Nuprl Lemma : face_lattice-fset-join-eq-1
∀I:fset(ℕ). ∀s:fset(Point(face_lattice(I))).
(\/(s) = 1 ∈ Point(face_lattice(I)) ⇐⇒ ∃x:Point(face_lattice(I)). (x ∈ s ∧ (x = 1 ∈ Point(face_lattice(I)))))
Proof
Definitions occuring in Statement :
face_lattice-deq: face_lattice-deq(),
face_lattice: face_lattice(I),
lattice-fset-join: \/(s),
lattice-1: 1,
lattice-point: Point(l),
fset-member: a ∈ s,
fset: fset(T),
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
guard: {T},
exists: ∃x:A. B[x],
squash: ↓T,
true: True,
cand: A c∧ B,
sq_stable: SqStable(P),
decidable: Dec(P),
or: P ∨ Q,
empty-fset: {},
lattice-fset-join: \/(s),
top: Top,
fset-add: fset-add(eq;x;s),
uiff: uiff(P;Q)
Lemmas referenced :
fset-induction,
lattice-point_wf,
face_lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
face_lattice-deq_wf,
iff_wf,
lattice-fset-join_wf,
decidable__equal_face_lattice,
lattice-1_wf,
bdd-distributive-lattice_wf,
exists_wf,
fset-member_wf,
fset_wf,
sq_stable__iff,
sq_stable__equal,
empty-fset_wf,
fset-add_wf,
not_wf,
nat_wf,
squash_wf,
true_wf,
deq_wf,
iff_weakening_equal,
decidable__fset-member,
face-lattice-0-not-1,
reduce_nil_lemma,
member-empty-fset,
bdd-distributive-lattice-subtype-bdd-lattice,
fset-singleton_wf,
face_lattice-1-join-irreducible,
lattice-fset-join-union,
lattice-fset-join-singleton,
fset-union_wf,
member-fset-union,
member-fset-singleton,
or_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
universeEquality,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
independent_pairFormation,
isect_memberFormation,
voidElimination,
productElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
unionElimination,
isect_memberEquality,
voidEquality,
inlFormation,
inrFormation,
addLevel,
orFunctionality,
promote_hyp
Latex:
\mforall{}I:fset(\mBbbN{}). \mforall{}s:fset(Point(face\_lattice(I))).
(\mbackslash{}/(s) = 1 \mLeftarrow{}{}\mRightarrow{} \mexists{}x:Point(face\_lattice(I)). (x \mmember{} s \mwedge{} (x = 1)))
Date html generated:
2017_10_05-AM-01_10_15
Last ObjectModification:
2017_03_02-PM-10_27_18
Theory : cubical!type!theory
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