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Theorem List for Metamath Proof Explorer - 33401-33500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremopltcon2b 33401 Contraposition law for strict ordering in orthoposets. (chsscon2 27533 analog.) (Contributed by NM, 5-Nov-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < ( 𝑌) ↔ 𝑌 < ( 𝑋)))
 
Theoremopexmid 33402 Law of excluded middle for orthoposets. (chjo 27546 analog.) (Contributed by NM, 13-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &    = (join‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → (𝑋 ( 𝑋)) = 1 )
 
Theoremopnoncon 33403 Law of contradiction for orthoposets. (chocin 27526 analog.) (Contributed by NM, 13-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → (𝑋 ( 𝑋)) = 0 )
 
TheoremriotaocN 33404* The orthocomplement of the unique poset element such that 𝜓. (riotaneg 10753 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   (𝑥 = ( 𝑦) → (𝜑𝜓))       ((𝐾 ∈ OP ∧ ∃!𝑥𝐵 𝜑) → (𝑥𝐵 𝜑) = ( ‘(𝑦𝐵 𝜓)))
 
TheoremcmtfvalN 33405* Value of commutes relation. (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐵𝑥 = ((𝑥 𝑦) (𝑥 ( 𝑦))))})
 
TheoremcmtvalN 33406 Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 27615 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
 
Theoremisolat 33407 The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)
(𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))
 
Theoremollat 33408 An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)
(𝐾 ∈ OL → 𝐾 ∈ Lat)
 
Theoremolop 33409 An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)
(𝐾 ∈ OL → 𝐾 ∈ OP)
 
TheoremolposN 33410 An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
(𝐾 ∈ OL → 𝐾 ∈ Poset)
 
TheoremisolatiN 33411 Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
𝐾 ∈ Lat    &   𝐾 ∈ OP       𝐾 ∈ OL
 
Theoremoldmm1 33412 De Morgan's law for meet in an ortholattice. (chdmm1 27556 analog.) (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌)) = (( 𝑋) ( 𝑌)))
 
Theoremoldmm2 33413 De Morgan's law for meet in an ortholattice. (chdmm2 27557 analog.) (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) 𝑌)) = (𝑋 ( 𝑌)))
 
Theoremoldmm3N 33414 De Morgan's law for meet in an ortholattice. (chdmm3 27558 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) 𝑌))
 
Theoremoldmm4 33415 De Morgan's law for meet in an ortholattice. (chdmm4 27559 analog.) (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) ( 𝑌))) = (𝑋 𝑌))
 
Theoremoldmj1 33416 De Morgan's law for join in an ortholattice. (chdmj1 27560 analog.) (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌)) = (( 𝑋) ( 𝑌)))
 
Theoremoldmj2 33417 De Morgan's law for join in an ortholattice. (chdmj2 27561 analog.) (Contributed by NM, 7-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) 𝑌)) = (𝑋 ( 𝑌)))
 
Theoremoldmj3 33418 De Morgan's law for join in an ortholattice. (chdmj3 27562 analog.) (Contributed by NM, 7-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) 𝑌))
 
Theoremoldmj4 33419 De Morgan's law for join in an ortholattice. (chdmj4 27563 analog.) (Contributed by NM, 7-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) ( 𝑌))) = (𝑋 𝑌))
 
Theoremolj01 33420 An ortholattice element joined with zero equals itself. (chj0 27528 analog.) (Contributed by NM, 19-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋 0 ) = 𝑋)
 
Theoremolj02 33421 An ortholattice element joined with zero equals itself. (Contributed by NM, 28-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵) → ( 0 𝑋) = 𝑋)
 
Theoremolm11 33422 The meet of an ortholattice element with one equals itself. (chm1i 27487 analog.) (Contributed by NM, 22-May-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋 1 ) = 𝑋)
 
Theoremolm12 33423 The meet of an ortholattice element with one equals itself. (Contributed by NM, 22-May-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵) → ( 1 𝑋) = 𝑋)
 
TheoremlatmassOLD 33424 Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3688 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
 
Theoremlatm12 33425 A rearrangement of lattice meet. (in12 3689 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = (𝑌 (𝑋 𝑍)))
 
Theoremlatm32 33426 A rearrangement of lattice meet. (in12 3689 analog.) (Contributed by NM, 13-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) 𝑌))
 
Theoremlatmrot 33427 Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑍 𝑋) 𝑌))
 
Theoremlatm4 33428 Rearrangement of lattice meet of 4 classes. (in4 3694 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 𝑌) (𝑍 𝑊)) = ((𝑋 𝑍) (𝑌 𝑊)))
 
TheoremlatmmdiN 33429 Lattice meet distributes over itself. (inindi 3695 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))
 
Theoremlatmmdir 33430 Lattice meet distributes over itself. (inindir 3696 analog.) (Contributed by NM, 6-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) (𝑌 𝑍)))
 
Theoremolm01 33431 Meet with lattice zero is zero. (chm0 27522 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋 0 ) = 0 )
 
Theoremolm02 33432 Meet with lattice zero is zero. (Contributed by NM, 9-Oct-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵) → ( 0 𝑋) = 0 )
 
Theoremisoml 33433* The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
 
TheoremisomliN 33434* Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
𝐾 ∈ OL    &   𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   ((𝑥𝐵𝑦𝐵) → (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥)))))       𝐾 ∈ OML
 
Theoremomlol 33435 An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)
(𝐾 ∈ OML → 𝐾 ∈ OL)
 
Theoremomlop 33436 An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)
(𝐾 ∈ OML → 𝐾 ∈ OP)
 
Theoremomllat 33437 An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)
(𝐾 ∈ OML → 𝐾 ∈ Lat)
 
Theoremomllaw 33438 The orthomodular law. (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))
 
Theoremomllaw2N 33439 Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 27616 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 (( 𝑋) 𝑌)) = 𝑌))
 
Theoremomllaw3 33440 Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 27467 analog.) (Contributed by NM, 19-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌 ∧ (𝑌 ( 𝑋)) = 0 ) → 𝑋 = 𝑌))
 
Theoremomllaw4 33441 Orthomodular law equivalent. Remark in [Holland95] p. 223. (Contributed by NM, 19-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (( ‘(( 𝑋) 𝑌)) 𝑌) = 𝑋))
 
Theoremomllaw5N 33442 The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 27644 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (( 𝑋) (𝑋 𝑌))) = (𝑋 𝑌))
 
TheoremcmtcomlemN 33443 Lemma for cmtcomN 33444. (cmcmlem 27622 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌𝐶𝑋))
 
TheoremcmtcomN 33444 Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 27623 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌𝐶𝑋))
 
Theoremcmt2N 33445 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 27624 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋𝐶( 𝑌)))
 
Theoremcmt3N 33446 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 27626 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ( 𝑋)𝐶𝑌))
 
Theoremcmt4N 33447 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 27626 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ( 𝑋)𝐶( 𝑌)))
 
Theoremcmtbr2N 33448 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 27627 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋 𝑌) (𝑋 ( 𝑌)))))
 
Theoremcmtbr3N 33449 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 27639 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) = (𝑋 𝑌)))
 
Theoremcmtbr4N 33450 Alternate definition for the commutes relation. (cmbr4i 27632 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 (( 𝑋) 𝑌)) 𝑌))
 
TheoremlecmtN 33451 Ordered elements commute. (lecmi 27633 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑋𝐶𝑌))
 
TheoremcmtidN 33452 Any element commutes with itself. (cmidi 27641 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵) → 𝑋𝐶𝑋)
 
Theoremomlfh1N 33453 Foulis-Holland Theorem, part 1. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Part of Theorem 5 in [Kalmbach] p. 25. (fh1 27649 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))
 
Theoremomlfh3N 33454 Foulis-Holland Theorem, part 3. Dual of omlfh1N 33453. (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))
 
Theoremomlmod1i2N 33455 Analogue of modular law atmod1i2 34053 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑍𝑌𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) 𝑍))
 
TheoremomlspjN 33456 Contraction of a Sasaki projection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → ((𝑋 ( 𝑌)) 𝑌) = 𝑋)
 
20.22.10  Atomic lattices with covering property
 
Syntaxccvr 33457 Extend class notation with covers relation.
class
 
Syntaxcatm 33458 Extend class notation with atoms.
class Atoms
 
Syntaxcal 33459 Extend class notation with atomic lattices.
class AtLat
 
Syntaxclc 33460 Extend class notation with lattices with the covering property.
class CvLat
 
Definitiondf-covers 33461* Define the covers relation ("is covered by") for posets. "𝑎 is covered by 𝑏 " means that 𝑎 is strictly less than 𝑏 and there is nothing in between. See cvrval 33464 for the relation form. (Contributed by NM, 18-Sep-2011.)
⋖ = (𝑝 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (Base‘𝑝) ∧ 𝑏 ∈ (Base‘𝑝)) ∧ 𝑎(lt‘𝑝)𝑏 ∧ ¬ ∃𝑧 ∈ (Base‘𝑝)(𝑎(lt‘𝑝)𝑧𝑧(lt‘𝑝)𝑏))})
 
Definitiondf-ats 33462* Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.)
Atoms = (𝑝 ∈ V ↦ {𝑎 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑎})
 
Theoremcvrfval 33463* Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       (𝐾𝐴𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧𝐵 (𝑥 < 𝑧𝑧 < 𝑦))})
 
Theoremcvrval 33464* Binary relation expressing 𝐵 covers 𝐴, which means that 𝐵 is larger than 𝐴 and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (cvbr 28313 analog.) (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))
 
Theoremcvrlt 33465 The covers relation implies the less-than relation. (cvpss 28316 analog.) (Contributed by NM, 8-Oct-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       (((𝐾𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
 
Theoremcvrnbtwn 33466 There is no element between the two arguments of the covers relation. (cvnbtwn 28317 analog.) (Contributed by NM, 18-Oct-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))
 
Theoremncvr1 33467 No element covers the lattice unit. (Contributed by NM, 8-Jul-2013.)
𝐵 = (Base‘𝐾)    &    1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → ¬ 1 𝐶𝑋)
 
TheoremcvrletrN 33468 Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑌 𝑍) → 𝑋 < 𝑍))
 
Theoremcvrval2 33469* Binary relation expressing 𝑌 covers 𝑋. Definition of covers in [Kalmbach] p. 15. (cvbr2 28314 analog.) (Contributed by NM, 16-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ∀𝑧𝐵 ((𝑋 < 𝑧𝑧 𝑌) → 𝑧 = 𝑌))))
 
Theoremcvrnbtwn2 33470 The covers relation implies no in-betweenness. (cvnbtwn2 28318 analog.) (Contributed by NM, 17-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍𝑍 𝑌) ↔ 𝑍 = 𝑌))
 
Theoremcvrnbtwn3 33471 The covers relation implies no in-betweenness. (cvnbtwn3 28319 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 < 𝑌) ↔ 𝑋 = 𝑍))
 
Theoremcvrcon3b 33472 Contraposition law for the covers relation. (cvcon3 28315 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ( 𝑌)𝐶( 𝑋)))
 
Theoremcvrle 33473 The covers relation implies the less-than-or-equal relation. (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       (((𝐾𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 𝑌)
 
Theoremcvrnbtwn4 33474 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 28320 analog.) (Contributed by NM, 18-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 𝑍𝑍 𝑌) ↔ (𝑋 = 𝑍𝑍 = 𝑌)))
 
Theoremcvrnle 33475 The covers relation implies the negation of the reverse less-than-or-equal relation. (Contributed by NM, 18-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → ¬ 𝑌 𝑋)
 
Theoremcvrne 33476 The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       (((𝐾𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝑌)
 
TheoremcvrnrefN 33477 The covers relation is not reflexive. (cvnref 28322 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴𝑋𝐵) → ¬ 𝑋𝐶𝑋)
 
Theoremcvrcmp 33478 If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ Poset ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑍𝐶𝑋𝑍𝐶𝑌)) → (𝑋 𝑌𝑋 = 𝑌))
 
Theoremcvrcmp2 33479 If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ OP ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑍𝑌𝐶𝑍)) → (𝑋 𝑌𝑋 = 𝑌))
 
Theorempats 33480* The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾𝐷𝐴 = {𝑥𝐵0 𝐶𝑥})
 
Theoremisat 33481 The predicate "is an atom". (ela 28370 analog.) (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾𝐷 → (𝑃𝐴 ↔ (𝑃𝐵0 𝐶𝑃)))
 
Theoremisat2 33482 The predicate "is an atom". (elatcv0 28372 analog.) (Contributed by NM, 18-Jun-2012.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾𝐷𝑃𝐵) → (𝑃𝐴0 𝐶𝑃))
 
Theorematcvr0 33483 An atom covers zero. (atcv0 28373 analog.) (Contributed by NM, 4-Nov-2011.)
0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾𝐷𝑃𝐴) → 0 𝐶𝑃)
 
Theorematbase 33484 An atom is a member of the lattice base set (i.e. a lattice element). (atelch 28375 analog.) (Contributed by NM, 10-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝑃𝐴𝑃𝐵)
 
Theorematssbase 33485 The set of atoms is a subset of the base set. (atssch 28374 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝐴 = (Atoms‘𝐾)       𝐴𝐵
 
Theorem0ltat 33486 An atom is greater than zero. (Contributed by NM, 4-Jul-2012.)
0 = (0.‘𝐾)    &    < = (lt‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ OP ∧ 𝑃𝐴) → 0 < 𝑃)
 
Theoremleatb 33487 A poset element less than or equal to an atom equals either zero or the atom. (atss 28377 analog.) (Contributed by NM, 17-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴) → (𝑋 𝑃 ↔ (𝑋 = 𝑃𝑋 = 0 )))
 
Theoremleat 33488 A poset element less than or equal to an atom equals either zero or the atom. (Contributed by NM, 15-Oct-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴) ∧ 𝑋 𝑃) → (𝑋 = 𝑃𝑋 = 0 ))
 
Theoremleat2 33489 A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴) ∧ (𝑋0𝑋 𝑃)) → 𝑋 = 𝑃)
 
Theoremleat3 33490 A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ OP ∧ 𝑋𝐵𝑃𝐴) ∧ 𝑋 𝑃) → (𝑋𝐴𝑋 = 0 ))
 
Theoremmeetat 33491 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-Aug-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑃𝐴) → ((𝑋 𝑃) = 𝑃 ∨ (𝑋 𝑃) = 0 ))
 
Theoremmeetat2 33492 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 30-Aug-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑃𝐴) → ((𝑋 𝑃) ∈ 𝐴 ∨ (𝑋 𝑃) = 0 ))
 
Definitiondf-atl 33493* Define the class of atomic lattices, in which every nonzero element is greater than or equal to an atom. We also ensure the existence of a lattice zero, since a lattice by itself may not have a zero. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.)
AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑝 ∈ (Atoms‘𝑘)𝑝(le‘𝑘)𝑥))}
 
Theoremisatl 33494* The predicate "is an atomic lattice." Every nonzero element is less than or equal to an atom. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
 
Theorematllat 33495 An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)
(𝐾 ∈ AtLat → 𝐾 ∈ Lat)
 
Theorematlpos 33496 An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.)
(𝐾 ∈ AtLat → 𝐾 ∈ Poset)
 
Theorematl0dm 33497 Condition necessary for zero element to exist. (Contributed by NM, 14-Sep-2018.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)       (𝐾 ∈ AtLat → 𝐵 ∈ dom 𝐺)
 
Theorematl0cl 33498 An atomic lattice has a zero element. We can use this in place of op0cl 33379 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)       (𝐾 ∈ AtLat → 0𝐵)
 
Theorematl0le 33499 Orthoposet zero is less than or equal to any element. (ch0le 27472 analog.) (Contributed by NM, 12-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐵) → 0 𝑋)
 
Theorematlle0 33500 An element less than or equal to zero equals zero. (chle0 27474 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐵) → (𝑋 0𝑋 = 0 ))
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
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