Nuprl Lemma : fps-scalar-mul-rng-add
∀[X:Type]. ∀[r:CRng]. ∀[c1,c2:|r|]. ∀[f:PowerSeries(X;r)]. ((c1 +r c2)*f = ((c1)*f+(c2)*f) ∈ PowerSeries(X;r))
Proof
Definitions occuring in Statement :
fps-scalar-mul: (c)*f,
fps-add: (f+g),
power-series: PowerSeries(X;r),
uall: ∀[x:A]. B[x],
infix_ap: x f y,
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
infix_ap: x f y,
crng: CRng,
rng: Rng,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
power-series: PowerSeries(X;r),
true: True,
fps-scalar-mul: (c)*f,
fps-add: (f+g),
fps-coeff: f[b],
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
fps-ext,
fps-scalar-mul_wf,
rng_plus_wf,
fps-add_wf,
bag_wf,
power-series_wf,
rng_car_wf,
crng_wf,
rng_times_wf,
infix_ap_wf,
equal_wf,
squash_wf,
true_wf,
rng_times_over_plus,
crng_times_comm,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
cumulativity,
applyEquality,
setElimination,
rename,
because_Cache,
hypothesis,
productElimination,
independent_isectElimination,
lambdaFormation,
sqequalRule,
isect_memberEquality,
axiomEquality,
universeEquality,
natural_numberEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[X:Type]. \mforall{}[r:CRng]. \mforall{}[c1,c2:|r|]. \mforall{}[f:PowerSeries(X;r)]. ((c1 +r c2)*f = ((c1)*f+(c2)*f))
Date html generated:
2018_05_21-PM-09_57_29
Last ObjectModification:
2017_07_26-PM-06_33_23
Theory : power!series
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