Nuprl Lemma : series-converges-rmul
∀c:ℝ. ∀x:ℕ ⟶ ℝ. (Σn.x[n]↓ ⇒ Σn.x[n] * c↓)
Proof
Definitions occuring in Statement :
series-converges: Σn.x[n]↓,
rmul: a * b,
real: ℝ,
nat: ℕ,
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,
series-converges: Σn.x[n]↓,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ
Lemmas referenced :
rmul_wf,
series-sum-linear3,
nat_wf,
series-sum_wf,
series-converges_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
cut,
lemma_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
independent_functionElimination,
functionEquality
Latex:
\mforall{}c:\mBbbR{}. \mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}. (\mSigma{}n.x[n]\mdownarrow{} {}\mRightarrow{} \mSigma{}n.x[n] * c\mdownarrow{})
Date html generated:
2016_05_18-AM-07_58_49
Last ObjectModification:
2015_12_28-AM-01_09_15
Theory : reals
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