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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | oldmm4 39201 | De Morgan's law for meet in an ortholattice. (chdmm4 31556 analog.) (Contributed by NM, 6-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ ⊥ = (oc‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌))) = (𝑋 ∨ 𝑌)) | ||
Theorem | oldmj1 39202 | De Morgan's law for join in an ortholattice. (chdmj1 31557 analog.) (Contributed by NM, 6-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ ⊥ = (oc‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∨ 𝑌)) = (( ⊥ ‘𝑋) ∧ ( ⊥ ‘𝑌))) | ||
Theorem | oldmj2 39203 | De Morgan's law for join in an ortholattice. (chdmj2 31558 analog.) (Contributed by NM, 7-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ ⊥ = (oc‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ ( ⊥ ‘𝑌))) | ||
Theorem | oldmj3 39204 | De Morgan's law for join in an ortholattice. (chdmj3 31559 analog.) (Contributed by NM, 7-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ ⊥ = (oc‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∨ ( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∧ 𝑌)) | ||
Theorem | oldmj4 39205 | De Morgan's law for join in an ortholattice. (chdmj4 31560 analog.) (Contributed by NM, 7-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ ⊥ = (oc‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌))) = (𝑋 ∧ 𝑌)) | ||
Theorem | olj01 39206 | An ortholattice element joined with zero equals itself. (chj0 31525 analog.) (Contributed by NM, 19-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 0 = (0.‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) | ||
Theorem | olj02 39207 | An ortholattice element joined with zero equals itself. (Contributed by NM, 28-Jan-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 0 = (0.‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( 0 ∨ 𝑋) = 𝑋) | ||
Theorem | olm11 39208 | The meet of an ortholattice element with one equals itself. (chm1i 31484 analog.) (Contributed by NM, 22-May-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 1 = (1.‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 1 ) = 𝑋) | ||
Theorem | olm12 39209 | 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 ∧ 𝑋 ∈ 𝐵) → ( 1 ∧ 𝑋) = 𝑋) | ||
Theorem | latmassOLD 39210 | Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 4235 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = (𝑋 ∧ (𝑌 ∧ 𝑍))) | ||
Theorem | latm12 39211 | A rearrangement of lattice meet. (in12 4236 analog.) (Contributed by NM, 8-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ (𝑌 ∧ 𝑍)) = (𝑌 ∧ (𝑋 ∧ 𝑍))) | ||
Theorem | latm32 39212 | A rearrangement of lattice meet. (in12 4236 analog.) (Contributed by NM, 13-Nov-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = ((𝑋 ∧ 𝑍) ∧ 𝑌)) | ||
Theorem | latmrot 39213 | Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = ((𝑍 ∧ 𝑋) ∧ 𝑌)) | ||
Theorem | latm4 39214 | Rearrangement of lattice meet of 4 classes. (in4 4241 analog.) (Contributed by NM, 8-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ (𝑍 ∧ 𝑊)) = ((𝑋 ∧ 𝑍) ∧ (𝑌 ∧ 𝑊))) | ||
Theorem | latmmdiN 39215 | Lattice meet distributes over itself. (inindi 4242 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ (𝑌 ∧ 𝑍)) = ((𝑋 ∧ 𝑌) ∧ (𝑋 ∧ 𝑍))) | ||
Theorem | latmmdir 39216 | Lattice meet distributes over itself. (inindir 4243 analog.) (Contributed by NM, 6-Jun-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = ((𝑋 ∧ 𝑍) ∧ (𝑌 ∧ 𝑍))) | ||
Theorem | olm01 39217 | Meet with lattice zero is zero. (chm0 31519 analog.) (Contributed by NM, 8-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 0 = (0.‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 0 ) = 0 ) | ||
Theorem | olm02 39218 | Meet with lattice zero is zero. (Contributed by NM, 9-Oct-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 0 = (0.‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( 0 ∧ 𝑋) = 0 ) | ||
Theorem | isoml 39219* | 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 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))))) | ||
Theorem | isomliN 39220* | 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) & ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))) ⇒ ⊢ 𝐾 ∈ OML | ||
Theorem | omlol 39221 | An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.) |
⊢ (𝐾 ∈ OML → 𝐾 ∈ OL) | ||
Theorem | omlop 39222 | An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.) |
⊢ (𝐾 ∈ OML → 𝐾 ∈ OP) | ||
Theorem | omllat 39223 | An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.) |
⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat) | ||
Theorem | omllaw 39224 | The orthomodular law. (Contributed by NM, 18-Sep-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ ⊥ = (oc‘𝐾) ⇒ ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑌 = (𝑋 ∨ (𝑌 ∧ ( ⊥ ‘𝑋))))) | ||
Theorem | omllaw2N 39225 | Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 31613 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ ⊥ = (oc‘𝐾) ⇒ ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ 𝑌)) = 𝑌)) | ||
Theorem | omllaw3 39226 | Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 31464 analog.) (Contributed by NM, 19-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ ⊥ = (oc‘𝐾) & ⊢ 0 = (0.‘𝐾) ⇒ ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ (𝑌 ∧ ( ⊥ ‘𝑋)) = 0 ) → 𝑋 = 𝑌)) | ||
Theorem | omllaw4 39227 | 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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (( ⊥ ‘(( ⊥ ‘𝑋) ∧ 𝑌)) ∧ 𝑌) = 𝑋)) | ||
Theorem | omllaw5N 39228 | The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 31641 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ ⊥ = (oc‘𝐾) ⇒ ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ (( ⊥ ‘𝑋) ∧ (𝑋 ∨ 𝑌))) = (𝑋 ∨ 𝑌)) | ||
Theorem | cmtcomlemN 39229 | Lemma for cmtcomN 39230. (cmcmlem 31619 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐶 = (cm‘𝐾) ⇒ ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → 𝑌𝐶𝑋)) | ||
Theorem | cmtcomN 39230 | Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 31620 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐶 = (cm‘𝐾) ⇒ ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑌𝐶𝑋)) | ||
Theorem | cmt2N 39231 | Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 31621 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ⊥ = (oc‘𝐾) & ⊢ 𝐶 = (cm‘𝐾) ⇒ ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋𝐶( ⊥ ‘𝑌))) | ||
Theorem | cmt3N 39232 | Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 31623 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ⊥ = (oc‘𝐾) & ⊢ 𝐶 = (cm‘𝐾) ⇒ ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ ( ⊥ ‘𝑋)𝐶𝑌)) | ||
Theorem | cmt4N 39233 | Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 31623 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ⊥ = (oc‘𝐾) & ⊢ 𝐶 = (cm‘𝐾) ⇒ ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ ( ⊥ ‘𝑋)𝐶( ⊥ ‘𝑌))) | ||
Theorem | cmtbr2N 39234 | Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 31624 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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋 = ((𝑋 ∨ 𝑌) ∧ (𝑋 ∨ ( ⊥ ‘𝑌))))) | ||
Theorem | cmtbr3N 39235 | Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 31636 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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) = (𝑋 ∧ 𝑌))) | ||
Theorem | cmtbr4N 39236 | Alternate definition for the commutes relation. (cmbr4i 31629 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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 ∧ (( ⊥ ‘𝑋) ∨ 𝑌)) ≤ 𝑌)) | ||
Theorem | lecmtN 39237 | Ordered elements commute. (lecmi 31630 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐶 = (cm‘𝐾) ⇒ ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → 𝑋𝐶𝑌)) | ||
Theorem | cmtidN 39238 | Any element commutes with itself. (cmidi 31638 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐶 = (cm‘𝐾) ⇒ ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → 𝑋𝐶𝑋) | ||
Theorem | omlfh1N 39239 | 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 31646 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐶 = (cm‘𝐾) ⇒ ⊢ ((𝐾 ∈ OML ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋𝐶𝑌 ∧ 𝑋𝐶𝑍)) → (𝑋 ∧ (𝑌 ∨ 𝑍)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍))) | ||
Theorem | omlfh3N 39240 | Foulis-Holland Theorem, part 3. Dual of omlfh1N 39239. (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐶 = (cm‘𝐾) ⇒ ⊢ ((𝐾 ∈ OML ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋𝐶𝑌 ∧ 𝑋𝐶𝑍)) → (𝑋 ∨ (𝑌 ∧ 𝑍)) = ((𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑍))) | ||
Theorem | omlmod1i2N 39241 | Analogue of modular law atmod1i2 39841 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 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ≤ 𝑍 ∧ 𝑌𝐶𝑍)) → (𝑋 ∨ (𝑌 ∧ 𝑍)) = ((𝑋 ∨ 𝑌) ∧ 𝑍)) | ||
Theorem | omlspjN 39242 | 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 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → ((𝑋 ∨ ( ⊥ ‘𝑌)) ∧ 𝑌) = 𝑋) | ||
Syntax | ccvr 39243 | Extend class notation with covers relation. |
class ⋖ | ||
Syntax | catm 39244 | Extend class notation with atoms. |
class Atoms | ||
Syntax | cal 39245 | Extend class notation with atomic lattices. |
class AtLat | ||
Syntax | clc 39246 | Extend class notation with lattices with the covering property. |
class CvLat | ||
Definition | df-covers 39247* | 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 39250 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)𝑧 ∧ 𝑧(lt‘𝑝)𝑏))}) | ||
Definition | df-ats 39248* | Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.) |
⊢ Atoms = (𝑝 ∈ V ↦ {𝑎 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑎}) | ||
Theorem | cvrfval 39249* | Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝐴 → 𝐶 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))}) | ||
Theorem | cvrval 39250* | 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 32310 analog.) (Contributed by NM, 18-Sep-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑋 < 𝑧 ∧ 𝑧 < 𝑌)))) | ||
Theorem | cvrlt 39251 | The covers relation implies the less-than relation. (cvpss 32313 analog.) (Contributed by NM, 8-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌) | ||
Theorem | cvrnbtwn 39252 | There is no element between the two arguments of the covers relation. (cvnbtwn 32314 analog.) (Contributed by NM, 18-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌)) | ||
Theorem | ncvr1 39253 | No element covers the lattice unity. (Contributed by NM, 8-Jul-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 1 = (1.‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ¬ 1 𝐶𝑋) | ||
Theorem | cvrletrN 39254 | 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 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋𝐶𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 < 𝑍)) | ||
Theorem | cvrval2 39255* | Binary relation expressing 𝑌 covers 𝑋. Definition of covers in [Kalmbach] p. 15. (cvbr2 32311 analog.) (Contributed by NM, 16-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ∀𝑧 ∈ 𝐵 ((𝑋 < 𝑧 ∧ 𝑧 ≤ 𝑌) → 𝑧 = 𝑌)))) | ||
Theorem | cvrnbtwn2 39256 | The covers relation implies no in-betweenness. (cvnbtwn2 32315 analog.) (Contributed by NM, 17-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 = 𝑌)) | ||
Theorem | cvrnbtwn3 39257 | The covers relation implies no in-betweenness. (cvnbtwn3 32316 analog.) (Contributed by NM, 4-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ↔ 𝑋 = 𝑍)) | ||
Theorem | cvrcon3b 39258 | Contraposition law for the covers relation. (cvcon3 32312 analog.) (Contributed by NM, 4-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ⊥ = (oc‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ ( ⊥ ‘𝑌)𝐶( ⊥ ‘𝑋))) | ||
Theorem | cvrle 39259 | The covers relation implies the "less than or equal to" relation. (Contributed by NM, 12-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 ≤ 𝑌) | ||
Theorem | cvrnbtwn4 39260 | The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 32317 analog.) (Contributed by NM, 18-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) ↔ (𝑋 = 𝑍 ∨ 𝑍 = 𝑌))) | ||
Theorem | cvrnle 39261 | The covers relation implies the negation of the converse "less than or equal to" relation. (Contributed by NM, 18-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ 𝑌 ≤ 𝑋) | ||
Theorem | cvrne 39262 | The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 ≠ 𝑌) | ||
Theorem | cvrnrefN 39263 | The covers relation is not reflexive. (cvnref 32319 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ¬ 𝑋𝐶𝑋) | ||
Theorem | cvrcmp 39264 | 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 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑍𝐶𝑋 ∧ 𝑍𝐶𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌)) | ||
Theorem | cvrcmp2 39265 | 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 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋𝐶𝑍 ∧ 𝑌𝐶𝑍)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌)) | ||
Theorem | pats 39266* | The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝐷 → 𝐴 = {𝑥 ∈ 𝐵 ∣ 0 𝐶𝑥}) | ||
Theorem | isat 39267 | The predicate "is an atom". (ela 32367 analog.) (Contributed by NM, 18-Sep-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 0 𝐶𝑃))) | ||
Theorem | isat2 39268 | The predicate "is an atom". (elatcv0 32369 analog.) (Contributed by NM, 18-Jun-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐵) → (𝑃 ∈ 𝐴 ↔ 0 𝐶𝑃)) | ||
Theorem | atcvr0 39269 | An atom covers zero. (atcv0 32370 analog.) (Contributed by NM, 4-Nov-2011.) |
⊢ 0 = (0.‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐴) → 0 𝐶𝑃) | ||
Theorem | atbase 39270 | An atom is a member of the lattice base set (i.e. a lattice element). (atelch 32372 analog.) (Contributed by NM, 10-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) | ||
Theorem | atssbase 39271 | The set of atoms is a subset of the base set. (atssch 32371 analog.) (Contributed by NM, 21-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ 𝐴 ⊆ 𝐵 | ||
Theorem | 0ltat 39272 | An atom is greater than zero. (Contributed by NM, 4-Jul-2012.) |
⊢ 0 = (0.‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴) → 0 < 𝑃) | ||
Theorem | leatb 39273 | A poset element less than or equal to an atom equals either zero or the atom. (atss 32374 analog.) (Contributed by NM, 17-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≤ 𝑃 ↔ (𝑋 = 𝑃 ∨ 𝑋 = 0 ))) | ||
Theorem | leat 39274 | 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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑋 ≤ 𝑃) → (𝑋 = 𝑃 ∨ 𝑋 = 0 )) | ||
Theorem | leat2 39275 | 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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ≠ 0 ∧ 𝑋 ≤ 𝑃)) → 𝑋 = 𝑃) | ||
Theorem | leat3 39276 | 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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑋 ≤ 𝑃) → (𝑋 ∈ 𝐴 ∨ 𝑋 = 0 )) | ||
Theorem | meetat 39277 | 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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) = 𝑃 ∨ (𝑋 ∧ 𝑃) = 0 )) | ||
Theorem | meetat2 39278 | 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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ((𝑋 ∧ 𝑃) ∈ 𝐴 ∨ (𝑋 ∧ 𝑃) = 0 )) | ||
Definition | df-atl 39279* | 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))} | ||
Theorem | isatl 39280* | 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 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))) | ||
Theorem | atllat 39281 | An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.) |
⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat) | ||
Theorem | atlpos 39282 | An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.) |
⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | ||
Theorem | atl0dm 39283 | 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) | ||
Theorem | atl0cl 39284 | An atomic lattice has a zero element. We can use this in place of op0cl 39165 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 0 = (0.‘𝐾) ⇒ ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) | ||
Theorem | atl0le 39285 | Orthoposet zero is less than or equal to any element. (ch0le 31469 analog.) (Contributed by NM, 12-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋) | ||
Theorem | atlle0 39286 | An element less than or equal to zero equals zero. (chle0 31471 analog.) (Contributed by NM, 21-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → (𝑋 ≤ 0 ↔ 𝑋 = 0 )) | ||
Theorem | atlltn0 39287 | A lattice element greater than zero is nonzero. (Contributed by NM, 1-Jun-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ 0 = (0.‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ 𝑋 ≠ 0 )) | ||
Theorem | isat3 39288* | The predicate "is an atom". (elat2 32368 analog.) (Contributed by NM, 27-Apr-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (𝐾 ∈ AtLat → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ 𝐵 ∧ 𝑃 ≠ 0 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑃 → (𝑥 = 𝑃 ∨ 𝑥 = 0 ))))) | ||
Theorem | atn0 39289 | An atom is not zero. (atne0 32373 analog.) (Contributed by NM, 5-Nov-2012.) |
⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ 0 ) | ||
Theorem | atnle0 39290 | An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.) |
⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ≤ 0 ) | ||
Theorem | atlen0 39291 | A lattice element is nonzero if an atom is under it. (Contributed by NM, 26-May-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑋 ≠ 0 ) | ||
Theorem | atcmp 39292 | If two atoms are comparable, they are equal. (atsseq 32375 analog.) (Contributed by NM, 13-Oct-2011.) |
⊢ ≤ = (le‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≤ 𝑄 ↔ 𝑃 = 𝑄)) | ||
Theorem | atncmp 39293 | Frequently-used variation of atcmp 39292. (Contributed by NM, 29-Jun-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑃 ≤ 𝑄 ↔ 𝑃 ≠ 𝑄)) | ||
Theorem | atnlt 39294 | Two atoms cannot satisfy the less than relation. (Contributed by NM, 7-Feb-2012.) |
⊢ < = (lt‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃 < 𝑄) | ||
Theorem | atcvreq0 39295 | An element covered by an atom must be zero. (atcveq0 32376 analog.) (Contributed by NM, 4-Nov-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋𝐶𝑃 ↔ 𝑋 = 0 )) | ||
Theorem | atncvrN 39296 | Two atoms cannot satisfy the covering relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.) |
⊢ 𝐶 = ( ⋖ ‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃𝐶𝑄) | ||
Theorem | atlex 39297* | Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 32388 analog.) (Contributed by NM, 21-Oct-2011.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋) | ||
Theorem | atnle 39298 | Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 32404 analog.) (Contributed by NM, 5-Nov-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 )) | ||
Theorem | atnem0 39299 | The meet of distinct atoms is zero. (atnemeq0 32405 analog.) (Contributed by NM, 5-Nov-2012.) |
⊢ ∧ = (meet‘𝐾) & ⊢ 0 = (0.‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 ↔ (𝑃 ∧ 𝑄) = 0 )) | ||
Theorem | atlatmstc 39300* | An atomic, complete, orthomodular lattice is atomistic i.e. every element is the join of the atoms under it. See remark before Proposition 1 in [Kalmbach] p. 140; also remark in [BeltramettiCassinelli] p. 98. (hatomistici 32390 analog.) (Contributed by NM, 5-Nov-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 1 = (lub‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋) |
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