Nuprl Lemma : free-vs-unique
∀S:Type. ∀K:CRng. ∀V:VectorSpace(K).
(∃i:S ⟶ Point(V). ∀vs:VectorSpace(K). ∀f:S ⟶ Point(vs). ∃!h:V ⟶ vs. ∀s:S. ((h (i s)) = (f s) ∈ Point(vs))
⇐⇒ V ≅ free-vs(K;S))
Proof
Definitions occuring in Statement :
free-vs: free-vs(K;S),
vs-iso: A ≅ B,
vs-map: A ⟶ B,
vector-space: VectorSpace(K),
vs-point: Point(vs),
exists!: ∃!x:T. P[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
crng: CRng,
rng: Rng,
so_lambda: λ2x.t[x],
prop: ℙ,
vs-map: A ⟶ B,
so_apply: x[s],
rev_implies: P ⇐ Q,
exists!: ∃!x:T. P[x],
cand: A c∧ B,
free-vs-inc: <s>,
single-bag: {x},
cons: [a / b],
compose: f o g,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
vs-iso: A ≅ B
Lemmas referenced :
vs-point_wf,
exists!_wf,
vs-map_wf,
equal_wf,
vs-iso_wf,
free-vs_wf,
vector-space_wf,
crng_wf,
istype-universe,
free-vs-property,
free-vs-inc_wf,
vs-add_wf,
vs-mul_wf,
rng_car_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
vs-map-compose
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
sqequalRule,
productIsType,
functionIsType,
universeIsType,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesis,
inhabitedIsType,
because_Cache,
lambdaEquality_alt,
functionEquality,
applyEquality,
dependent_functionElimination,
instantiate,
universeEquality,
productElimination,
dependent_set_memberEquality_alt,
equalityIsType1,
independent_functionElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
applyLambdaEquality,
dependent_pairFormation_alt,
functionExtensionality,
hyp_replacement,
functionExtensionality_alt
Latex:
\mforall{}S:Type. \mforall{}K:CRng. \mforall{}V:VectorSpace(K).
(\mexists{}i:S {}\mrightarrow{} Point(V). \mforall{}vs:VectorSpace(K). \mforall{}f:S {}\mrightarrow{} Point(vs). \mexists{}!h:V {}\mrightarrow{} vs. \mforall{}s:S. ((h (i s)) = (f s))
\mLeftarrow{}{}\mRightarrow{} V \mcong{} free-vs(K;S))
Date html generated:
2019_10_31-AM-06_29_57
Last ObjectModification:
2019_08_02-PM-04_05_04
Theory : linear!algebra
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