Nuprl Lemma : double-sum-in-vs
∀[n,m:ℤ]. ∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[l,u:{n..m + 1-} ⟶ ℤ]. ∀[h:x:{n..m + 1-} ⟶ {l[x]..u[x] + 1-} ⟶ Point(vs)].
(Σ{Σ{h[x;y] | l[x]≤y≤u[x]} | n≤x≤m}
= Σ{h[fst(p);snd(p)] | p ∈ ⋃x∈[n, m + 1).bag-map(λy.<x, y>;[l[x], u[x] + 1))}
∈ Point(vs))
Proof
Definitions occuring in Statement :
sum-in-vs: Σ{f[i] | n≤i≤m},
vs-bag-add: Σ{f[b] | b ∈ bs},
vector-space: VectorSpace(K),
vs-point: Point(vs),
from-upto: [n, m),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
lambda: λx.A[x],
function: x:A ⟶ B[x],
pair: <a, b>,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
rng: Rng,
bag-combine: ⋃x∈bs.f[x],
bag-map: bag-map(f;bs)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
sum-in-vs: Σ{f[i] | n≤i≤m},
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
uimplies: b supposing a,
int_seg: {i..j-},
lelt: i ≤ j < k,
rng: Rng,
all: ∀x:A. B[x]
Lemmas referenced :
vs-double-bag-add,
int_seg_wf,
from-upto_wf,
list-subtype-bag,
le_wf,
less_than_wf,
istype-le,
istype-less_than,
vs-point_wf,
vector-space_wf,
rng_wf,
istype-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
lambdaEquality_alt,
applyEquality,
addEquality,
natural_numberEquality,
hypothesis,
universeIsType,
setEquality,
intEquality,
productEquality,
independent_isectElimination,
setIsType,
inhabitedIsType,
productIsType,
functionIsType,
setElimination,
rename,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
dependent_functionElimination
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[l,u:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbZ{}]. \mforall{}[h:x:\{n..m + 1\msupminus{}\}
{}\mrightarrow{} \{l[x]..u[x] + 1\msupminus{}\}
{}\mrightarrow{} Point(vs)].
(\mSigma{}\{\mSigma{}\{h[x;y] | l[x]\mleq{}y\mleq{}u[x]\} | n\mleq{}x\mleq{}m\}
= \mSigma{}\{h[fst(p);snd(p)] | p \mmember{} \mcup{}x\mmember{}[n, m + 1).bag-map(\mlambda{}y.<x, y>[l[x], u[x] + 1))\})
Date html generated:
2019_10_31-AM-06_26_31
Last ObjectModification:
2019_08_09-PM-01_31_53
Theory : linear!algebra
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