Nuprl Lemma : vs-lift-inc
∀[S:Type]. ∀[K:Rng]. ∀[s:S]. ∀[vs:VectorSpace(K)]. ∀[f:S ⟶ Point(vs)]. (vs-lift(vs;f;<s>) = (f s) ∈ Point(vs))
Proof
Definitions occuring in Statement :
free-vs-inc: <s>,
vs-lift: vs-lift(vs;f;fs),
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
comm: Comm(T;op),
ident: Ident(T;op;id),
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
infix_ap: x f y,
assoc: Assoc(T;op),
monoid_p: IsMonoid(T;op;id),
cand: A c∧ B,
and: P ∧ Q,
uimplies: b supposing a,
rng: Rng,
prop: ℙ,
squash: ↓T,
vs-bag-add: Σ{f[b] | b ∈ bs},
vs-lift: vs-lift(vs;f;fs),
free-vs-inc: <s>,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_wf,
vector-space_wf,
vs-mul-one,
rng_one_wf,
vs-mul_wf,
rng_car_wf,
vs-add-comm,
vs-mon_ident,
iff_weakening_equal,
vs-mon_assoc,
vs-0_wf,
vs-add_wf,
bag-summation-single,
vs-point_wf,
true_wf,
squash_wf,
equal_wf
Rules used in proof :
dependent_functionElimination,
functionEquality,
universeEquality,
functionExtensionality,
spreadEquality,
cumulativity,
productEquality,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
independent_functionElimination,
productElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
independent_isectElimination,
rename,
setElimination,
because_Cache,
equalitySymmetry,
hypothesis,
equalityTransitivity,
hypothesisEquality,
isectElimination,
extract_by_obid,
imageElimination,
sqequalHypSubstitution,
lambdaEquality,
thin,
applyEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[S:Type]. \mforall{}[K:Rng]. \mforall{}[s:S]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[f:S {}\mrightarrow{} Point(vs)]. (vs-lift(vs;f;<s>) = (f s))
Date html generated:
2018_05_22-PM-09_46_19
Last ObjectModification:
2018_01_09-PM-00_27_56
Theory : linear!algebra
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