Nuprl Lemma : vs-lift_wf-relative
∀[S,T:Type].
∀[K:CRng]. ∀[vs:VectorSpace(K)]. ∀[f:S ⟶ Point(vs)].
λx.vs-lift(vs;f;x) ∈ relative-free-vs(K;S;T) ⟶ vs supposing ∀t:T. (↓(f t) = 0 ∈ Point(vs))
supposing strong-subtype(T;S)
Proof
Definitions occuring in Statement :
relative-free-vs: relative-free-vs(K;S;T),
vs-lift: vs-lift(vs;f;fs),
vs-map: A ⟶ B,
vs-0: 0,
vector-space: VectorSpace(K),
vs-point: Point(vs),
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
squash: ↓T,
member: t ∈ T,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
crng: CRng,
rng: Rng,
subtype_rel: A ⊆r B,
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
prop: ℙ,
relative-free-vs: relative-free-vs(K;S;T),
so_lambda: λ2x.t[x],
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],
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
exists: ∃x:A. B[x],
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
istype: istype(T),
vs-subspace: vs-subspace(K;vs;x.P[x]),
vs-map: A ⟶ B,
sub-vs: (v:vs | P[v])
Lemmas referenced :
squash_wf,
equal_wf,
vs-point_wf,
vs-0_wf,
vector-space_wf,
crng_wf,
strong-subtype_wf,
istype-universe,
vs-lift_wf-vs-map,
fs-in-subtype-subspace,
vs-map-quotient,
free-vs_wf,
fs-in-subtype_wf,
subtype_rel_self,
formal-sum_wf,
rec_select_update_lemma,
istype-void,
basic-formal-sum_wf,
bfs-equiv_wf,
fs-in-subtype-basic,
subtype_quotient,
bfs-equiv-rel,
true_wf,
vs-lift_wf2,
iff_weakening_equal,
subtype_rel_dep_function,
free-vs-map-into-subspace,
rng_car_wf,
vs-lift-inc,
vs-zero-add,
vs-zero-mul,
vs-add_wf,
rng_sig_wf,
vs-mul_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
functionIsType,
universeIsType,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
applyEquality,
productElimination,
dependent_functionElimination,
inhabitedIsType,
instantiate,
universeEquality,
independent_isectElimination,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
lambdaFormation_alt,
isect_memberEquality_alt,
voidElimination,
pointwiseFunctionalityForEquality,
pertypeElimination,
promote_hyp,
productIsType,
equalityIstype,
sqequalBase,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
independent_pairFormation
Latex:
\mforall{}[S,T:Type].
\mforall{}[K:CRng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[f:S {}\mrightarrow{} Point(vs)].
\mlambda{}x.vs-lift(vs;f;x) \mmember{} relative-free-vs(K;S;T) {}\mrightarrow{} vs supposing \mforall{}t:T. (\mdownarrow{}(f t) = 0)
supposing strong-subtype(T;S)
Date html generated:
2019_10_31-AM-06_31_36
Last ObjectModification:
2019_08_20-PM-05_16_55
Theory : linear!algebra
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