Nuprl Lemma : vs-sum_wf
∀[K:Rng]. ∀[S:Type]. ∀[V:S ⟶ VectorSpace(K)]. (Σs:S. V[s] ∈ VectorSpace(K))
Proof
Definitions occuring in Statement :
vs-sum: Σs:S. V[s],
vector-space: VectorSpace(K),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type,
rng: Rng
Definitions unfolded in proof :
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
all: ∀x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
so_apply: x[s],
rng: Rng,
vs-sum: Σs:S. V[s],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_wf,
vector-space_wf,
vs-mul-add,
rng_times_wf,
infix_ap_wf,
vs-mul-mul,
vs-mul-zero,
vs-mul-one,
vs-mul-linear,
vs-add-comm,
iff_weakening_equal,
vs-mon_assoc,
true_wf,
squash_wf,
equal_wf,
rng_car_wf,
vs-mul_wf,
vs-add_wf,
vs-0_wf,
vs-point_wf,
mk-vs_wf
Rules used in proof :
isect_memberEquality,
dependent_functionElimination,
axiomEquality,
independent_pairFormation,
independent_functionElimination,
productElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaFormation,
independent_isectElimination,
lambdaEquality,
functionExtensionality,
applyEquality,
hypothesisEquality,
cumulativity,
functionEquality,
hypothesis,
because_Cache,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[K:Rng]. \mforall{}[S:Type]. \mforall{}[V:S {}\mrightarrow{} VectorSpace(K)]. (\mSigma{}s:S. V[s] \mmember{} VectorSpace(K))
Date html generated:
2018_05_22-PM-09_42_33
Last ObjectModification:
2018_01_09-PM-01_03_17
Theory : linear!algebra
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