Nuprl Lemma : fps-deriv-scalar-mul
∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f:PowerSeries(X;r)]. ∀[c:|r|]. ∀[x:X].
(d(c)*f/dx = (c)*df/dx ∈ PowerSeries(X;r))
Proof
Definitions occuring in Statement :
fps-deriv: df/dx,
fps-scalar-mul: (c)*f,
power-series: PowerSeries(X;r),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
fps-deriv: df/dx,
fps-scalar-mul: (c)*f,
fps-coeff: f[b],
crng: CRng,
subtype_rel: A ⊆r B,
nat: ℕ,
power-series: PowerSeries(X;r),
rng: Rng
Lemmas referenced :
fps-ext,
fps-deriv_wf,
fps-scalar-mul_wf,
crng_times_ac_1,
int-to-ring_wf,
bag-count_wf,
nat_wf,
cons-bag_wf,
bag_wf,
rng_car_wf,
power-series_wf,
crng_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
lambdaFormation,
sqequalRule,
setElimination,
rename,
addEquality,
applyEquality,
lambdaEquality,
natural_numberEquality,
isect_memberEquality,
axiomEquality,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f:PowerSeries(X;r)]. \mforall{}[c:|r|]. \mforall{}[x:X].
(d(c)*f/dx = (c)*df/dx)
Date html generated:
2018_05_21-PM-10_16_11
Last ObjectModification:
2018_05_19-PM-04_17_31
Theory : power!series
Home
Index