Nuprl Lemma : vs-iso-iff-kernel-0
∀[K:Rng]. ∀[A,B:VectorSpace(K)].
(A ≅ B ⇐⇒ ∃f:A ⟶ B. ((∀a:Point(A). (a ∈ Ker(f) ⇐⇒ a = 0 ∈ Point(A))) ∧ Surj(Point(A);Point(B);f)))
Proof
Definitions occuring in Statement :
vs-iso: A ≅ B,
vs-map-kernel: a ∈ Ker(f),
vs-map: A ⟶ B,
vs-0: 0,
vector-space: VectorSpace(K),
vs-point: Point(vs),
surject: Surj(A;B;f),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
all: ∀x:A. B[x],
so_apply: x[s],
vs-map: A ⟶ B,
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
rng: Rng,
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
exists: ∃x:A. B[x],
vs-iso: A ≅ B,
vs-map-kernel: a ∈ Ker(f),
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
surject: Surj(A;B;f),
pi1: fst(t),
inject: Inj(A;B;f)
Lemmas referenced :
rng_wf,
vector-space_wf,
surject_wf,
vs-0_wf,
equal_wf,
vs-map-kernel_wf,
iff_wf,
vs-point_wf,
all_wf,
vs-map_wf,
exists_wf,
vs-iso_wf,
iff_weakening_equal,
true_wf,
squash_wf,
vs-map-0,
rng_sig_wf,
vs-map-kernel-zero,
vs-mul_wf,
vs-add_wf,
rng_car_wf
Rules used in proof :
dependent_functionElimination,
productEquality,
lambdaEquality,
sqequalRule,
because_Cache,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_pairFormation,
productElimination,
axiomEquality,
independent_pairEquality,
independent_functionElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
applyEquality,
levelHypothesis,
equalityUniverse,
cumulativity,
functionEquality,
functionExtensionality,
promote_hyp,
dependent_set_memberEquality
Latex:
\mforall{}[K:Rng]. \mforall{}[A,B:VectorSpace(K)].
(A \mcong{} B \mLeftarrow{}{}\mRightarrow{} \mexists{}f:A {}\mrightarrow{} B. ((\mforall{}a:Point(A). (a \mmember{} Ker(f) \mLeftarrow{}{}\mRightarrow{} a = 0)) \mwedge{} Surj(Point(A);Point(B);f)))
Date html generated:
2018_05_22-PM-09_43_24
Last ObjectModification:
2018_01_09-PM-02_36_49
Theory : linear!algebra
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