Nuprl Lemma : vs-map-kernel-zero
∀[K:Rng]. ∀[A,B:VectorSpace(K)]. ∀[f:A ⟶ B].
(∀a:Point(A). (a ∈ Ker(f) ⇐⇒ a = 0 ∈ Point(A)) ⇐⇒ Inj(Point(A);Point(B);f))
Proof
Definitions occuring in Statement :
vs-map-kernel: a ∈ Ker(f),
vs-map: A ⟶ B,
vs-0: 0,
vector-space: VectorSpace(K),
vs-point: Point(vs),
inject: Inj(A;B;f),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
vs-map-kernel: a ∈ Ker(f),
inject: Inj(A;B;f),
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
so_apply: x[s],
vs-map: A ⟶ B,
so_lambda: λ2x.t[x],
rng: Rng,
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T
Lemmas referenced :
rng_wf,
vector-space_wf,
vs-map_wf,
inject_wf,
vs-0_wf,
equal_wf,
vs-map-kernel_wf,
iff_wf,
vs-point_wf,
all_wf,
vs-subtract_wf,
equal-iff-vs-subtract-is-0,
vs-subtract-self,
iff_weakening_equal,
vs-map-subtract,
true_wf,
squash_wf,
vs-map-0,
rng_sig_wf
Rules used in proof :
isect_memberEquality,
applyEquality,
axiomEquality,
dependent_functionElimination,
independent_pairEquality,
productElimination,
because_Cache,
lambdaEquality,
sqequalRule,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
levelHypothesis,
equalityUniverse,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
addLevel,
cumulativity,
functionEquality
Latex:
\mforall{}[K:Rng]. \mforall{}[A,B:VectorSpace(K)]. \mforall{}[f:A {}\mrightarrow{} B].
(\mforall{}a:Point(A). (a \mmember{} Ker(f) \mLeftarrow{}{}\mRightarrow{} a = 0) \mLeftarrow{}{}\mRightarrow{} Inj(Point(A);Point(B);f))
Date html generated:
2018_05_22-PM-09_43_12
Last ObjectModification:
2018_01_09-PM-02_30_59
Theory : linear!algebra
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