Nuprl Lemma : distributive-lattice-distrib
∀[L:DistributiveLattice]. ∀[a,b,c:Point(L)].
((a ∧ b ∨ c = a ∧ b ∨ a ∧ c ∈ Point(L)) ∧ (b ∨ c ∧ a = b ∧ a ∨ c ∧ a ∈ Point(L)))
Proof
Definitions occuring in Statement :
distributive-lattice: DistributiveLattice,
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,
and: P ∧ Q,
cand: A c∧ B,
distributive-lattice: DistributiveLattice,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
lattice-axioms: lattice-axioms(l)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
lattice-point_wf,
lattice-join_wf,
lattice-meet_wf,
iff_weakening_equal,
distributive-lattice_wf,
lattice-structure_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
applyEquality,
lambdaEquality,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
because_Cache,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
independent_pairFormation,
independent_pairEquality,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[L:DistributiveLattice]. \mforall{}[a,b,c:Point(L)].
((a \mwedge{} b \mvee{} c = a \mwedge{} b \mvee{} a \mwedge{} c) \mwedge{} (b \mvee{} c \mwedge{} a = b \mwedge{} a \mvee{} c \mwedge{} a))
Date html generated:
2020_05_20-AM-08_25_48
Last ObjectModification:
2017_07_28-AM-09_12_57
Theory : lattices
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