Nuprl Lemma : lattice-le-iff
∀[l:Lattice]. ∀[a,b:Point(l)]. uiff(a ≤ b;b = a ∨ b ∈ Point(l))
Proof
Definitions occuring in Statement :
lattice-le: a ≤ b,
lattice: Lattice,
lattice-join: a ∨ b,
lattice-point: Point(l),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
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,
lattice-le: a ≤ b,
lattice: Lattice,
prop: ℙ,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
lattice_properties,
lattice-le_wf,
equal_wf,
lattice-point_wf,
lattice-join_wf,
lattice_wf,
iff_weakening_equal,
squash_wf,
true_wf,
lattice-structure_wf,
lattice-meet_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesisEquality,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
thin,
hypothesis,
setElimination,
rename,
productElimination,
sqequalRule,
axiomEquality,
because_Cache,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
applyEquality,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
universeEquality
Latex:
\mforall{}[l:Lattice]. \mforall{}[a,b:Point(l)]. uiff(a \mleq{} b;b = a \mvee{} b)
Date html generated:
2020_05_20-AM-08_25_29
Last ObjectModification:
2017_07_28-AM-09_12_52
Theory : lattices
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