Nuprl Lemma : sub-vs_wf
∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[P:Point(vs) ⟶ ℙ]. (v:vs | P[v]) ∈ VectorSpace(K) supposing vs-subspace(K;vs;x.P[x])
Proof
Definitions occuring in Statement :
sub-vs: (v:vs | P[v]),
vs-subspace: vs-subspace(K;vs;x.P[x]),
vector-space: VectorSpace(K),
vs-point: Point(vs),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
rng: Rng
Definitions unfolded in proof :
infix_ap: x f y,
label: ...$L... t,
so_lambda: λ2x.t[x],
prop: ℙ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
squash: ↓T,
cand: A c∧ B,
so_apply: x[s1;s2],
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
rng: Rng,
subtype_rel: A ⊆r B,
so_apply: x[s],
and: P ∧ Q,
sub-vs: (v:vs | P[v]),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
vs-subspace: vs-subspace(K;vs;x.P[x])
Lemmas referenced :
rng_plus_wf,
rng_wf,
vector-space_wf,
all_wf,
vs-mul-add,
rng_times_wf,
infix_ap_wf,
vs-mul-mul,
vs-mul-zero,
vs-mul-one,
vs-mul-linear,
trivial-equal,
vs-add-comm,
set_wf,
iff_weakening_equal,
vs-mon_assoc,
equal_wf,
rng_car_wf,
vs-mul_wf,
vs-add_wf,
vs-point_wf,
vs-0_wf,
mk-vs_wf
Rules used in proof :
universeEquality,
cumulativity,
isect_memberEquality,
functionEquality,
productEquality,
axiomEquality,
applyLambdaEquality,
independent_pairFormation,
equalitySymmetry,
equalityTransitivity,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
lambdaFormation,
lambdaEquality,
functionExtensionality,
rename,
setElimination,
dependent_set_memberEquality,
hypothesis,
hypothesisEquality,
applyEquality,
setEquality,
because_Cache,
isectElimination,
extract_by_obid,
thin,
productElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[P:Point(vs) {}\mrightarrow{} \mBbbP{}].
(v:vs | P[v]) \mmember{} VectorSpace(K) supposing vs-subspace(K;vs;x.P[x])
Date html generated:
2018_05_22-PM-09_42_28
Last ObjectModification:
2018_01_09-PM-01_03_23
Theory : linear!algebra
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