Nuprl Lemma : vs-lift-add
∀[S:Type]. ∀[K:CRng]. ∀[vs:VectorSpace(K)]. ∀[f:S ⟶ Point(vs)]. ∀[u,v:Point(free-vs(K;S))].
(vs-lift(vs;f;u + v) = vs-lift(vs;f;u) + vs-lift(vs;f;v) ∈ Point(vs))
Proof
Definitions occuring in Statement :
free-vs: free-vs(K;S),
vs-lift: vs-lift(vs;f;fs),
vs-add: x + y,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
prop: ℙ,
and: P ∧ Q,
quotient: x,y:A//B[x; y],
rng: Rng,
crng: CRng,
formal-sum: formal-sum(K;S),
vs-add: x + y,
btrue: tt,
bfalse: ff,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
top: Top,
all: ∀x:A. B[x],
mk-vs: mk-vs,
vs-point: Point(vs),
free-vs: free-vs(K;S),
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-formal-sum: basic-formal-sum(K;S),
guard: {T},
subtype_rel: A ⊆r B,
formal-sum-add: x + y,
true: True,
implies: P ⇒ Q,
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
squash: ↓T
Lemmas referenced :
crng_wf,
vector-space_wf,
free-vs_wf,
basic-formal-sum_wf,
bfs-equiv_wf,
equal-wf-base,
vs-point_wf,
rec_select_update_lemma,
formal-sum_wf,
subtype_rel_self,
vs-lift_wf2,
vs-add_wf,
equal_wf,
vs-lift-append,
bfs-equiv-rel,
quotient-member-eq,
rng_sig_wf,
true_wf,
squash_wf
Rules used in proof :
universeEquality,
functionEquality,
axiomEquality,
cumulativity,
productEquality,
productElimination,
pertypeElimination,
hypothesisEquality,
because_Cache,
rename,
setElimination,
isectElimination,
pointwiseFunctionalityForEquality,
sqequalRule,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
dependent_functionElimination,
extract_by_obid,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
applyLambdaEquality,
hyp_replacement,
lambdaEquality,
applyEquality,
equalitySymmetry,
equalityTransitivity,
baseClosed,
imageMemberEquality,
natural_numberEquality,
independent_functionElimination,
independent_isectElimination,
imageElimination
Latex:
\mforall{}[S:Type]. \mforall{}[K:CRng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[f:S {}\mrightarrow{} Point(vs)]. \mforall{}[u,v:Point(free-vs(K;S))].
(vs-lift(vs;f;u + v) = vs-lift(vs;f;u) + vs-lift(vs;f;v))
Date html generated:
2018_05_22-PM-09_46_22
Last ObjectModification:
2018_01_09-PM-00_35_47
Theory : linear!algebra
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