Nuprl Lemma : lattice_properties
∀[l:Lattice]
((∀[a,b:Point(l)]. (a ∧ b = b ∧ a ∈ Point(l)))
∧ (∀[a,b:Point(l)]. (a ∨ b = b ∨ a ∈ Point(l)))
∧ (∀[a,b,c:Point(l)]. (a ∧ b ∧ c = a ∧ b ∧ c ∈ Point(l)))
∧ (∀[a,b,c:Point(l)]. (a ∨ b ∨ c = a ∨ b ∨ c ∈ Point(l)))
∧ (∀[a,b:Point(l)]. (a ∨ a ∧ b = a ∈ Point(l)))
∧ (∀[a,b:Point(l)]. (a ∧ a ∨ b = a ∈ Point(l))))
Proof
Definitions occuring in Statement :
lattice: Lattice,
lattice-join: a ∨ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
lattice: Lattice,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
lattice-axioms: lattice-axioms(l),
cand: A c∧ B,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
sq_stable__equal,
sq_stable__uall,
sq_stable__and,
squash_wf,
lattice-join_wf,
lattice-meet_wf,
equal_wf,
uall_wf,
lattice_wf,
lattice-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
sqequalRule,
productElimination,
independent_pairEquality,
isect_memberEquality,
isectElimination,
hypothesisEquality,
axiomEquality,
hypothesis,
lemma_by_obid,
lambdaEquality,
productEquality,
because_Cache,
independent_pairFormation,
independent_functionElimination,
dependent_functionElimination,
lambdaFormation,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[l:Lattice]
((\mforall{}[a,b:Point(l)]. (a \mwedge{} b = b \mwedge{} a))
\mwedge{} (\mforall{}[a,b:Point(l)]. (a \mvee{} b = b \mvee{} a))
\mwedge{} (\mforall{}[a,b,c:Point(l)]. (a \mwedge{} b \mwedge{} c = a \mwedge{} b \mwedge{} c))
\mwedge{} (\mforall{}[a,b,c:Point(l)]. (a \mvee{} b \mvee{} c = a \mvee{} b \mvee{} c))
\mwedge{} (\mforall{}[a,b:Point(l)]. (a \mvee{} a \mwedge{} b = a))
\mwedge{} (\mforall{}[a,b:Point(l)]. (a \mwedge{} a \mvee{} b = a)))
Date html generated:
2020_05_20-AM-08_23_43
Last ObjectModification:
2016_01_17-PM-00_54_32
Theory : lattices
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