎ β π β§ π β π») β§ πΉ β π·) β πΉ β πΌ) | ||
Theorem | ldil1o 39287 | A lattice dilation is a one-to-one onto function. (Contributed by NM, 19-Apr-2013.) |
β’ π΅ = (BaseβπΎ) & β’ π» = (LHypβπΎ) & β’ π· = ((LDilβπΎ)βπ) β β’ (((πΎ β π β§ π β π») β§ πΉ β π·) β πΉ:π΅β1-1-ontoβπ΅) | ||
Theorem | ldilval 39288 | Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ β€ = (leβπΎ) & β’ π» = (LHypβπΎ) & β’ π· = ((LDilβπΎ)βπ) β β’ (((πΎ β π β§ π β π») β§ πΉ β π· β§ (π β π΅ β§ π β€ π)) β (πΉβπ) = π) | ||
Theorem | idldil 39289 | The identity function is a lattice dilation. (Contributed by NM, 18-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ π» = (LHypβπΎ) & β’ π· = ((LDilβπΎ)βπ) β β’ ((πΎ β π΄ β§ π β π») β ( I βΎ π΅) β π·) | ||
Theorem | ldilcnv 39290 | The converse of a lattice dilation is a lattice dilation. (Contributed by NM, 10-May-2013.) |
β’ π» = (LHypβπΎ) & β’ π· = ((LDilβπΎ)βπ) β β’ (((πΎ β HL β§ π β π») β§ πΉ β π·) β β‘πΉ β π·) | ||
Theorem | ldilco 39291 | The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.) |
β’ π» = (LHypβπΎ) & β’ π· = ((LDilβπΎ)βπ) β β’ (((πΎ β π β§ π β π») β§ πΉ β π· β§ πΊ β π·) β (πΉ β πΊ) β π·) | ||
Theorem | ltrnfset 39292* | The set of all lattice translations for a lattice πΎ. (Contributed by NM, 11-May-2012.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) β β’ (πΎ β πΆ β (LTrnβπΎ) = (π€ β π» β¦ {π β ((LDilβπΎ)βπ€) β£ βπ β π΄ βπ β π΄ ((Β¬ π β€ π€ β§ Β¬ π β€ π€) β ((π β¨ (πβπ)) β§ π€) = ((π β¨ (πβπ)) β§ π€))})) | ||
Theorem | ltrnset 39293* | The set of lattice translations for a fiducial co-atom π. (Contributed by NM, 11-May-2012.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π· = ((LDilβπΎ)βπ) & β’ π = ((LTrnβπΎ)βπ) β β’ ((πΎ β π΅ β§ π β π») β π = {π β π· β£ βπ β π΄ βπ β π΄ ((Β¬ π β€ π β§ Β¬ π β€ π) β ((π β¨ (πβπ)) β§ π) = ((π β¨ (πβπ)) β§ π))}) | ||
Theorem | isltrn 39294* | The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π· = ((LDilβπΎ)βπ) & β’ π = ((LTrnβπΎ)βπ) β β’ ((πΎ β π΅ β§ π β π») β (πΉ β π β (πΉ β π· β§ βπ β π΄ βπ β π΄ ((Β¬ π β€ π β§ Β¬ π β€ π) β ((π β¨ (πΉβπ)) β§ π) = ((π β¨ (πΉβπ)) β§ π))))) | ||
Theorem | isltrn2N 39295* | The predicate "is a lattice translation". Version of isltrn 39294 that considers only different π and π. TODO: Can this eliminate some separate proofs for the π = π case? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π· = ((LDilβπΎ)βπ) & β’ π = ((LTrnβπΎ)βπ) β β’ ((πΎ β π΅ β§ π β π») β (πΉ β π β (πΉ β π· β§ βπ β π΄ βπ β π΄ ((Β¬ π β€ π β§ Β¬ π β€ π β§ π β π) β ((π β¨ (πΉβπ)) β§ π) = ((π β¨ (πΉβπ)) β§ π))))) | ||
Theorem | ltrnu 39296 | Uniqueness property of a lattice translation value for atoms not under the fiducial co-atom π. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 20-May-2012.) |
β’ β€ = (leβπΎ) & β’ β¨ = (joinβπΎ) & β’ β§ = (meetβπΎ) & β’ π΄ = (AtomsβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) β β’ ((((πΎ β π β§ π β π») β§ πΉ β π) β§ (π β π΄ β§ Β¬ π β€ π) β§ (π β π΄ β§ Β¬ π β€ π)) β ((π β¨ (πΉβπ)) β§ π) = ((π β¨ (πΉβπ)) β§ π)) | ||
Theorem | ltrnldil 39297 | A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.) |
β’ π» = (LHypβπΎ) & β’ π· = ((LDilβπΎ)βπ) & β’ π = ((LTrnβπΎ)βπ) β β’ (((πΎ β π β§ π β π») β§ πΉ β π) β πΉ β π·) | ||
Theorem | ltrnlaut 39298 | A lattice translation is a lattice automorphism. (Contributed by NM, 20-May-2012.) |
β’ π» = (LHypβπΎ) & β’ πΌ = (LAutβπΎ) & β’ π = ((LTrnβπΎ)βπ) β β’ (((πΎ β π β§ π β π») β§ πΉ β π) β πΉ β πΌ) | ||
Theorem | ltrn1o 39299 | A lattice translation is a one-to-one onto function. (Contributed by NM, 20-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) β β’ (((πΎ β π β§ π β π») β§ πΉ β π) β πΉ:π΅β1-1-ontoβπ΅) | ||
Theorem | ltrncl 39300 | Closure of a lattice translation. (Contributed by NM, 20-May-2012.) |
β’ π΅ = (BaseβπΎ) & β’ π» = (LHypβπΎ) & β’ π = ((LTrnβπΎ)βπ) β β’ (((πΎ β π β§ π β π») β§ πΉ β π β§ π β π΅) β (πΉβπ) β π΅) |
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