Nuprl Lemma : free-vs-subspace
∀[S:Type]. ∀[K:CRng]. ∀[P:formal-sum(K;S) ⟶ ℙ].
((P[{}]
∧ (∀fs,y:formal-sum(K;S). (P[fs] ⇒ P[y] ⇒ P[fs + y]))
∧ (∀fs:formal-sum(K;S). ∀a:|K|. (P[fs] ⇒ P[a * fs])))
⇒ vs-subspace(K;free-vs(K;S);fs.P[fs]))
Proof
Definitions occuring in Statement :
free-vs: free-vs(K;S),
formal-sum-add: x + y,
formal-sum: formal-sum(K;S),
formal-sum-mul: k * x,
vs-subspace: vs-subspace(K;vs;x.P[x]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
crng: CRng,
rng_car: |r|,
empty-bag: {}
Definitions unfolded in proof :
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
rng: Rng,
crng: CRng,
so_apply: x[s],
prop: ℙ,
cand: A c∧ B,
btrue: tt,
bfalse: ff,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
top: Top,
member: t ∈ T,
all: ∀x:A. B[x],
vs-0: 0,
vs-add: x + y,
vs-point: Point(vs),
vs-mul: a * x,
mk-vs: mk-vs,
vs-subspace: vs-subspace(K;vs;x.P[x]),
free-vs: free-vs(K;S),
and: P ∧ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
formal-sum: formal-sum(K;S),
basic-formal-sum: basic-formal-sum(K;S)
Lemmas referenced :
crng_wf,
formal-sum-mul_wf,
rng_car_wf,
formal-sum-add_wf,
all_wf,
formal-sum_wf,
rec_select_update_lemma,
bfs-equiv-rel,
bfs-equiv_wf,
basic-formal-sum_wf,
subtype_quotient,
empty-bag_wf
Rules used in proof :
universeEquality,
functionEquality,
lambdaEquality,
cumulativity,
because_Cache,
rename,
setElimination,
isectElimination,
hypothesisEquality,
functionExtensionality,
applyEquality,
productEquality,
independent_pairFormation,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
sqequalRule,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_isectElimination
Latex:
\mforall{}[S:Type]. \mforall{}[K:CRng]. \mforall{}[P:formal-sum(K;S) {}\mrightarrow{} \mBbbP{}].
((P[\{\}]
\mwedge{} (\mforall{}fs,y:formal-sum(K;S). (P[fs] {}\mRightarrow{} P[y] {}\mRightarrow{} P[fs + y]))
\mwedge{} (\mforall{}fs:formal-sum(K;S). \mforall{}a:|K|. (P[fs] {}\mRightarrow{} P[a * fs])))
{}\mRightarrow{} vs-subspace(K;free-vs(K;S);fs.P[fs]))
Date html generated:
2018_05_22-PM-09_46_56
Last ObjectModification:
2018_01_09-PM-05_49_55
Theory : linear!algebra
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