Nuprl Lemma : implies-vs-subspace
∀K:Rng. ∀vs:VectorSpace(K).
∀[P:Point(vs) ⟶ ℙ]. (P[0] ⇒ (∀x,y:Point(vs). (P[x] ⇒ P[y] ⇒ (∀k:|K|. P[k * x + y]))) ⇒ vs-subspace(K;vs;x.P[x]))
Proof
Definitions occuring in Statement :
vs-subspace: vs-subspace(K;vs;x.P[x]),
vs-mul: a * x,
vs-add: x + y,
vs-0: 0,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
rng: Rng,
rng_car: |r|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
vs-subspace: vs-subspace(K;vs;x.P[x]),
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
rng: Rng,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
squash: ↓T,
true: True
Lemmas referenced :
vs-point_wf,
rng_car_wf,
all_wf,
vs-mul_wf,
vs-add_wf,
vs-0_wf,
vector-space_wf,
rng_wf,
rng_one_wf,
vs-mul-linear,
subtype_rel_self,
iff_weakening_equal,
squash_wf,
true_wf,
rng_sig_wf,
vs-mul-one,
vs-mon_ident
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
independent_pairFormation,
hypothesis,
cut,
applyEquality,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
instantiate,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}K:Rng. \mforall{}vs:VectorSpace(K).
\mforall{}[P:Point(vs) {}\mrightarrow{} \mBbbP{}]
(P[0]
{}\mRightarrow{} (\mforall{}x,y:Point(vs). (P[x] {}\mRightarrow{} P[y] {}\mRightarrow{} (\mforall{}k:|K|. P[k * x + y])))
{}\mRightarrow{} vs-subspace(K;vs;x.P[x]))
Date html generated:
2018_05_22-PM-09_42_02
Last ObjectModification:
2018_05_20-PM-10_41_50
Theory : linear!algebra
Home
Index