Nuprl Lemma : compact-dist-zero-in-complete
∀[X:Type]
∀d:metric(X)
(mcomplete(X with d)
⇒ (∀[A:Type]. (metric-subspace(X;d;A) ⇒ (∀c:mcompact(A;d). ∀x:X. (dist(x;A) = r0 ⇐⇒ x ∈ A)))))
Proof
Definitions occuring in Statement :
compact-dist: dist(x;A),
mcompact: mcompact(X;d),
mcomplete: mcomplete(M),
metric-subspace: metric-subspace(X;d;A),
mk-metric-space: X with d,
metric: metric(X),
req: x = y,
int-to-real: r(n),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
member: t ∈ T,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
sq_exists: ∃x:A [B[x]],
rless: x < y,
m-closed-subspace: m-closed-subspace(X;d;A),
mcompact: mcompact(X;d),
label: ...$L... t,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
nat_plus: ℕ+,
exists: ∃x:A. B[x],
istype: istype(T),
metric-subspace: metric-subspace(X;d;A),
subtype_rel: A ⊆r B,
respects-equality: respects-equality(S;T),
rev_implies: P ⇐ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
req_weakening,
mdist-same,
rless_functionality,
int_term_value_mul_lemma,
itermMultiply_wf,
rless-int-fractions2,
rleq_wf,
rleq_weakening_rless,
istype-base,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
rless-int,
rdiv_wf,
mdist_wf,
rless_wf,
nat_plus_wf,
istype-universe,
metric_wf,
mk-metric-space_wf,
mcomplete_wf,
metric-subspace_wf,
mcompact_wf,
metric-on-subtype,
req_witness,
int-to-real_wf,
compact-dist_wf,
req_wf,
compact-dist-zero,
strong-subtype-iff-respects-equality,
m-closed-iff-complete
Rules used in proof :
multiplyEquality,
sqequalBase,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
inrFormation_alt,
rename,
setElimination,
closedConclusion,
productIsType,
functionIsType,
universeEquality,
instantiate,
applyEquality,
inhabitedIsType,
functionIsTypeImplies,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
lambdaEquality_alt,
independent_pairEquality,
equalityIstype,
sqequalRule,
promote_hyp,
independent_pairFormation,
natural_numberEquality,
universeIsType,
because_Cache,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
lambdaFormation_alt,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}[X:Type]
\mforall{}d:metric(X)
(mcomplete(X with d)
{}\mRightarrow{} (\mforall{}[A:Type]
(metric-subspace(X;d;A) {}\mRightarrow{} (\mforall{}c:mcompact(A;d). \mforall{}x:X. (dist(x;A) = r0 \mLeftarrow{}{}\mRightarrow{} x \mmember{} A)))))
Date html generated:
2019_10_31-AM-05_59_36
Last ObjectModification:
2019_10_30-PM-04_11_03
Theory : reals
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