Nuprl Lemma : vs-lift-neg-bfs
∀[K:CRng]. ∀[S:Type]. ∀[fs:basic-formal-sum(K;S)]. ∀[vs:VectorSpace(K)]. ∀[f:S ⟶ Point(vs)].
(vs-lift(vs;f;-(fs)) = -K 1 * vs-lift(vs;f;fs) ∈ Point(vs))
Proof
Definitions occuring in Statement :
neg-bfs: -(fs),
vs-lift: vs-lift(vs;f;fs),
basic-formal-sum: basic-formal-sum(K;S),
vs-mul: a * x,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_one: 1,
rng_minus: -r
Definitions unfolded in proof :
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
true: True,
prop: ℙ,
squash: ↓T,
so_apply: x[s],
so_lambda: λ2x.t[x],
rng: Rng,
crng: CRng,
all: ∀x:A. B[x],
basic-formal-sum: basic-formal-sum(K;S),
subtype_rel: A ⊆r B,
top: Top,
vs-bag-add: Σ{f[b] | b ∈ bs},
neg-bfs: -(fs),
vs-lift: vs-lift(vs;f;fs),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
crng_wf,
basic-formal-sum_wf,
vector-space_wf,
iff_weakening_equal,
rng_one_wf,
vs-bag-add-mul,
rng_minus_wf,
vs-mul_wf,
vs-bag-add_wf,
vs-point_wf,
true_wf,
squash_wf,
equal_wf,
rng_car_wf,
bag-subtype-list,
bag-summation-map,
vs-mul-mul,
rng_wf,
bag_wf,
rng_times_over_minus,
rng_times_one
Rules used in proof :
axiomEquality,
functionEquality,
independent_functionElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
independent_pairEquality,
spreadEquality,
functionExtensionality,
productElimination,
because_Cache,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
cumulativity,
hypothesis,
rename,
setElimination,
productEquality,
dependent_functionElimination,
applyEquality,
hypothesisEquality,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
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)) = -K 1 * vs-lift(vs;f;fs))
Date html generated:
2018_05_22-PM-09_47_23
Last ObjectModification:
2018_01_09-PM-01_27_06
Theory : linear!algebra
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