Nuprl Lemma : series-sum_functionality
∀x,y:ℕ ⟶ ℝ. ∀a,b:ℝ. ({Σn.x[n] = a ⇒ Σn.y[n] = b}) supposing ((a = b) and (∀n:ℕ. (x[n] = y[n])))
Proof
Definitions occuring in Statement :
series-sum: Σn.x[n] = a,
req: x = y,
real: ℝ,
nat: ℕ,
uimplies: b supposing a,
guard: {T},
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],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q,
guard: {T},
series-sum: Σn.x[n] = a,
so_lambda: λ2x.t[x],
nat: ℕ,
subtype_rel: A ⊆r B,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
nat_wf,
converges-to_functionality,
rsum_wf,
int_seg_subtype_nat,
false_wf,
int_seg_wf,
req_wf,
all_wf,
real_wf,
le_wf,
req_weakening,
req_functionality,
rsum_functionality2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
functionExtensionality,
hypothesis,
independent_functionElimination,
rename,
natural_numberEquality,
setElimination,
addEquality,
independent_isectElimination,
independent_pairFormation,
because_Cache,
functionEquality,
dependent_set_memberEquality,
intEquality,
productElimination
Latex:
\mforall{}x,y:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}a,b:\mBbbR{}. (\{\mSigma{}n.x[n] = a {}\mRightarrow{} \mSigma{}n.y[n] = b\}) supposing ((a = b) and (\mforall{}n:\mBbbN{}. (x[n] = y[n])))
Date html generated:
2016_10_26-AM-09_19_27
Last ObjectModification:
2016_08_26-PM-01_50_35
Theory : reals
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