Nuprl Lemma : all-vs-quotient-of-free
∀K:CRng. ∀V:VectorSpace(K).
∃S:Type. ∃P:Point(free-vs(K;S)) ⟶ ℙ. (vs-subspace(K;free-vs(K;S);x.P[x]) ∧ V ≅ free-vs(K;S)//a.P[a])
Proof
Definitions occuring in Statement :
free-vs: free-vs(K;S),
vs-quotient: vs//z.P[z],
vs-iso: A ≅ B,
vs-subspace: vs-subspace(K;vs;x.P[x]),
vector-space: VectorSpace(K),
vs-point: Point(vs),
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
crng: CRng
Definitions unfolded in proof :
uimplies: b supposing a,
so_apply: x[s],
and: P ∧ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
rng: Rng,
crng: CRng,
uall: ∀[x:A]. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
vs-map: A ⟶ B,
exists!: ∃!x:T. P[x],
surject: Surj(A;B;f)
Lemmas referenced :
crng_wf,
vector-space_wf,
vs-quotient_wf,
vs-iso_wf,
vs-subspace_wf,
exists_wf,
free-vs_wf,
implies-iso-vs-quotient,
vs-point_wf,
surject_wf,
free-vs-property,
equal_wf,
free-vs-inc_wf
Rules used in proof :
independent_isectElimination,
functionExtensionality,
productEquality,
sqequalRule,
universeEquality,
lambdaEquality,
applyEquality,
cumulativity,
functionEquality,
instantiate,
independent_functionElimination,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
because_Cache,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
dependent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination
Latex:
\mforall{}K:CRng. \mforall{}V:VectorSpace(K).
\mexists{}S:Type
\mexists{}P:Point(free-vs(K;S)) {}\mrightarrow{} \mBbbP{}. (vs-subspace(K;free-vs(K;S);x.P[x]) \mwedge{} V \mcong{} free-vs(K;S)//a.P[a])
Date html generated:
2018_05_22-PM-09_46_52
Last ObjectModification:
2018_01_09-PM-05_27_33
Theory : linear!algebra
Home
Index