Nuprl Lemma : vs-mul-add
∀[K:RngSig]. ∀[vs:VectorSpace(K)]. ∀[a,b:|K|]. ∀[x:Point(vs)]. (a +K b * x = a * x + b * x ∈ Point(vs))
Proof
Definitions occuring in Statement :
vs-mul: a * x,
vs-add: x + y,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
infix_ap: x f y,
equal: s = t ∈ T,
rng_plus: +r,
rng_car: |r|,
rng_sig: RngSig
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
vs-add: x + y,
vs-mul: a * x,
infix_ap: x f y,
guard: {T},
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
and: P ∧ Q,
btrue: tt,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
subtype_rel: A ⊆r B,
record-select: r.x,
record+: record+,
vector-space: VectorSpace(K)
Lemmas referenced :
rng_sig_wf,
vector-space_wf,
rng_car_wf,
vs-point_wf,
rng_plus_wf,
rng_times_wf,
infix_ap_wf,
rng_zero_wf,
rng_one_wf,
equal_wf,
all_wf,
subtype_rel_self
Rules used in proof :
dependent_functionElimination,
because_Cache,
axiomEquality,
isect_memberEquality,
sqequalRule,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesis,
hypothesisEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
imageElimination,
baseClosed,
imageMemberEquality,
applyLambdaEquality,
rename,
setElimination,
equalitySymmetry,
equalityTransitivity,
functionExtensionality,
lambdaEquality,
productEquality,
functionEquality,
setEquality,
universeEquality,
instantiate,
tokenEquality,
applyEquality,
dependentIntersectionEqElimination,
dependentIntersectionElimination
Latex:
\mforall{}[K:RngSig]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[a,b:|K|]. \mforall{}[x:Point(vs)]. (a +K b * x = a * x + b * x)
Date html generated:
2018_05_22-PM-09_40_46
Last ObjectModification:
2018_01_09-PM-01_05_09
Theory : linear!algebra
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