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Theorem List for Metamath Proof Explorer - 33501-33600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremopcon3b 33501 Contraposition law for orthoposets. (chcon3i 27709 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ( 𝑌) = ( 𝑋)))

Theoremopcon2b 33502 Orthocomplement contraposition law. (negcon2 10213 analog.) (Contributed by NM, 16-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ( 𝑌) ↔ 𝑌 = ( 𝑋)))

Theoremopcon1b 33503 Orthocomplement contraposition law. (negcon1 10212 analog.) (Contributed by NM, 24-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) = 𝑌 ↔ ( 𝑌) = 𝑋))

Theoremoplecon3 33504 Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → ( 𝑌) ( 𝑋)))

Theoremoplecon3b 33505 Contraposition law for orthoposets. (chsscon3 27743 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ ( 𝑌) ( 𝑋)))

Theoremoplecon1b 33506 Contraposition law for strict ordering in orthoposets. (chsscon1 27744 analog.) (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) 𝑌 ↔ ( 𝑌) 𝑋))

Theoremopoc1 33507 Orthocomplement of orthoposet unit. (Contributed by NM, 24-Jan-2012.)
0 = (0.‘𝐾)    &    1 = (1.‘𝐾)    &    = (oc‘𝐾)       (𝐾 ∈ OP → ( 1 ) = 0 )

Theoremopoc0 33508 Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)
0 = (0.‘𝐾)    &    1 = (1.‘𝐾)    &    = (oc‘𝐾)       (𝐾 ∈ OP → ( 0 ) = 1 )

Theoremopltcon3b 33509 Contraposition law for strict ordering in orthoposets. (chpsscon3 27746 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ( 𝑌) < ( 𝑋)))

Theoremopltcon1b 33510 Contraposition law for strict ordering in orthoposets. (chpsscon1 27747 analog.) (Contributed by NM, 5-Nov-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (( 𝑋) < 𝑌 ↔ ( 𝑌) < 𝑋))

Theoremopltcon2b 33511 Contraposition law for strict ordering in orthoposets. (chsscon2 27745 analog.) (Contributed by NM, 5-Nov-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < ( 𝑌) ↔ 𝑌 < ( 𝑋)))

Theoremopexmid 33512 Law of excluded middle for orthoposets. (chjo 27758 analog.) (Contributed by NM, 13-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &    = (join‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → (𝑋 ( 𝑋)) = 1 )

Theoremopnoncon 33513 Law of contradiction for orthoposets. (chocin 27738 analog.) (Contributed by NM, 13-Sep-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵) → (𝑋 ( 𝑋)) = 0 )

TheoremriotaocN 33514* The orthocomplement of the unique poset element such that 𝜓. (riotaneg 10879 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 33515* 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 33516 Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 27827 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 33517 The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)
(𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))

Theoremollat 33518 An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)
(𝐾 ∈ OL → 𝐾 ∈ Lat)

Theoremolop 33519 An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)
(𝐾 ∈ OL → 𝐾 ∈ OP)

TheoremolposN 33520 An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
(𝐾 ∈ OL → 𝐾 ∈ Poset)

TheoremisolatiN 33521 Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
𝐾 ∈ Lat    &   𝐾 ∈ OP       𝐾 ∈ OL

Theoremoldmm1 33522 De Morgan's law for meet in an ortholattice. (chdmm1 27768 analog.) (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌)) = (( 𝑋) ( 𝑌)))

Theoremoldmm2 33523 De Morgan's law for meet in an ortholattice. (chdmm2 27769 analog.) (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) 𝑌)) = (𝑋 ( 𝑌)))

Theoremoldmm3N 33524 De Morgan's law for meet in an ortholattice. (chdmm3 27770 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 33525 De Morgan's law for meet in an ortholattice. (chdmm4 27771 analog.) (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) ( 𝑌))) = (𝑋 𝑌))

Theoremoldmj1 33526 De Morgan's law for join in an ortholattice. (chdmj1 27772 analog.) (Contributed by NM, 6-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 𝑌)) = (( 𝑋) ( 𝑌)))

Theoremoldmj2 33527 De Morgan's law for join in an ortholattice. (chdmj2 27773 analog.) (Contributed by NM, 7-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) 𝑌)) = (𝑋 ( 𝑌)))

Theoremoldmj3 33528 De Morgan's law for join in an ortholattice. (chdmj3 27774 analog.) (Contributed by NM, 7-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 ( 𝑌))) = (( 𝑋) 𝑌))

Theoremoldmj4 33529 De Morgan's law for join in an ortholattice. (chdmj4 27775 analog.) (Contributed by NM, 7-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ( ‘(( 𝑋) ( 𝑌))) = (𝑋 𝑌))

Theoremolj01 33530 An ortholattice element joined with zero equals itself. (chj0 27740 analog.) (Contributed by NM, 19-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋 0 ) = 𝑋)

Theoremolj02 33531 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 33532 The meet of an ortholattice element with one equals itself. (chm1i 27699 analog.) (Contributed by NM, 22-May-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    1 = (1.‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋 1 ) = 𝑋)

Theoremolm12 33533 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 33534 Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3785 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 33535 A rearrangement of lattice meet. (in12 3786 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = (𝑌 (𝑋 𝑍)))

Theoremlatm32 33536 A rearrangement of lattice meet. (in12 3786 analog.) (Contributed by NM, 13-Nov-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) 𝑌))

Theoremlatmrot 33537 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 33538 Rearrangement of lattice meet of 4 classes. (in4 3791 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 33539 Lattice meet distributes over itself. (inindi 3792 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 33540 Lattice meet distributes over itself. (inindir 3793 analog.) (Contributed by NM, 6-Jun-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)       ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) (𝑌 𝑍)))

Theoremolm01 33541 Meet with lattice zero is zero. (chm0 27734 analog.) (Contributed by NM, 8-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)       ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋 0 ) = 0 )

Theoremolm02 33542 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 33543* 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 33544* 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 33545 An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)
(𝐾 ∈ OML → 𝐾 ∈ OL)

Theoremomlop 33546 An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)
(𝐾 ∈ OML → 𝐾 ∈ OP)

Theoremomllat 33547 An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)
(𝐾 ∈ OML → 𝐾 ∈ Lat)

Theoremomllaw 33548 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 33549 Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 27828 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 33550 Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 27679 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 33551 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 33552 The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 27856 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 33553 Lemma for cmtcomN 33554. (cmcmlem 27834 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌𝐶𝑋))

TheoremcmtcomN 33554 Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 27835 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌𝐶𝑋))

Theoremcmt2N 33555 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 27836 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 33556 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 27838 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 33557 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 27838 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 33558 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 27839 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 33559 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 27851 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 33560 Alternate definition for the commutes relation. (cmbr4i 27844 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 33561 Ordered elements commute. (lecmi 27845 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 33562 Any element commutes with itself. (cmidi 27853 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = (cm‘𝐾)       ((𝐾 ∈ OML ∧ 𝑋𝐵) → 𝑋𝐶𝑋)

Theoremomlfh1N 33563 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 27861 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 33564 Foulis-Holland Theorem, part 3. Dual of omlfh1N 33563. (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 33565 Analogue of modular law atmod1i2 34163 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 33566 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)

21.22.10  Atomic lattices with covering property

Syntaxccvr 33567 Extend class notation with covers relation.
class

Syntaxcatm 33568 Extend class notation with atoms.
class Atoms

Syntaxcal 33569 Extend class notation with atomic lattices.
class AtLat

Syntaxclc 33570 Extend class notation with lattices with the covering property.
class CvLat

Definitiondf-covers 33571* 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 33574 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 33572* Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.)
Atoms = (𝑝 ∈ V ↦ {𝑎 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑎})

Theoremcvrfval 33573* 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 33574* 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 28525 analog.) (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧𝐵 (𝑋 < 𝑧𝑧 < 𝑌))))

Theoremcvrlt 33575 The covers relation implies the less-than relation. (cvpss 28528 analog.) (Contributed by NM, 8-Oct-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       (((𝐾𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)

Theoremcvrnbtwn 33576 There is no element between the two arguments of the covers relation. (cvnbtwn 28529 analog.) (Contributed by NM, 18-Oct-2011.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍𝑍 < 𝑌))

Theoremncvr1 33577 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 33578 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 33579* Binary relation expressing 𝑌 covers 𝑋. Definition of covers in [Kalmbach] p. 15. (cvbr2 28526 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 33580 The covers relation implies no in-betweenness. (cvnbtwn2 28530 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 33581 The covers relation implies no in-betweenness. (cvnbtwn3 28531 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 33582 Contraposition law for the covers relation. (cvcon3 28527 analog.) (Contributed by NM, 4-Nov-2011.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ( 𝑌)𝐶( 𝑋)))

Theoremcvrle 33583 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 33584 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 28532 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 33585 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 33586 The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       (((𝐾𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝑌)

TheoremcvrnrefN 33587 The covers relation is not reflexive. (cvnref 28534 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)       ((𝐾𝐴𝑋𝐵) → ¬ 𝑋𝐶𝑋)

Theoremcvrcmp 33588 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 33589 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 33590* 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 33591 The predicate "is an atom". (ela 28582 analog.) (Contributed by NM, 18-Sep-2011.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝐾𝐷 → (𝑃𝐴 ↔ (𝑃𝐵0 𝐶𝑃)))

Theoremisat2 33592 The predicate "is an atom". (elatcv0 28584 analog.) (Contributed by NM, 18-Jun-2012.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾𝐷𝑃𝐵) → (𝑃𝐴0 𝐶𝑃))

Theorematcvr0 33593 An atom covers zero. (atcv0 28585 analog.) (Contributed by NM, 4-Nov-2011.)
0 = (0.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾𝐷𝑃𝐴) → 0 𝐶𝑃)

Theorematbase 33594 An atom is a member of the lattice base set (i.e. a lattice element). (atelch 28587 analog.) (Contributed by NM, 10-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (𝑃𝐴𝑃𝐵)

Theorematssbase 33595 The set of atoms is a subset of the base set. (atssch 28586 analog.) (Contributed by NM, 21-Oct-2011.)
𝐵 = (Base‘𝐾)    &   𝐴 = (Atoms‘𝐾)       𝐴𝐵

Theorem0ltat 33596 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 33597 A poset element less than or equal to an atom equals either zero or the atom. (atss 28589 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 33598 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 33599 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 33600 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 ))

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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 42301-42360
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