Nuprl Lemma : free-vs-maps-eq
∀[S:Type]. ∀[K:CRng]. ∀[vs:VectorSpace(K)]. ∀[f,g:free-vs(K;S) ⟶ vs].
f = g ∈ free-vs(K;S) ⟶ vs supposing ∀s:S. ((f <s>) = (g <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),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
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],
uimplies: b supposing a,
crng: CRng,
rng: Rng,
vs-map: A ⟶ B,
exists!: ∃!x:T. P[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
free-vs-property,
vs-point_wf,
free-vs-inc_wf,
vs-map_wf,
free-vs_wf,
istype-universe,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
functionIsType,
universeIsType,
equalityIstype,
setElimination,
rename,
applyEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
because_Cache,
instantiate,
universeEquality,
lambdaEquality_alt,
productElimination,
independent_functionElimination,
lambdaFormation_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}[S:Type]. \mforall{}[K:CRng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[f,g:free-vs(K;S) {}\mrightarrow{} vs].
f = g supposing \mforall{}s:S. ((f <s>) = (g <s>))
Date html generated:
2019_10_31-AM-06_29_47
Last ObjectModification:
2019_08_01-AM-10_59_41
Theory : linear!algebra
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