P. 393 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39201-39300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	oplecon3b 39201	Contraposition law for orthoposets. (chsscon3 31519 analog.) (Contributed by NM, 4-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘𝑋)))
Theorem	oplecon1b 39202	Contraposition law for strict ordering in orthoposets. (chsscon1 31520 analog.) (Contributed by NM, 6-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ 𝑋))
Theorem	opoc1 39203	Orthocomplement of orthoposet unity. (Contributed by NM, 24-Jan-2012.)
⊢ 0 = (0.‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = 0 )
Theorem	opoc0 39204	Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)
⊢ 0 = (0.‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 0 ) = 1 )
Theorem	opltcon3b 39205	Contraposition law for strict ordering in orthoposets. (chpsscon3 31522 analog.) (Contributed by NM, 4-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ ( ⊥ ‘𝑌) < ( ⊥ ‘𝑋)))
Theorem	opltcon1b 39206	Contraposition law for strict ordering in orthoposets. (chpsscon1 31523 analog.) (Contributed by NM, 5-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) < 𝑌 ↔ ( ⊥ ‘𝑌) < 𝑋))
Theorem	opltcon2b 39207	Contraposition law for strict ordering in orthoposets. (chsscon2 31521 analog.) (Contributed by NM, 5-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ 𝑌 < ( ⊥ ‘𝑋)))
Theorem	opexmid 39208	Law of excluded middle for orthoposets. (chjo 31534 analog.) (Contributed by NM, 13-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ ( ⊥ ‘𝑋)) = 1 )
Theorem	opnoncon 39209	Law of contradiction for orthoposets. (chocin 31514 analog.) (Contributed by NM, 13-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ ( ⊥ ‘𝑋)) = 0 )
Theorem	riotaocN 39210*	The orthocomplement of the unique poset element such that 𝜓. (riotaneg 12247 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ (𝑥 = ( ⊥ ‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐾 ∈ OP ∧ ∃!𝑥 ∈ 𝐵 𝜑) → (℩𝑥 ∈ 𝐵 𝜑) = ( ⊥ ‘(℩𝑦 ∈ 𝐵 𝜓)))
Theorem	cmtfvalN 39211*	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)    ⇒   ⊢ (𝐾 ∈ 𝐴 → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ ( ⊥ ‘𝑦))))})
Theorem	cmtvalN 39212	Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 31603 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ 𝑋 = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ ( ⊥ ‘𝑌)))))
Theorem	isolat 39213	The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)
⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))
Theorem	ollat 39214	An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)
⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat)
Theorem	olop 39215	An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)
⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
Theorem	olposN 39216	An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
⊢ (𝐾 ∈ OL → 𝐾 ∈ Poset)
Theorem	isolatiN 39217	Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
⊢ 𝐾 ∈ Lat    &   ⊢ 𝐾 ∈ OP    ⇒   ⊢ 𝐾 ∈ OL
Theorem	oldmm1 39218	De Morgan's law for meet in an ortholattice. (chdmm1 31544 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	oldmm2 39219	De Morgan's law for meet in an ortholattice. (chdmm2 31545 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	oldmm3N 39220	De Morgan's law for meet in an ortholattice. (chdmm3 31546 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘(𝑋 ∧ ( ⊥ ‘𝑌))) = (( ⊥ ‘𝑋) ∨ 𝑌))
Theorem	oldmm4 39221	De Morgan's law for meet in an ortholattice. (chdmm4 31547 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 39222	De Morgan's law for join in an ortholattice. (chdmj1 31548 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 39223	De Morgan's law for join in an ortholattice. (chdmj2 31549 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 39224	De Morgan's law for join in an ortholattice. (chdmj3 31550 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 39225	De Morgan's law for join in an ortholattice. (chdmj4 31551 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 39226	An ortholattice element joined with zero equals itself. (chj0 31516 analog.) (Contributed by NM, 19-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋)
Theorem	olj02 39227	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 39228	The meet of an ortholattice element with one equals itself. (chm1i 31475 analog.) (Contributed by NM, 22-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 1 ) = 𝑋)
Theorem	olm12 39229	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 39230	Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 4228 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 39231	A rearrangement of lattice meet. (in12 4229 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 39232	A rearrangement of lattice meet. (in12 4229 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 39233	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 39234	Rearrangement of lattice meet of 4 classes. (in4 4234 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 39235	Lattice meet distributes over itself. (inindi 4235 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 39236	Lattice meet distributes over itself. (inindir 4236 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 39237	Meet with lattice zero is zero. (chm0 31510 analog.) (Contributed by NM, 8-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 0 ) = 0 )
Theorem	olm02 39238	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 39239*	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 39240*	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 39241	An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)
⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
Theorem	omlop 39242	An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)
⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
Theorem	omllat 39243	An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)
⊢ (𝐾 ∈ OML → 𝐾 ∈ Lat)
Theorem	omllaw 39244	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 39245	Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 31604 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 39246	Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 31455 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 39247	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 39248	The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 31632 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 39249	Lemma for cmtcomN 39250. (cmcmlem 31610 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 39250	Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 31611 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 39251	Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 31612 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 39252	Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 31614 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 39253	Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 31614 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 39254	Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 31615 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 39255	Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 31627 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 39256	Alternate definition for the commutes relation. (cmbr4i 31620 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 39257	Ordered elements commute. (lecmi 31621 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 39258	Any element commutes with itself. (cmidi 31629 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = (cm‘𝐾)    ⇒   ⊢ ((𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵) → 𝑋𝐶𝑋)
Theorem	omlfh1N 39259	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 31637 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 39260	Foulis-Holland Theorem, part 3. Dual of omlfh1N 39259. (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 39261	Analogue of modular law atmod1i2 39861 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 39262	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)
21.28.10  Atomic lattices with covering property
Syntax	ccvr 39263	Extend class notation with covers relation.
class ⋖
Syntax	catm 39264	Extend class notation with atoms.
class Atoms
Syntax	cal 39265	Extend class notation with atomic lattices.
class AtLat
Syntax	clc 39266	Extend class notation with lattices with the covering property.
class CvLat
Definition	df-covers 39267*	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 39270 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 39268*	Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.)
⊢ Atoms = (𝑝 ∈ V ↦ {𝑎 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑎})
Theorem	cvrfval 39269*	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 39270*	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 32301 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 39271	The covers relation implies the less-than relation. (cvpss 32304 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 39272	There is no element between the two arguments of the covers relation. (cvnbtwn 32305 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 39273	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 39274	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 39275*	Binary relation expressing 𝑌 covers 𝑋. Definition of covers in [Kalmbach] p. 15. (cvbr2 32302 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 39276	The covers relation implies no in-betweenness. (cvnbtwn2 32306 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 39277	The covers relation implies no in-betweenness. (cvnbtwn3 32307 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 39278	Contraposition law for the covers relation. (cvcon3 32303 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 39279	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 39280	The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 32308 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 39281	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 39282	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 39283	The covers relation is not reflexive. (cvnref 32310 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ¬ 𝑋𝐶𝑋)
Theorem	cvrcmp 39284	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 39285	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 39286*	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 39287	The predicate "is an atom". (ela 32358 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 39288	The predicate "is an atom". (elatcv0 32360 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 39289	An atom covers zero. (atcv0 32361 analog.) (Contributed by NM, 4-Nov-2011.)
⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐴) → 0 𝐶𝑃)
Theorem	atbase 39290	An atom is a member of the lattice base set (i.e. a lattice element). (atelch 32363 analog.) (Contributed by NM, 10-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
Theorem	atssbase 39291	The set of atoms is a subset of the base set. (atssch 32362 analog.) (Contributed by NM, 21-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	0ltat 39292	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 39293	A poset element less than or equal to an atom equals either zero or the atom. (atss 32365 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 39294	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 39295	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 39296	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 39297	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 39298	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 39299*	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 39300*	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)))