Nuprl Lemma : lattice-join-0
∀[l:BoundedLattice]. ∀[x:Point(l)]. ((0 ∨ x = x ∈ Point(l)) ∧ (x ∨ 0 = x ∈ Point(l)))
Proof
Definitions occuring in Statement :
bdd-lattice: BoundedLattice,
lattice-0: 0,
lattice-join: 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,
bdd-lattice: BoundedLattice,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
guard: {T},
lattice-axioms: lattice-axioms(l),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
bounded-lattice-axioms: bounded-lattice-axioms(l)
Lemmas referenced :
lattice-point_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
bdd-lattice_wf,
lattice-0_wf,
and_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
hypothesis,
independent_pairFormation,
sqequalRule,
independent_pairEquality,
axiomEquality,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
independent_isectElimination,
isect_memberEquality,
because_Cache,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
equalityTransitivity,
applyLambdaEquality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[l:BoundedLattice]. \mforall{}[x:Point(l)]. ((0 \mvee{} x = x) \mwedge{} (x \mvee{} 0 = x))
Date html generated:
2020_05_20-AM-08_25_59
Last ObjectModification:
2017_07_28-AM-09_13_03
Theory : lattices
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