Nuprl Lemma : vs-bag-add-append
∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[S:Type]. ∀[f:S ⟶ Point(vs)]. ∀[bs,cs:bag(S)].
(Σ{f[b] | b ∈ bs + cs} = Σ{f[b] | b ∈ bs} + Σ{f[b] | b ∈ cs} ∈ Point(vs))
Proof
Definitions occuring in Statement :
vs-bag-add: Σ{f[b] | b ∈ bs},
vs-add: x + y,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
bag-append: as + bs,
bag: bag(T)
Definitions unfolded in proof :
all: ∀x:A. B[x],
comm: Comm(T;op),
ident: Ident(T;op;id),
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
infix_ap: x f y,
assoc: Assoc(T;op),
monoid_p: IsMonoid(T;op;id),
cand: A c∧ B,
and: P ∧ Q,
uimplies: b supposing a,
rng: Rng,
vs-bag-add: Σ{f[b] | b ∈ bs},
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_wf,
vector-space_wf,
bag_wf,
vs-add-comm,
vs-mon_ident,
iff_weakening_equal,
vs-mon_assoc,
true_wf,
squash_wf,
equal_wf,
vs-0_wf,
vs-add_wf,
vs-point_wf,
bag-summation-append
Rules used in proof :
dependent_functionElimination,
functionEquality,
cumulativity,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
independent_functionElimination,
productElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
applyEquality,
sqequalRule,
independent_pairFormation,
independent_isectElimination,
lambdaEquality,
hypothesisEquality,
hypothesis,
because_Cache,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[S:Type]. \mforall{}[f:S {}\mrightarrow{} Point(vs)]. \mforall{}[bs,cs:bag(S)].
(\mSigma{}\{f[b] | b \mmember{} bs + cs\} = \mSigma{}\{f[b] | b \mmember{} bs\} + \mSigma{}\{f[b] | b \mmember{} cs\})
Date html generated:
2018_05_22-PM-09_41_30
Last ObjectModification:
2018_01_09-PM-01_03_58
Theory : linear!algebra
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