Nuprl Lemma : vs-map-image-iff-surject
∀[K:Rng]. ∀[A,B:VectorSpace(K)]. ∀[f:A ⟶ B]. (∀b:Point(B). b ∈ Img(f) ⇐⇒ Surj(Point(A);Point(B);f))
Proof
Definitions occuring in Statement :
vs-map-image: b ∈ Img(f),
vs-map: A ⟶ B,
vector-space: VectorSpace(K),
vs-point: Point(vs),
surject: Surj(A;B;f),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rng: Rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
surject: Surj(A;B;f),
exists: ∃x:A. B[x],
vs-map-image: b ∈ Img(f),
all: ∀x:A. B[x],
member: t ∈ T,
rng: Rng,
prop: ℙ,
rev_implies: P ⇐ Q,
vs-map: A ⟶ B
Lemmas referenced :
vs-point_wf,
vs-map-image_wf,
surject_wf,
vs-map_wf,
vector-space_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
independent_pairFormation,
lambdaFormation_alt,
sqequalRule,
sqequalHypSubstitution,
hypothesis,
functionIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
because_Cache,
inhabitedIsType,
dependent_functionElimination
Latex:
\mforall{}[K:Rng]. \mforall{}[A,B:VectorSpace(K)]. \mforall{}[f:A {}\mrightarrow{} B].
(\mforall{}b:Point(B). b \mmember{} Img(f) \mLeftarrow{}{}\mRightarrow{} Surj(Point(A);Point(B);f))
Date html generated:
2019_10_31-AM-06_27_27
Last ObjectModification:
2019_08_21-PM-06_22_16
Theory : linear!algebra
Home
Index