Nuprl Lemma : mul_over_plus_fps
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[a,b,c:PowerSeries(X;r)].
(((a*(b+c)) = ((a*b)+(a*c)) ∈ PowerSeries(X;r)) ∧ (((b+c)*a) = ((b*a)+(c*a)) ∈ PowerSeries(X;r)))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-mul: (f*g),
fps-add: (f+g),
power-series: PowerSeries(X;r),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
fps-rng: fps-rng(r),
rng_car: |r|,
pi1: fst(t),
rng_times: *,
pi2: snd(t),
rng_plus: +r,
infix_ap: x f y,
and: P ∧ Q
Lemmas referenced :
rng_times_over_plus,
fps-rng_wf,
crng_subtype_rng,
crng_wf,
deq_wf,
valueall-type_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
applyEquality,
sqequalRule,
isect_memberEquality_alt,
productElimination,
independent_pairEquality,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
universeIsType,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[a,b,c:PowerSeries(X;r)].
(((a*(b+c)) = ((a*b)+(a*c))) \mwedge{} (((b+c)*a) = ((b*a)+(c*a))))
supposing valueall-type(X)
Date html generated:
2020_05_20-AM-09_05_27
Last ObjectModification:
2020_02_03-PM-02_56_22
Theory : power!series
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