Nuprl Lemma : lattice-1-le-iff
∀[l:BoundedLattice]. ∀[b:Point(l)]. uiff(1 ≤ b;b = 1 ∈ Point(l))
Proof
Definitions occuring in Statement :
bdd-lattice: BoundedLattice,
lattice-1: 1,
lattice-le: 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,
bdd-lattice: BoundedLattice,
prop: ℙ,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
lattice-le: a ≤ b,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
true: True,
guard: {T}
Lemmas referenced :
iff_weakening_uiff,
lattice-le_wf,
lattice-1_wf,
equal_wf,
lattice-join_wf,
lattice-le-iff,
bdd-lattice-subtype-lattice,
lattice-point_wf,
uiff_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,
squash_wf,
true_wf,
lattice-join-1,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
addLevel,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
independent_isectElimination,
extract_by_obid,
isectElimination,
because_Cache,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_functionElimination,
applyEquality,
sqequalRule,
lambdaEquality,
cumulativity,
universeEquality,
axiomEquality,
instantiate,
independent_pairEquality,
isect_memberEquality,
productEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[l:BoundedLattice]. \mforall{}[b:Point(l)]. uiff(1 \mleq{} b;b = 1)
Date html generated:
2020_05_20-AM-08_26_14
Last ObjectModification:
2017_07_28-AM-09_13_11
Theory : lattices
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