Nuprl Lemma : cauchy-mlimit_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[cmplt:mcomplete(X with d)]. ∀[x:ℕ ⟶ X]. ∀[c:mcauchy(d;n.x n)].
(cauchy-mlimit(cmplt;x;c) ∈ X)
Proof
Definitions occuring in Statement :
cauchy-mlimit: cauchy-mlimit(cmplt;x;c),
mcomplete: mcomplete(M),
mcauchy: mcauchy(d;n.x[n]),
mk-metric-space: X with d,
metric: metric(X),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cauchy-mlimit: cauchy-mlimit(cmplt;x;c),
subtype_rel: A ⊆r B,
mcomplete: mcomplete(M),
mk-metric-space: X with d,
all: ∀x:A. B[x],
implies: P ⇒ Q,
metric: metric(X),
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
mconverges: x[n]↓ as n→∞,
exists: ∃x:A. B[x],
pi1: fst(t)
Lemmas referenced :
subtype_rel_self,
nat_wf,
mcauchy_wf,
istype-nat,
mconverges_wf,
subtype_rel_function,
mcomplete_wf,
mk-metric-space_wf,
metric_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
applyEquality,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
hypothesis,
setElimination,
rename,
because_Cache,
lambdaEquality_alt,
independent_isectElimination,
inhabitedIsType,
lambdaFormation_alt,
productElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
functionIsType,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[cmplt:mcomplete(X with d)]. \mforall{}[x:\mBbbN{} {}\mrightarrow{} X]. \mforall{}[c:mcauchy(d;n.x n)].
(cauchy-mlimit(cmplt;x;c) \mmember{} X)
Date html generated:
2019_10_30-AM-06_42_51
Last ObjectModification:
2019_10_02-AM-10_55_19
Theory : reals
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