Nuprl Lemma : fps-product-append
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[T:Type]. ∀[f:T ⟶ PowerSeries(X;r)]. ∀[b,c:bag(T)].
(Π(x∈b + c).f[x] = (Π(x∈b).f[x]*Π(x∈c).f[x]) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-product: Π(x∈b).f[x],
fps-mul: (f*g),
power-series: PowerSeries(X;r),
bag-append: as + bs,
bag: bag(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fps-product: Π(x∈b).f[x],
bag-product: Πx ∈ b. f[x],
squash: ↓T,
prop: ℙ,
and: P ∧ Q,
cand: A c∧ B,
monoid_p: IsMonoid(T;op;id),
assoc: Assoc(T;op),
infix_ap: x f y,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
ident: Ident(T;op;id),
comm: Comm(T;op),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
power-series_wf,
bag-summation-append,
fps-mul_wf,
fps-one_wf,
mul_assoc_fps,
iff_weakening_equal,
fps-mul-comm,
bag_wf,
crng_wf,
deq_wf,
valueall-type_wf,
mul_one_fps,
bag-summation_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
cumulativity,
independent_isectElimination,
independent_pairFormation,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
isect_memberEquality,
axiomEquality,
independent_pairEquality,
functionExtensionality,
universeEquality,
functionEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} PowerSeries(X;r)]. \mforall{}[b,c:bag(T)].
(\mPi{}(x\mmember{}b + c).f[x] = (\mPi{}(x\mmember{}b).f[x]*\mPi{}(x\mmember{}c).f[x]))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_57_14
Last ObjectModification:
2017_07_26-PM-06_33_16
Theory : power!series
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