Nuprl Lemma : fun-series-sum_wf
∀[I:Interval]. ∀[f:ℕ ⟶ I ⟶ℝ]. ∀cnv:Σn.f[n;x]↓ for x ∈ I. ∀z:{z:ℝ| z ∈ I} . (Σn.f[n](z) ∈ ℝ)
Proof
Definitions occuring in Statement :
fun-series-sum: Σn.f[n](z),
fun-series-converges: Σn.f[n; x]↓ for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
real: ℝ,
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
fun-series-sum: Σn.f[n](z),
fun-series-converges: Σn.f[n; x]↓ for x ∈ I,
fun-converges: λn.f[n; x]↓ for x ∈ I),
exists: ∃x:A. B[x],
pi1: fst(t),
rfun: I ⟶ℝ,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B
Lemmas referenced :
i-member_wf,
set_wf,
real_wf,
fun-series-converges_wf,
rfun_wf,
nat_wf,
interval_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
setElimination,
thin,
rename,
sqequalRule,
sqequalHypSubstitution,
productElimination,
applyEquality,
hypothesisEquality,
dependent_set_memberEquality,
hypothesis,
lemma_by_obid,
isectElimination,
lambdaEquality,
setEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
functionEquality,
isect_memberEquality
Latex:
\mforall{}[I:Interval]. \mforall{}[f:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}]. \mforall{}cnv:\mSigma{}n.f[n;x]\mdownarrow{} for x \mmember{} I. \mforall{}z:\{z:\mBbbR{}| z \mmember{} I\} . (\mSigma{}n.f[n](z) \mmember{} \mBbbR{})
Date html generated:
2016_05_18-AM-09_55_35
Last ObjectModification:
2015_12_27-PM-11_07_38
Theory : reals
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