Nuprl Lemma : vs-subspace_wf
∀[K:RngSig]. ∀[vs:VectorSpace(K)]. ∀[P:Point(vs) ⟶ ℙ]. (vs-subspace(K;vs;x.P[x]) ∈ ℙ)
Proof
Definitions occuring in Statement :
vs-subspace: vs-subspace(K;vs;x.P[x]),
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
rng_sig: RngSig
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
and: P ∧ Q,
prop: ℙ,
vs-subspace: vs-subspace(K;vs;x.P[x]),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_sig_wf,
vector-space_wf,
vs-mul_wf,
rng_car_wf,
vs-add_wf,
all_wf,
vs-0_wf,
vs-point_wf
Rules used in proof :
dependent_functionElimination,
isect_memberEquality,
universeEquality,
cumulativity,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
functionEquality,
lambdaEquality,
because_Cache,
hypothesis,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesisEquality,
functionExtensionality,
applyEquality,
productEquality,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[K:RngSig]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[P:Point(vs) {}\mrightarrow{} \mBbbP{}]. (vs-subspace(K;vs;x.P[x]) \mmember{} \mBbbP{})
Date html generated:
2018_05_22-PM-09_41_59
Last ObjectModification:
2018_01_09-AM-10_43_58
Theory : linear!algebra
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