Nuprl Lemma : vs-map-quotient-kernel
∀[K:CRng]. ∀[A,B:VectorSpace(K)]. ∀[f:A ⟶ B]. (f ∈ A//z.z ∈ Ker(f) ⟶ B)
Proof
Definitions occuring in Statement :
vs-quotient: vs//z.P[z],
vs-map-kernel: a ∈ Ker(f),
vs-map: A ⟶ B,
vector-space: VectorSpace(K),
uall: ∀[x:A]. B[x],
member: t ∈ T,
crng: CRng
Definitions unfolded in proof :
all: ∀x:A. B[x],
rng: Rng,
crng: CRng,
and: P ∧ Q,
vs-map: A ⟶ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
cand: A c∧ B,
quotient: x,y:A//B[x; y],
btrue: tt,
bfalse: ff,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
top: Top,
mk-vs: mk-vs,
vs-point: Point(vs),
vs-quotient: vs//z.P[z],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
vs-subtract: (x - y),
vs-neg: -(x),
vs-map-kernel: a ∈ Ker(f),
eq-mod-subspace: x = y mod (z.P[z]),
guard: {T},
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
vs-add: x + y,
record-update: r[x := v],
record-select: r.x,
vs-mul: a * x
Lemmas referenced :
crng_wf,
vector-space_wf,
vs-map_wf,
vs-map-kernel-is-subspace,
vs-map-kernel_wf,
vs-quotient_wf,
vs-mul_wf,
rng_car_wf,
vs-add_wf,
vs-point_wf,
equal_wf,
all_wf,
eq-mod-subspace_wf,
equal-wf-base,
rec_select_update_lemma,
equal-iff-vs-subtract-is-0,
iff_weakening_equal,
vs-0_wf,
vs-map-subtract,
true_wf,
squash_wf,
subtype_rel_self,
rng_sig_wf
Rules used in proof :
dependent_functionElimination,
isect_memberEquality,
hypothesisEquality,
because_Cache,
isectElimination,
extract_by_obid,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
sqequalRule,
hypothesis,
productElimination,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_isectElimination,
functionExtensionality,
applyEquality,
lambdaEquality,
productEquality,
dependent_set_memberEquality,
independent_pairFormation,
pertypeElimination,
pointwiseFunctionalityForEquality,
voidEquality,
voidElimination,
independent_functionElimination,
equalityElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
imageElimination,
lambdaFormation
Latex:
\mforall{}[K:CRng]. \mforall{}[A,B:VectorSpace(K)]. \mforall{}[f:A {}\mrightarrow{} B]. (f \mmember{} A//z.z \mmember{} Ker(f) {}\mrightarrow{} B)
Date html generated:
2018_05_22-PM-09_44_13
Last ObjectModification:
2018_01_09-PM-04_47_20
Theory : linear!algebra
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