Nuprl Lemma : rng_lsum-upto
∀[n:ℤ]. ∀[r:Rng]. ∀[f:ℕn ⟶ |r|]. (Σ{r} x ∈ upto(n). f[x] = (Σ(r) 0 ≤ i < n. f[i]) ∈ |r|)
Proof
Definitions occuring in Statement :
rng_lsum: Σ{r} x ∈ as. f[x],
upto: upto(n),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
rng_sum: rng_sum,
rng: Rng,
rng_car: |r|
Definitions unfolded in proof :
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
so_apply: x[s],
so_lambda: λ2x.t[x],
rng: Rng,
prop: ℙ,
squash: ↓T,
member: t ∈ T,
uall: ∀[x:A]. B[x],
upto: upto(n)
Lemmas referenced :
rng_wf,
iff_weakening_equal,
rng_sum_wf,
int_seg_wf,
rng_lsum-from-upto,
rng_car_wf,
true_wf,
squash_wf,
equal_wf
Rules used in proof :
intEquality,
axiomEquality,
isect_memberEquality,
functionEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
because_Cache,
functionExtensionality,
natural_numberEquality,
rename,
setElimination,
universeEquality,
equalitySymmetry,
hypothesis,
equalityTransitivity,
hypothesisEquality,
isectElimination,
extract_by_obid,
imageElimination,
sqequalHypSubstitution,
lambdaEquality,
thin,
applyEquality,
cut,
introduction,
isect_memberFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[r:Rng]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} |r|]. (\mSigma{}\{r\} x \mmember{} upto(n). f[x] = (\mSigma{}(r) 0 \mleq{} i < n. f[i]))
Date html generated:
2018_05_21-PM-09_33_13
Last ObjectModification:
2017_12_15-AM-10_10_49
Theory : matrices
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