Nuprl Lemma : eq-mod-subspace-equiv
∀K:CRng. ∀vs:VectorSpace(K). ∀P:Point(vs) ⟶ ℙ.
(vs-subspace(K;vs;z.P[z]) ⇒ EquivRel(Point(vs);x,y.x = y mod (z.P[z])))
Proof
Definitions occuring in Statement :
eq-mod-subspace: x = y mod (z.P[z]),
vs-subspace: vs-subspace(K;vs;x.P[x]),
vector-space: VectorSpace(K),
vs-point: Point(vs),
equiv_rel: EquivRel(T;x,y.E[x; y]),
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
crng: CRng
Definitions unfolded in proof :
so_lambda: λ2x.t[x],
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
cand: A c∧ B,
rng: Rng,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
crng: CRng,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
refl: Refl(T;x,y.E[x; y]),
equiv_rel: EquivRel(T;x,y.E[x; y]),
eq-mod-subspace: x = y mod (z.P[z]),
and: P ∧ Q,
vs-subspace: vs-subspace(K;vs;x.P[x]),
implies: P ⇒ Q,
all: ∀x:A. B[x],
vs-neg: -(x),
true: True,
squash: ↓T
Lemmas referenced :
crng_wf,
vector-space_wf,
vs-subspace_wf,
vs-neg_wf,
vs-add_wf,
vs-point_wf,
iff_weakening_equal,
vs-add-neg,
rng_one_wf,
rng_minus_wf,
vs-mul-linear,
vs-neg-neg,
rng_sig_wf,
true_wf,
squash_wf,
vs-add-comm,
equal_wf,
iff_transitivity,
vs-add-assoc,
vs-add-cancel,
vs-neg-add,
vs-mon_ident
Rules used in proof :
dependent_functionElimination,
cumulativity,
functionEquality,
functionExtensionality,
because_Cache,
independent_functionElimination,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
universeEquality,
lambdaEquality,
hypothesis,
rename,
setElimination,
isectElimination,
extract_by_obid,
introduction,
hypothesisEquality,
applyEquality,
cut,
independent_pairFormation,
sqequalRule,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination
Latex:
\mforall{}K:CRng. \mforall{}vs:VectorSpace(K). \mforall{}P:Point(vs) {}\mrightarrow{} \mBbbP{}.
(vs-subspace(K;vs;z.P[z]) {}\mRightarrow{} EquivRel(Point(vs);x,y.x = y mod (z.P[z])))
Date html generated:
2018_05_22-PM-09_43_58
Last ObjectModification:
2018_01_09-PM-01_01_00
Theory : linear!algebra
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