Nuprl Lemma : sum-in-vs-neg
∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[n,m:ℤ]. ∀[f:{n..m + 1-} ⟶ Point(vs)].
(Σ{-(f[i]) | n≤i≤m} = -(Σ{f[i] | n≤i≤m}) ∈ Point(vs))
Proof
Definitions occuring in Statement :
sum-in-vs: Σ{f[i] | n≤i≤m},
vs-neg: -(x),
vector-space: VectorSpace(K),
vs-point: Point(vs),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
vs-neg: -(x),
rng: Rng,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
int_seg_wf,
vs-point_wf,
istype-int,
vector-space_wf,
rng_wf,
sum-in-vs_wf,
vs-mul_wf,
rng_minus_wf,
rng_one_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
sum-in-vs-mul,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
hypothesis,
functionIsType,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
addEquality,
natural_numberEquality,
setElimination,
rename,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
dependent_functionElimination,
because_Cache,
lambdaEquality_alt,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[n,m:\mBbbZ{}]. \mforall{}[f:\{n..m + 1\msupminus{}\} {}\mrightarrow{} Point(vs)].
(\mSigma{}\{-(f[i]) | n\mleq{}i\mleq{}m\} = -(\mSigma{}\{f[i] | n\mleq{}i\mleq{}m\}))
Date html generated:
2019_10_31-AM-06_26_11
Last ObjectModification:
2019_08_08-PM-02_29_08
Theory : linear!algebra
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