Nuprl Lemma : cauchy-limit_wf
∀[x:ℕ ⟶ ℝ]. ∀[c:cauchy(n.x[n])]. (cauchy-limit(n.x[n];c) ∈ ℝ)
Proof
Definitions occuring in Statement :
cauchy-limit: cauchy-limit(n.x[n];c),
cauchy: cauchy(n.x[n]),
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_apply: x[s],
cauchy-limit: cauchy-limit(n.x[n];c),
converges-iff-cauchy-ext,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
pi2: snd(t),
pi1: fst(t),
subtype_rel: A ⊆r B,
converges: x[n]↓ as n→∞,
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
converges-iff-cauchy-ext,
all_wf,
nat_wf,
real_wf,
iff_wf,
converges_wf,
cauchy_wf,
equal_wf,
pi1_wf_top,
exists_wf,
converges-to_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
thin,
instantiate,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
functionEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
isect_memberEquality,
Error :applyLambdaEquality,
productElimination,
independent_pairEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[x:\mBbbN{} {}\mrightarrow{} \mBbbR{}]. \mforall{}[c:cauchy(n.x[n])]. (cauchy-limit(n.x[n];c) \mmember{} \mBbbR{})
Date html generated:
2016_10_26-AM-09_15_48
Last ObjectModification:
2016_08_29-PM-06_11_28
Theory : reals
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