Nuprl Lemma : free-vs-dim-1
∀S:Type. (S ⇒ (∀K:CRng. free-vs(K;S) ≅ one-dim-vs(K) supposing ∀x,y:S. (x = y ∈ S)))
Proof
Definitions occuring in Statement :
free-vs: free-vs(K;S),
vs-iso: A ≅ B,
one-dim-vs: one-dim-vs(K),
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
crng: CRng,
iff: P ⇐⇒ Q,
and: P ∧ Q,
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,
rng_car: |r|,
pi1: fst(t),
exists: ∃x:A. B[x],
rng: Rng,
so_lambda: λ2x.t[x],
prop: ℙ,
vs-map: A ⟶ B,
subtype_rel: A ⊆r B,
so_apply: x[s],
exists!: ∃!x:T. P[x],
cand: A c∧ B,
squash: ↓T,
true: True,
guard: {T},
rev_implies: P ⇐ Q
Lemmas referenced :
free-vs-unique,
one-dim-vs_wf,
rng_one_wf,
vs-point_wf,
vector-space_wf,
exists!_wf,
vs-map_wf,
equal_wf,
subtype_rel_self,
vs-iso_inversion,
free-vs_wf,
crng_wf,
istype-universe,
unique-one-dim-vs-map,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
functionIsTypeImplies,
inhabitedIsType,
rename,
extract_by_obid,
isectElimination,
setElimination,
productElimination,
independent_functionElimination,
dependent_pairFormation_alt,
functionIsType,
because_Cache,
universeIsType,
functionEquality,
applyEquality,
equalityIstype,
instantiate,
universeEquality,
independent_pairFormation,
promote_hyp,
productIsType,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}S:Type. (S {}\mRightarrow{} (\mforall{}K:CRng. free-vs(K;S) \mcong{} one-dim-vs(K) supposing \mforall{}x,y:S. (x = y)))
Date html generated:
2019_10_31-AM-06_30_39
Last ObjectModification:
2019_08_02-PM-04_04_40
Theory : linear!algebra
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