Nuprl Lemma : subtype-vs-quotient
∀[K:CRng]. ∀[vs:VectorSpace(K)]. ∀[P:Point(vs) ⟶ ℙ]. Point(vs) ⊆r Point(vs//z.P[z]) supposing vs-subspace(K;vs;z.P[z])
Proof
Definitions occuring in Statement :
vs-quotient: vs//z.P[z],
vs-subspace: vs-subspace(K;vs;x.P[x]),
vector-space: VectorSpace(K),
vs-point: Point(vs),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
function: x:A ⟶ B[x],
crng: CRng
Definitions unfolded in proof :
prop: ℙ,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
so_apply: x[s1;s2],
so_apply: x[s],
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
rng: Rng,
crng: CRng,
btrue: tt,
bfalse: ff,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
top: Top,
all: ∀x:A. B[x],
mk-vs: mk-vs,
vs-point: Point(vs),
vs-quotient: vs//z.P[z],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
crng_wf,
vector-space_wf,
vs-subspace_wf,
eq-mod-subspace-equiv,
eq-mod-subspace_wf,
vs-point_wf,
subtype_quotient,
rec_select_update_lemma
Rules used in proof :
universeEquality,
cumulativity,
functionEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
independent_functionElimination,
independent_isectElimination,
functionExtensionality,
applyEquality,
because_Cache,
lambdaEquality,
hypothesisEquality,
rename,
setElimination,
isectElimination,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[K:CRng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[P:Point(vs) {}\mrightarrow{} \mBbbP{}].
Point(vs) \msubseteq{}r Point(vs//z.P[z]) supposing vs-subspace(K;vs;z.P[z])
Date html generated:
2018_05_22-PM-09_44_07
Last ObjectModification:
2018_01_09-PM-04_25_35
Theory : linear!algebra
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