Nuprl Lemma : vs-hom_wf
∀[K:CRng]. ∀[A,B:VectorSpace(K)]. (Hom(A;B) ∈ VectorSpace(K))
Proof
Definitions occuring in Statement :
vs-hom: Hom(A;B),
vector-space: VectorSpace(K),
uall: ∀[x:A]. B[x],
member: t ∈ T,
crng: CRng
Definitions unfolded in proof :
infix_ap: x f y,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
so_apply: x[s],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
all: ∀x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
vs-map: A ⟶ B,
rng: Rng,
crng: CRng,
vs-hom: Hom(A;B),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
crng_times_ac_1,
crng_wf,
vector-space_wf,
rng_plus_wf,
vs-mul-add,
rng_zero_wf,
vs-mul-zero,
vs-mul-one,
rng_one_wf,
vs-add-comm,
vs-mon_assoc,
rng_times_wf,
infix_ap_wf,
crng_times_comm,
vs-mul-mul,
vs-mul-linear,
vs-ac_1,
vs-add-assoc,
vs-mul_wf,
vs-add_wf,
all_wf,
rng_car_wf,
vs-zero-mul,
iff_weakening_equal,
vs-zero-add,
true_wf,
squash_wf,
equal_wf,
vs-point_wf,
vs-0_wf,
vs-map_wf,
mk-vs_wf,
rng_times_assoc
Rules used in proof :
isect_memberEquality,
axiomEquality,
dependent_functionElimination,
functionExtensionality,
productEquality,
independent_pairFormation,
independent_functionElimination,
productElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
applyEquality,
lambdaFormation,
lambdaEquality,
dependent_set_memberEquality,
hypothesisEquality,
hypothesis,
because_Cache,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[K:CRng]. \mforall{}[A,B:VectorSpace(K)]. (Hom(A;B) \mmember{} VectorSpace(K))
Date html generated:
2018_05_22-PM-09_43_31
Last ObjectModification:
2018_01_09-PM-01_02_51
Theory : linear!algebra
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