Nuprl Lemma : lattice-meet-eq-1
∀[l:BoundedLattice]. ∀[x,y:Point(l)]. uiff(x ∧ y = 1 ∈ Point(l);(x = 1 ∈ Point(l)) ∧ (y = 1 ∈ Point(l)))
Proof
Definitions occuring in Statement :
bdd-lattice: BoundedLattice,
lattice-1: 1,
lattice-meet: a ∧ b,
lattice-point: Point(l),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
bdd-lattice: BoundedLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
lattice-point_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-1_wf,
squash_wf,
true_wf,
iff_weakening_equal,
lattice-meet-idempotent,
bdd-lattice-subtype-lattice,
bdd-lattice_wf,
lattice-join_wf,
lattice_properties,
lattice-1-join
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
independent_isectElimination,
because_Cache,
setElimination,
rename,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
isect_memberEquality,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}[l:BoundedLattice]. \mforall{}[x,y:Point(l)]. uiff(x \mwedge{} y = 1;(x = 1) \mwedge{} (y = 1))
Date html generated:
2020_05_20-AM-08_26_12
Last ObjectModification:
2017_07_28-AM-09_13_10
Theory : lattices
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