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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