Nuprl Lemma : vs-quotient_wf
∀[K:CRng]. ∀[vs:VectorSpace(K)]. ∀[P:Point(vs) ⟶ ℙ]. vs//z.P[z] ∈ VectorSpace(K) supposing vs-subspace(K;vs;z.P[z])
Proof
Definitions occuring in Statement :
vs-quotient: vs//z.P[z],
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],
member: t ∈ T,
function: x:A ⟶ B[x],
crng: CRng
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
implies: P ⇒ Q,
rng: Rng,
crng: CRng,
so_apply: x[s],
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
vs-quotient: vs//z.P[z],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
quotient: x,y:A//B[x; y],
infix_ap: x f y,
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
equiv_rel: EquivRel(T;x,y.E[x; y]),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
squash: ↓T,
cand: A c∧ B
Lemmas referenced :
rng_car_wf,
vs-0_wf,
quotient_wf,
crng_wf,
vector-space_wf,
vs-subspace_wf,
eq-mod-subspace_wf,
subtype_quotient,
vs-point_wf,
eq-mod-subspace-equiv,
mk-vs_wf,
equal-wf-base,
vs-add_functionality_eq-mod,
vs-add_wf,
quotient-member-eq,
vs-mul_functionality_eq-mod,
vs-mul_wf,
vs-mul-add,
rng_plus_wf,
infix_ap_wf,
vs-mul-mul,
rng_times_wf,
vs-mul-zero,
rng_zero_wf,
vs-mul-one,
rng_one_wf,
vs-mul-linear,
true_wf,
squash_wf,
vs-add-comm,
iff_weakening_equal,
vs-mon_assoc,
equal_wf
Rules used in proof :
universeEquality,
cumulativity,
functionEquality,
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
hypothesis,
rename,
setElimination,
isectElimination,
functionExtensionality,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productEquality,
productElimination,
pertypeElimination,
pointwiseFunctionalityForEquality,
independent_pairFormation,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
lambdaFormation
Latex:
\mforall{}[K:CRng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[P:Point(vs) {}\mrightarrow{} \mBbbP{}].
vs//z.P[z] \mmember{} VectorSpace(K) supposing vs-subspace(K;vs;z.P[z])
Date html generated:
2018_05_22-PM-09_44_04
Last ObjectModification:
2018_01_09-PM-01_00_51
Theory : linear!algebra
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