Nuprl Lemma : vs-lift-null-formal-sum
∀[K:CRng]. ∀[S:Type]. ∀[fs:basic-formal-sum(K;S)].
∀[vs:VectorSpace(K)]. ∀[f:S ⟶ Point(vs)]. (vs-lift(vs;f;fs) = 0 ∈ Point(vs)) supposing null-formal-sum(K;S;fs)
Proof
Definitions occuring in Statement :
null-formal-sum: null-formal-sum(K;S;fs),
vs-lift: vs-lift(vs;f;fs),
basic-formal-sum: basic-formal-sum(K;S),
vs-0: 0,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
true: True,
subtype_rel: A ⊆r B,
basic-formal-sum: basic-formal-sum(K;S),
prop: ℙ,
all: ∀x:A. B[x],
rng: Rng,
crng: CRng,
exists: ∃x:A. B[x],
null-formal-sum: null-formal-sum(K;S;fs),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
formal-sum: formal-sum(K;S),
vs-neg: -(x)
Lemmas referenced :
vs-0_wf,
vs-lift_wf2,
zero-bfs_wf,
neg-bfs_wf,
rng_car_wf,
bag-append_wf,
crng_wf,
basic-formal-sum_wf,
null-formal-sum_wf,
vector-space_wf,
vs-point_wf,
equal_wf,
squash_wf,
true_wf,
vs-lift-append,
vs-add_wf,
rng_sig_wf,
vs-lift-neg-bfs,
vs-lift-zero-bfs,
iff_weakening_equal,
bfs-equiv-rel,
bfs-equiv_wf,
subtype_quotient,
vs-grp_inv_assoc,
rng_one_wf,
rng_minus_wf,
vs-mul_wf,
vs-add-assoc,
bag_wf,
formal-sum_wf
Rules used in proof :
natural_numberEquality,
functionExtensionality,
lambdaEquality,
applyEquality,
productEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
because_Cache,
axiomEquality,
isect_memberEquality,
sqequalRule,
rename,
setElimination,
isectElimination,
extract_by_obid,
hypothesisEquality,
cumulativity,
functionEquality,
hypothesis,
thin,
productElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
imageElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
levelHypothesis,
equalityUniverse,
lambdaFormation
Latex:
\mforall{}[K:CRng]. \mforall{}[S:Type]. \mforall{}[fs:basic-formal-sum(K;S)].
\mforall{}[vs:VectorSpace(K)]. \mforall{}[f:S {}\mrightarrow{} Point(vs)]. (vs-lift(vs;f;fs) = 0)
supposing null-formal-sum(K;S;fs)
Date html generated:
2018_05_22-PM-09_47_27
Last ObjectModification:
2018_01_09-PM-00_45_29
Theory : linear!algebra
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