Nuprl Lemma : vs-map-into-subspace
∀[K:Rng]. ∀[A,B:VectorSpace(K)]. ∀[f:A ⟶ B]. ∀[P:Point(B) ⟶ ℙ].
(f ∈ A ⟶ (b:B | P[b])) supposing ((∀a:Point(A). P[f a]) and vs-subspace(K;B;b.P[b]))
Proof
Definitions occuring in Statement :
vs-map: A ⟶ B,
sub-vs: (v:vs | P[v]),
vs-subspace: vs-subspace(K;vs;x.P[x]),
vector-space: VectorSpace(K),
vs-point: Point(vs),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
rng: Rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
rng: Rng,
so_apply: x[s],
vs-map: A ⟶ B,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
sub-vs: (v:vs | P[v]),
vs-point: Point(vs),
mk-vs: mk-vs,
top: Top,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
vs-mul: a * x,
vs-add: x + y,
guard: {T},
vs-subspace: vs-subspace(K;vs;x.P[x]),
implies: P ⇒ Q
Lemmas referenced :
vs-point_wf,
subtype_rel_self,
vs-subspace_wf,
vs-map_wf,
vector-space_wf,
rng_wf,
sub-vs_wf,
vs-add_wf,
rng_car_wf,
vs-mul_wf,
rec_select_update_lemma,
istype-void
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsType,
universeIsType,
extract_by_obid,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
applyEquality,
instantiate,
universeEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
lambdaEquality_alt,
dependent_functionElimination,
dependent_set_memberEquality_alt,
productIsType,
because_Cache,
equalityIstype,
productElimination,
independent_isectElimination,
functionExtensionality,
voidElimination,
independent_pairFormation,
promote_hyp,
lambdaFormation_alt,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[K:Rng]. \mforall{}[A,B:VectorSpace(K)]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[P:Point(B) {}\mrightarrow{} \mBbbP{}].
(f \mmember{} A {}\mrightarrow{} (b:B | P[b])) supposing ((\mforall{}a:Point(A). P[f a]) and vs-subspace(K;B;b.P[b]))
Date html generated:
2019_10_31-AM-06_27_31
Last ObjectModification:
2019_08_12-PM-03_27_39
Theory : linear!algebra
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