Nuprl Lemma : vs-map-image-is-subspace
∀[K:Rng]. ∀A:VectorSpace(K). ∀[B:VectorSpace(K)]. ∀[f:A ⟶ B]. vs-subspace(K;B;b.b ∈ Img(f))
Proof
Definitions occuring in Statement :
vs-map-image: b ∈ Img(f),
vs-map: A ⟶ B,
vs-subspace: vs-subspace(K;vs;x.P[x]),
vector-space: VectorSpace(K),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
rng: Rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
vs-map-image: b ∈ Img(f),
vs-subspace: vs-subspace(K;vs;x.P[x]),
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
rng: Rng,
vs-map: A ⟶ B,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
vs-point_wf,
rng_car_wf,
vs-map_wf,
vector-space_wf,
rng_wf,
vs-0_wf,
vs-map-0,
rng_properties,
vs-add_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
rng_sig_wf,
vs-mul_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalRule,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
productElimination,
thin,
productIsType,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
because_Cache,
hypothesisEquality,
equalityIstype,
inhabitedIsType,
applyEquality,
dependent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[K:Rng]. \mforall{}A:VectorSpace(K). \mforall{}[B:VectorSpace(K)]. \mforall{}[f:A {}\mrightarrow{} B]. vs-subspace(K;B;b.b \mmember{} Img(f))
Date html generated:
2019_10_31-AM-06_27_24
Last ObjectModification:
2019_08_12-PM-01_12_04
Theory : linear!algebra
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