Nuprl Lemma : sum-in-vs-add
∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[n,m:ℤ]. ∀[f,g:{n..m + 1-} ⟶ Point(vs)].
(Σ{f[i] + g[i] | n≤i≤m} = Σ{f[i] | n≤i≤m} + Σ{g[i] | n≤i≤m} ∈ Point(vs))
Proof
Definitions occuring in Statement :
sum-in-vs: Σ{f[i] | n≤i≤m},
vs-add: x + y,
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 :
sum-in-vs: Σ{f[i] | n≤i≤m},
uall: ∀[x:A]. B[x],
member: t ∈ T,
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],
so_apply: x[s],
so_lambda: λ2x.t[x],
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
from-upto_wf,
list-subtype-bag,
le_wf,
less_than_wf,
int_seg_wf,
istype-le,
istype-less_than,
vs-point_wf,
istype-int,
vector-space_wf,
rng_wf,
vs-add_wf,
vs-bag-add_wf,
equal_wf,
vs-bag-add-add,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
addEquality,
natural_numberEquality,
hypothesis,
applyEquality,
setEquality,
intEquality,
productEquality,
independent_isectElimination,
lambdaEquality_alt,
setIsType,
inhabitedIsType,
productIsType,
because_Cache,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
functionIsType,
universeIsType,
setElimination,
rename,
dependent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination
Latex:
\mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[n,m:\mBbbZ{}]. \mforall{}[f,g:\{n..m + 1\msupminus{}\} {}\mrightarrow{} Point(vs)].
(\mSigma{}\{f[i] + g[i] | n\mleq{}i\mleq{}m\} = \mSigma{}\{f[i] | n\mleq{}i\mleq{}m\} + \mSigma{}\{g[i] | n\mleq{}i\mleq{}m\})
Date html generated:
2019_10_31-AM-06_26_06
Last ObjectModification:
2019_08_08-PM-02_17_25
Theory : linear!algebra
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