Nuprl Lemma : vs-lift_wf-vs-map
∀[S:Type]. ∀[K:CRng]. ∀[vs:VectorSpace(K)]. ∀[f:S ⟶ Point(vs)]. (λx.vs-lift(vs;f;x) ∈ free-vs(K;S) ⟶ vs)
Proof
Definitions occuring in Statement :
free-vs: free-vs(K;S),
vs-lift: vs-lift(vs;f;fs),
vs-map: A ⟶ B,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
member: t ∈ T,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
vs-map: A ⟶ B,
subtype_rel: A ⊆r B,
vs-point: Point(vs),
record-select: r.x,
free-vs: free-vs(K;S),
mk-vs: mk-vs,
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
formal-sum: formal-sum(K;S),
quotient: x,y:A//B[x; y],
crng: CRng,
rng: Rng,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x]
Lemmas referenced :
vs-lift_wf2,
subtype_rel_self,
formal-sum_wf,
vs-point_wf,
free-vs_wf,
vs-lift-add,
vs-lift-mul,
rng_car_wf,
vs-add_wf,
vs-mul_wf,
vector-space_wf,
crng_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
setElimination,
rename,
because_Cache,
hypothesis,
universeIsType,
lambdaFormation_alt,
inhabitedIsType,
independent_pairFormation,
productIsType,
functionIsType,
equalityIstype,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
isectIsTypeImplies,
dependent_functionElimination,
instantiate,
universeEquality
Latex:
\mforall{}[S:Type]. \mforall{}[K:CRng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[f:S {}\mrightarrow{} Point(vs)].
(\mlambda{}x.vs-lift(vs;f;x) \mmember{} free-vs(K;S) {}\mrightarrow{} vs)
Date html generated:
2019_10_31-AM-06_29_28
Last ObjectModification:
2019_07_31-PM-04_16_05
Theory : linear!algebra
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