Nuprl Lemma : series-sum-linear1
∀x,y:ℕ ⟶ ℝ. ∀a,b:ℝ. (Σn.x[n] = a ⇒ Σn.y[n] = b ⇒ Σn.x[n] + y[n] = a + b)
Proof
Definitions occuring in Statement :
series-sum: Σn.x[n] = a,
radd: 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 :
series-sum: Σn.x[n] = a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
radd-limit,
rsum_wf,
int_seg_wf,
nat_wf,
converges-to_wf,
int_seg_subtype_nat,
false_wf,
real_wf,
radd_wf,
req_weakening,
converges-to_functionality,
req_functionality,
rsum_linearity1
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
lambdaEquality,
isectElimination,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
addEquality,
independent_functionElimination,
independent_isectElimination,
independent_pairFormation,
functionEquality,
productElimination
Latex:
\mforall{}x,y:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}a,b:\mBbbR{}. (\mSigma{}n.x[n] = a {}\mRightarrow{} \mSigma{}n.y[n] = b {}\mRightarrow{} \mSigma{}n.x[n] + y[n] = a + b)
Date html generated:
2016_05_18-AM-07_56_57
Last ObjectModification:
2015_12_28-AM-01_08_33
Theory : reals
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