Nuprl Lemma : eq-0-in-relative-free-vs
∀[K:CRng]. ∀[S,T:Type].
∀[x:Point(free-vs(K;S))]. x = 0 ∈ Point(relative-free-vs(K;S;T)) supposing ↓fs-in-subtype(K;S;T;x)
supposing strong-subtype(T;S)
Proof
Definitions occuring in Statement :
relative-free-vs: relative-free-vs(K;S;T),
free-vs: free-vs(K;S),
fs-in-subtype: fs-in-subtype(K;S;T;f),
vs-0: 0,
vs-point: Point(vs),
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
squash: ↓T,
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
relative-free-vs: relative-free-vs(K;S;T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
so_lambda: λ2x.t[x],
crng: CRng,
rng: Rng,
subtype_rel: A ⊆r B,
vs-point: Point(vs),
record-select: r.x,
free-vs: free-vs(K;S),
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,
formal-sum: formal-sum(K;S),
quotient: x,y:A//B[x; y],
so_apply: x[s],
all: ∀x:A. B[x],
prop: ℙ
Lemmas referenced :
eq-0-in-vs-quotient,
free-vs_wf,
fs-in-subtype_wf,
subtype_rel_self,
formal-sum_wf,
fs-in-subtype-subspace,
squash_wf,
vs-point_wf,
strong-subtype_wf,
istype-universe,
crng_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaEquality_alt,
setElimination,
rename,
independent_isectElimination,
applyEquality,
because_Cache,
inhabitedIsType,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
instantiate,
universeEquality
Latex:
\mforall{}[K:CRng]. \mforall{}[S,T:Type].
\mforall{}[x:Point(free-vs(K;S))]. x = 0 supposing \mdownarrow{}fs-in-subtype(K;S;T;x) supposing strong-subtype(T;S)
Date html generated:
2019_10_31-AM-06_30_34
Last ObjectModification:
2019_08_20-PM-06_01_49
Theory : linear!algebra
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