Nuprl Lemma : generated-subspace_wf
∀[K:RngSig]. ∀[vs:VectorSpace(K)]. ∀[P:Point(vs) ⟶ ℙ]. (Subspace(x.P[x]) ∈ Point(vs) ⟶ ℙ)
Proof
Definitions occuring in Statement :
generated-subspace: Subspace(v.P[v]),
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
generated-subspace: Subspace(v.P[v]),
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
prop: ℙ
Lemmas referenced :
or_wf,
equal_wf,
vs-point_wf,
vs-0_wf,
exists_wf,
l_tree_wf,
rng_car_wf,
vs-tree-val_wf,
l_tree_covariant,
top_wf,
subtype_rel_product,
vector-space_wf,
rng_sig_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
productEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination
Latex:
\mforall{}[K:RngSig]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[P:Point(vs) {}\mrightarrow{} \mBbbP{}]. (Subspace(x.P[x]) \mmember{} Point(vs) {}\mrightarrow{} \mBbbP{})
Date html generated:
2018_05_22-PM-09_42_12
Last ObjectModification:
2018_05_20-PM-10_42_03
Theory : linear!algebra
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