Nuprl Lemma : fps-elim-div
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)]. ∀[z:|r|]. ∀[x:X].
(fps-elim(x) (f÷g)) = (fps-elim(x) f÷fps-elim(x) g) ∈ PowerSeries(X;r)
supposing (¬((fps-elim(x) f) = 0 ∈ PowerSeries(X;r))) ∧ ((g[{}] * z) = 1 ∈ |r|)
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-elim: fps-elim(x),
fps-div: (f÷g),
fps-zero: 0,
fps-coeff: f[b],
power-series: PowerSeries(X;r),
empty-bag: {},
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
not: ¬A,
and: P ∧ Q,
apply: f a,
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_one: 1,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
infix_ap: x f y,
all: ∀x:A. B[x],
cand: A c∧ B,
fps-coeff: f[b],
fps-elim: fps-elim(x),
ifthenelse: if b then t else f fi ,
bag-deq-member: bag-deq-member(eq;x;b),
deq-member: x ∈b L,
reduce: reduce(f;k;as),
list_ind: list_ind,
empty-bag: {},
nil: [],
it: ⋅,
bfalse: ff,
crng: CRng,
rng: Rng
Lemmas referenced :
fps-div-property,
equal_wf,
squash_wf,
true_wf,
power-series_wf,
fps-elim_wf,
iff_weakening_equal,
fps-elim-hom,
fps-div_wf,
fps-div-unique,
not_wf,
fps-zero_wf,
rng_car_wf,
rng_times_wf,
fps-coeff_wf,
empty-bag_wf,
rng_one_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,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
cumulativity,
because_Cache,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination,
independent_pairFormation,
productEquality,
setElimination,
rename,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f,g:PowerSeries(X;r)]. \mforall{}[z:|r|]. \mforall{}[x:X].
(fps-elim(x) (f\mdiv{}g)) = (fps-elim(x) f\mdiv{}fps-elim(x) g)
supposing (\mneg{}((fps-elim(x) f) = 0)) \mwedge{} ((g[\{\}] * z) = 1)
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_59_12
Last ObjectModification:
2017_07_26-PM-06_33_43
Theory : power!series
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