Nuprl Lemma : vs-map-kernel-is-subspace
∀[K:Rng]. ∀[A,B:VectorSpace(K)]. ∀[f:A ⟶ B]. vs-subspace(K;A;a.a ∈ Ker(f))
Proof
Definitions occuring in Statement :
vs-map-kernel: a ∈ Ker(f),
vs-map: A ⟶ B,
vs-subspace: vs-subspace(K;vs;x.P[x]),
vector-space: VectorSpace(K),
uall: ∀[x:A]. B[x],
rng: Rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
vs-map-kernel: a ∈ Ker(f),
vs-subspace: vs-subspace(K;vs;x.P[x]),
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
vs-map: A ⟶ B,
rng: Rng,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ
Lemmas referenced :
vs-map-0,
vs-point_wf,
rng_car_wf,
vs-map_wf,
vector-space_wf,
rng_wf,
equal_wf,
vs-zero-add,
iff_weakening_equal,
vs-zero-mul,
squash_wf,
true_wf,
istype-universe,
vs-add_wf,
rng_sig_wf,
subtype_rel_self,
vs-mul_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_pairFormation,
lambdaFormation_alt,
setElimination,
rename,
productElimination,
equalityIsType1,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
applyEquality,
universeIsType,
because_Cache,
independent_pairEquality,
axiomEquality,
lambdaEquality_alt,
dependent_functionElimination,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
instantiate,
universeEquality
Latex:
\mforall{}[K:Rng]. \mforall{}[A,B:VectorSpace(K)]. \mforall{}[f:A {}\mrightarrow{} B]. vs-subspace(K;A;a.a \mmember{} Ker(f))
Date html generated:
2019_10_31-AM-06_27_15
Last ObjectModification:
2018_11_08-PM-05_58_45
Theory : linear!algebra
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