Nuprl Lemma : vs-lift-formal-sum-mul
∀[K:CRng]. ∀[vs:VectorSpace(K)]. ∀[S:Type]. ∀[f:S ⟶ Point(vs)]. ∀[k:|K|]. ∀[x:basic-formal-sum(K;S)].
(vs-lift(vs;f;k * x) = k * vs-lift(vs;f;x) ∈ Point(vs))
Proof
Definitions occuring in Statement :
vs-lift: vs-lift(vs;f;fs),
formal-sum-mul: k * x,
basic-formal-sum: basic-formal-sum(K;S),
vs-mul: a * x,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
infix_ap: x f y,
cand: A c∧ B,
all: ∀x:A. B[x],
top: Top,
vs-bag-add: Σ{f[b] | b ∈ bs},
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
so_apply: x[s],
so_lambda: λ2x.t[x],
rng: Rng,
crng: CRng,
prop: ℙ,
squash: ↓T,
formal-sum-mul: k * x,
vs-lift: vs-lift(vs;f;fs),
basic-formal-sum: basic-formal-sum(K;S),
member: t ∈ T,
uall: ∀[x:A]. B[x],
assoc: Assoc(T;op),
comm: Comm(T;op)
Lemmas referenced :
vs-add-comm,
vs-mon_assoc,
crng_wf,
vector-space_wf,
basic-formal-sum_wf,
vs-mul-mul,
vs-0_wf,
vs-add_wf,
bag_wf,
comm_wf,
assoc_wf,
bag-summation_wf,
bag-subtype-list,
bag-summation-map,
iff_weakening_equal,
vs-bag-add-mul,
rng_times_wf,
infix_ap_wf,
bag-map_wf,
vs-mul_wf,
rng_car_wf,
vs-bag-add_wf,
vs-point_wf,
true_wf,
squash_wf,
equal_wf
Rules used in proof :
axiomEquality,
functionEquality,
independent_pairFormation,
dependent_functionElimination,
voidEquality,
voidElimination,
isect_memberEquality,
independent_functionElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
spreadEquality,
independent_pairEquality,
functionExtensionality,
productElimination,
sqequalRule,
cumulativity,
because_Cache,
productEquality,
rename,
setElimination,
universeEquality,
equalitySymmetry,
hypothesis,
equalityTransitivity,
hypothesisEquality,
isectElimination,
extract_by_obid,
imageElimination,
lambdaEquality,
thin,
applyEquality,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[K:CRng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[S:Type]. \mforall{}[f:S {}\mrightarrow{} Point(vs)]. \mforall{}[k:|K|].
\mforall{}[x:basic-formal-sum(K;S)].
(vs-lift(vs;f;k * x) = k * vs-lift(vs;f;x))
Date html generated:
2018_05_22-PM-09_45_59
Last ObjectModification:
2018_01_09-PM-00_23_55
Theory : linear!algebra
Home
Index