Nuprl Lemma : continuous-series-sum
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. ∀cnv:Σn.f[n;x]↓ for x ∈ I.
((∀n:ℕ. f[n;x] continuous for x ∈ I) ⇒ Σn.f[n](y) continuous for y ∈ I)
Proof
Definitions occuring in Statement :
fun-series-sum: Σn.f[n](z),
fun-series-converges: Σn.f[n; x]↓ for x ∈ I,
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
interval: Interval,
nat: ℕ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
fun-series-converges: Σn.f[n; x]↓ for x ∈ I,
fun-converges: λn.f[n; x]↓ for x ∈ I),
exists: ∃x:A. B[x],
fun-series-sum: Σn.f[n](z),
pi1: fst(t),
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A
Lemmas referenced :
all_wf,
nat_wf,
continuous_wf,
rfun_wf,
real_wf,
i-member_wf,
fun-series-converges_wf,
interval_wf,
fun-converges-to-continuous,
rsum_wf,
int_seg_subtype_nat,
false_wf,
int_seg_wf,
continuous-sum
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
lemma_by_obid,
isectElimination,
hypothesis,
lambdaEquality,
hypothesisEquality,
applyEquality,
setEquality,
because_Cache,
functionEquality,
natural_numberEquality,
setElimination,
rename,
addEquality,
independent_isectElimination,
independent_pairFormation,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}I:Interval. \mforall{}f:\mBbbN{} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}. \mforall{}cnv:\mSigma{}n.f[n;x]\mdownarrow{} for x \mmember{} I.
((\mforall{}n:\mBbbN{}. f[n;x] continuous for x \mmember{} I) {}\mRightarrow{} \mSigma{}n.f[n](y) continuous for y \mmember{} I)
Date html generated:
2016_05_18-AM-09_55_53
Last ObjectModification:
2015_12_27-PM-11_07_52
Theory : reals
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