Nuprl Lemma : sum-in-vs_wf
∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[n,m:ℤ]. ∀[f:{n..m + 1-} ⟶ Point(vs)]. (Σ{f[i] | n≤i≤m} ∈ Point(vs))
Proof
Definitions occuring in Statement :
sum-in-vs: Σ{f[i] | n≤i≤m},
vector-space: VectorSpace(K),
vs-point: Point(vs),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
rng: Rng
Definitions unfolded in proof :
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,
sum-in-vs: Σ{f[i] | n≤i≤m},
rng: Rng,
all: ∀x:A. B[x]
Lemmas referenced :
vs-bag-add_wf,
int_seg_wf,
from-upto_wf,
list-subtype-bag,
le_wf,
less_than_wf,
istype-le,
istype-less_than,
vs-point_wf,
istype-int,
vector-space_wf,
rng_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
addEquality,
natural_numberEquality,
applyEquality,
setEquality,
intEquality,
productEquality,
independent_isectElimination,
lambdaEquality_alt,
sqequalRule,
setIsType,
inhabitedIsType,
productIsType,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionIsType,
universeIsType,
setElimination,
rename,
dependent_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\} \mmember{} Point(vs))
Date html generated:
2019_10_31-AM-06_26_00
Last ObjectModification:
2019_08_08-AM-11_57_38
Theory : linear!algebra
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