Nuprl Lemma : free-vs-property
∀[S:Type]. ∀K:CRng. ∀vs:VectorSpace(K). ∀f:S ⟶ Point(vs). ∃!h:free-vs(K;S) ⟶ vs. ∀s:S. ((h <s>) = (f s) ∈ Point(vs))
Proof
Definitions occuring in Statement :
free-vs-inc: <s>,
free-vs: free-vs(K;S),
vs-map: A ⟶ B,
vector-space: VectorSpace(K),
vs-point: Point(vs),
exists!: ∃!x:T. P[x],
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
exists!: ∃!x:T. P[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
crng: CRng,
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
rng: Rng,
uimplies: b supposing a,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
vs-map: A ⟶ B
Lemmas referenced :
vs-lift_wf-vs-map,
vs-lift-unique,
vs-lift-inc,
equal_wf,
squash_wf,
true_wf,
istype-universe,
vs-map_wf,
free-vs_wf,
subtype_rel_self,
iff_weakening_equal,
vs-point_wf,
free-vs-inc_wf,
vector-space_wf,
crng_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
setElimination,
rename,
universeIsType,
independent_pairFormation,
applyEquality,
lambdaEquality_alt,
imageElimination,
inhabitedIsType,
instantiate,
universeEquality,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
because_Cache,
functionIsType,
equalityIstype,
productIsType,
dependent_functionElimination
Latex:
\mforall{}[S:Type]
\mforall{}K:CRng. \mforall{}vs:VectorSpace(K). \mforall{}f:S {}\mrightarrow{} Point(vs). \mexists{}!h:free-vs(K;S) {}\mrightarrow{} vs. \mforall{}s:S. ((h <s>) = (f s))
Date html generated:
2019_10_31-AM-06_29_43
Last ObjectModification:
2019_08_02-PM-04_08_39
Theory : linear!algebra
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