Nuprl Lemma : unique-one-dim-vs-map
∀K:CRng. ∀vs:VectorSpace(K). ∀v:Point(vs). ∃!h:one-dim-vs(K) ⟶ vs. ((h 1) = v ∈ Point(vs))
Proof
Definitions occuring in Statement :
vs-map: A ⟶ B,
one-dim-vs: one-dim-vs(K),
vector-space: VectorSpace(K),
vs-point: Point(vs),
exists!: ∃!x:T. P[x],
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T,
crng: CRng,
rng_one: 1
Definitions unfolded in proof :
all: ∀x:A. B[x],
exists!: ∃!x:T. P[x],
exists: ∃x:A. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
crng: CRng,
rng: Rng,
implies: P ⇒ Q,
vs-map: A ⟶ B,
subtype_rel: A ⊆r B,
rng_car: |r|,
pi1: fst(t),
vs-point: Point(vs),
record-select: r.x,
one-dim-vs: one-dim-vs(K),
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,
vs-mul: a * x,
top: Top,
vs-add: x + y,
infix_ap: x f y,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
vs-mul-one,
rng_one_wf,
subtype_rel_self,
vs-point_wf,
one-dim-vs_wf,
vs-map_wf,
vector-space_wf,
crng_wf,
rec_select_update_lemma,
istype-void,
vs-mul_wf,
rng_car_wf,
vs-mul-add,
vs-mul-mul,
rng_plus_wf,
vs-add_wf,
rng_times_wf,
vs-map-eq,
rng_times_one,
equal_wf,
squash_wf,
true_wf,
istype-universe,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
dependent_pairFormation_alt,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_pairFormation,
equalityIstype,
because_Cache,
applyEquality,
universeIsType,
productIsType,
functionIsType,
inhabitedIsType,
dependent_functionElimination,
dependent_set_memberEquality_alt,
isect_memberEquality_alt,
voidElimination,
lambdaEquality_alt,
equalitySymmetry,
independent_isectElimination,
functionExtensionality,
productElimination,
imageElimination,
equalityTransitivity,
instantiate,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
applyLambdaEquality
Latex:
\mforall{}K:CRng. \mforall{}vs:VectorSpace(K). \mforall{}v:Point(vs). \mexists{}!h:one-dim-vs(K) {}\mrightarrow{} vs. ((h 1) = v)
Date html generated:
2019_10_31-AM-06_27_11
Last ObjectModification:
2019_08_02-PM-04_08_21
Theory : linear!algebra
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