Nuprl Lemma : vs-mul_functionality_eq-mod
∀K:CRng. ∀vs:VectorSpace(K). ∀P:Point(vs) ⟶ ℙ.
(vs-subspace(K;vs;z.P[z])
⇒ (∀x,x':Point(vs). ∀k,k':|K|. (x = x' mod (z.P[z]) ⇒ (k = k' ∈ |K|) ⇒ k * x = k' * x' 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]),
vs-mul: a * x,
vector-space: VectorSpace(K),
vs-point: Point(vs),
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
equal: s = t ∈ T,
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
eq-mod-subspace: x = y mod (z.P[z]),
vs-subspace: vs-subspace(K;vs;x.P[x]),
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
crng: CRng,
rng: Rng,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
vs-neg: -(x),
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y
Lemmas referenced :
eq-mod-subspace_wf,
rng_car_wf,
vs-point_wf,
vs-subspace_wf,
vector-space_wf,
crng_wf,
vs-add_wf,
vs-neg_wf,
vs-mul_wf,
rng_minus_wf,
rng_one_wf,
rng_times_wf,
vs-mul-mul,
iff_weakening_equal,
crng_times_comm,
vs-mul-linear,
squash_wf,
true_wf,
rng_sig_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
equalityIstype,
inhabitedIsType,
hypothesisEquality,
hypothesis,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
dependent_functionElimination,
sqequalRule,
lambdaEquality_alt,
applyEquality,
functionIsType,
universeEquality,
because_Cache,
independent_functionElimination,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
instantiate
Latex:
\mforall{}K:CRng. \mforall{}vs:VectorSpace(K). \mforall{}P:Point(vs) {}\mrightarrow{} \mBbbP{}.
(vs-subspace(K;vs;z.P[z])
{}\mRightarrow{} (\mforall{}x,x':Point(vs). \mforall{}k,k':|K|.
(x = x' mod (z.P[z]) {}\mRightarrow{} (k = k') {}\mRightarrow{} k * x = k' * x' mod (z.P[z]))))
Date html generated:
2020_05_20-PM-01_18_13
Last ObjectModification:
2020_01_06-PM-01_23_29
Theory : linear!algebra
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