Nuprl Lemma : vs-map-quotient
∀[K:CRng]. ∀[A,B:VectorSpace(K)]. ∀[P:Point(A) ⟶ ℙ].
∀[f:A ⟶ B]. f ∈ A//z.P[z] ⟶ B supposing ∀a:Point(A). (P[a] ⇒ ((f a) = 0 ∈ Point(B)))
supposing vs-subspace(K;A;z.P[z])
Proof
Definitions occuring in Statement :
vs-quotient: vs//z.P[z],
vs-map: A ⟶ B,
vs-subspace: vs-subspace(K;vs;x.P[x]),
vs-0: 0,
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],
implies: P ⇒ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
crng: CRng,
rng: Rng,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
vs-subspace: vs-subspace(K;vs;x.P[x]),
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
vs-map: A ⟶ B,
prop: ℙ,
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
vs-point: Point(vs),
record-select: r.x,
vs-quotient: vs//z.P[z],
mk-vs: mk-vs,
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
quotient: x,y:A//B[x; y],
eq-mod-subspace: x = y mod (z.P[z]),
vs-add: x + y,
vs-0: 0,
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
ext-eq: A ≡ B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
vs-mul: a * x
Lemmas referenced :
vs-map-quotients,
equal_wf,
vs-point_wf,
vs-0_wf,
rng_car_wf,
vs-map_wf,
vs-subspace_wf,
vector-space_wf,
crng_wf,
vs-zero-add,
iff_weakening_equal,
vs-zero-mul,
squash_wf,
true_wf,
istype-universe,
vs-add_wf,
rng_sig_wf,
subtype_rel_self,
vs-mul_wf,
vs-neg_wf,
vs-add-neg,
vs-add-cancel,
subtype_quotient,
quotient_wf,
vs-quotient_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
because_Cache,
universeIsType,
independent_isectElimination,
independent_pairFormation,
lambdaFormation_alt,
equalityIstype,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
applyEquality,
axiomEquality,
functionIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
universeEquality,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
instantiate,
pointwiseFunctionalityForEquality,
pertypeElimination,
promote_hyp,
productIsType,
sqequalBase,
applyLambdaEquality,
dependent_set_memberEquality_alt,
functionExtensionality,
hyp_replacement
Latex:
\mforall{}[K:CRng]. \mforall{}[A,B:VectorSpace(K)]. \mforall{}[P:Point(A) {}\mrightarrow{} \mBbbP{}].
\mforall{}[f:A {}\mrightarrow{} B]. f \mmember{} A//z.P[z] {}\mrightarrow{} B supposing \mforall{}a:Point(A). (P[a] {}\mRightarrow{} ((f a) = 0))
supposing vs-subspace(K;A;z.P[z])
Date html generated:
2019_10_31-AM-06_28_04
Last ObjectModification:
2019_08_20-PM-02_13_40
Theory : linear!algebra
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