Nuprl Lemma : vs-lift-append
∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[S:Type]. ∀[f:S ⟶ Point(vs)]. ∀[fs,fs':bag(|K| × S)].
(vs-lift(vs;f;fs + fs') = vs-lift(vs;f;fs) + vs-lift(vs;f;fs') ∈ Point(vs))
Proof
Definitions occuring in Statement :
vs-lift: vs-lift(vs;f;fs),
vs-add: x + y,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
rng_car: |r|,
bag-append: as + bs,
bag: bag(T)
Definitions unfolded in proof :
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
rng: Rng,
vs-lift: vs-lift(vs;f;fs),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_wf,
vector-space_wf,
vs-point_wf,
bag_wf,
vs-mul_wf,
rng_car_wf,
vs-bag-add-append
Rules used in proof :
dependent_functionElimination,
universeEquality,
functionEquality,
axiomEquality,
isect_memberEquality,
functionExtensionality,
applyEquality,
independent_pairEquality,
productElimination,
spreadEquality,
lambdaEquality,
sqequalRule,
cumulativity,
hypothesis,
because_Cache,
rename,
setElimination,
productEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[S:Type]. \mforall{}[f:S {}\mrightarrow{} Point(vs)]. \mforall{}[fs,fs':bag(|K| \mtimes{} S)].
(vs-lift(vs;f;fs + fs') = vs-lift(vs;f;fs) + vs-lift(vs;f;fs'))
Date html generated:
2018_05_22-PM-09_44_47
Last ObjectModification:
2018_01_09-AM-11_00_48
Theory : linear!algebra
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