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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