Nuprl Lemma : vs-lift-zero-bfs
∀[K:Rng]. ∀[S:Type]. ∀[ss:bag(S)]. ∀[vs:VectorSpace(K)]. ∀[f:S ⟶ Point(vs)]. (vs-lift(vs;f;0 * ss) = 0 ∈ Point(vs))
Proof
Definitions occuring in Statement :
vs-lift: vs-lift(vs;f;fs),
zero-bfs: 0 * ss,
vs-0: 0,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
bag: bag(T)
Definitions unfolded in proof :
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
true: True,
rng: Rng,
prop: ℙ,
squash: ↓T,
so_apply: x[s],
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
top: Top,
vs-bag-add: Σ{f[b] | b ∈ bs},
vs-lift: vs-lift(vs;f;fs),
zero-bfs: 0 * ss,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_wf,
bag_wf,
vector-space_wf,
vs-bag-add_wf,
vs-mul-zero,
iff_weakening_equal,
vs-0_wf,
rng_zero_wf,
vs-bag-add-mul,
vs-point_wf,
true_wf,
squash_wf,
equal_wf,
bag-subtype-list,
bag-summation-map
Rules used in proof :
axiomEquality,
functionEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
because_Cache,
cumulativity,
functionExtensionality,
rename,
setElimination,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
hypothesis,
dependent_functionElimination,
applyEquality,
hypothesisEquality,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[K:Rng]. \mforall{}[S:Type]. \mforall{}[ss:bag(S)]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[f:S {}\mrightarrow{} Point(vs)].
(vs-lift(vs;f;0 * ss) = 0)
Date html generated:
2018_05_22-PM-09_44_49
Last ObjectModification:
2018_01_09-PM-01_00_35
Theory : linear!algebra
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