Nuprl Lemma : m-closed-iff-complete
∀[X:Type]
∀d:metric(X)
(mcomplete(X with d) ⇒ (∀[A:Type]. (metric-subspace(X;d;A) ⇒ (m-closed-subspace(X;d;A) ⇐⇒ mcomplete(A with d)))))
Proof
Definitions occuring in Statement :
mcomplete: mcomplete(M),
m-closed-subspace: m-closed-subspace(X;d;A),
metric-subspace: metric-subspace(X;d;A),
mk-metric-space: X with d,
metric: metric(X),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
mcauchy: mcauchy(d;n.x[n]),
pi1: fst(t),
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
sq_stable: SqStable(P),
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
or: P ∨ Q,
rneq: x ≠ y,
nat_plus: ℕ+,
sq_exists: ∃x:A [B[x]],
mconverges-to: lim n→∞.x[n] = y,
m-closed-subspace: m-closed-subspace(X;d;A),
exists: ∃x:A. B[x],
mconverges: x[n]↓ as n→∞,
mk-metric-space: X with d,
mcomplete: mcomplete(M),
guard: {T},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
istype: istype(T),
cand: A c∧ B,
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
metric: metric(X),
and: P ∧ Q,
metric-subspace: metric-subspace(X;d;A),
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
int_term_value_subtract_lemma,
subtract_wf,
m-unique-limit,
radd-int-fractions,
mul_nat_plus,
mul_bounds_1b,
rleq-int-fractions,
radd_functionality,
radd_functionality_wrt_rleq,
mdist-triangle-inequality,
istype-le,
int_term_value_mul_lemma,
itermMultiply_wf,
istype-less_than,
int_term_value_add_lemma,
itermAdd_wf,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req_weakening,
mdist-symm,
rleq_functionality,
req-iff-rsub-is-0,
itermSubtract_wf,
rleq_weakening,
rleq_weakening_equal,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
rleq_functionality_wrt_implies,
rless_wf,
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,
nat_properties,
rless-int,
rdiv_wf,
mdist_wf,
sq_stable__rleq,
nat_plus_wf,
mconverges-to_wf,
istype-nat,
mcauchy_wf,
nat_wf,
istype-universe,
metric_wf,
metric-subspace_wf,
mk-metric-space_wf,
mcomplete_wf,
m-closed-subspace_wf,
int-to-real_wf,
req_wf,
radd_wf,
rleq_wf,
real_wf,
subtype_rel_dep_function
Rules used in proof :
multiplyEquality,
dependent_set_memberFormation_alt,
equalityIstype,
addEquality,
promote_hyp,
imageElimination,
baseClosed,
imageMemberEquality,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
unionElimination,
inrFormation_alt,
closedConclusion,
dependent_pairFormation_alt,
equalitySymmetry,
equalityTransitivity,
independent_functionElimination,
functionExtensionality,
dependent_functionElimination,
universeEquality,
instantiate,
natural_numberEquality,
functionIsType,
productIsType,
independent_pairFormation,
inhabitedIsType,
because_Cache,
independent_isectElimination,
universeIsType,
hypothesis,
functionEquality,
lambdaEquality_alt,
sqequalRule,
isectElimination,
extract_by_obid,
introduction,
applyEquality,
hypothesisEquality,
dependent_set_memberEquality_alt,
rename,
setElimination,
thin,
productElimination,
sqequalHypSubstitution,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[X:Type]
\mforall{}d:metric(X)
(mcomplete(X with d)
{}\mRightarrow{} (\mforall{}[A:Type]. (metric-subspace(X;d;A) {}\mRightarrow{} (m-closed-subspace(X;d;A) \mLeftarrow{}{}\mRightarrow{} mcomplete(A with d)))))
Date html generated:
2019_10_30-AM-06_49_51
Last ObjectModification:
2019_10_23-PM-07_07_13
Theory : reals
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