Nuprl Lemma : fps-scalar-mul-mul
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[c:|r|]. ∀[f,g:PowerSeries(X;r)].
(((c)*(f*g) = ((c)*f*g) ∈ PowerSeries(X;r)) ∧ ((c)*(f*g) = (f*(c)*g) ∈ PowerSeries(X;r)))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-scalar-mul: (c)*f,
fps-mul: (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,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
all: ∀x:A. B[x],
cand: A c∧ B,
dist_1op_2op_lr: Dist1op2opLR(A;1op;2op),
infix_ap: x f y
Lemmas referenced :
fps-scalar-mul-property,
power-series_wf,
crng_wf,
deq_wf,
valueall-type_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
productElimination,
dependent_functionElimination,
sqequalRule,
independent_pairFormation,
because_Cache,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[c:|r|]. \mforall{}[f,g:PowerSeries(X;r)].
(((c)*(f*g) = ((c)*f*g)) \mwedge{} ((c)*(f*g) = (f*(c)*g)))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_57_31
Last ObjectModification:
2018_05_19-PM-04_13_41
Theory : power!series
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