Nuprl Lemma : fps-div-one
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f:PowerSeries(X;r)]. ((f÷1) = f ∈ PowerSeries(X;r)) supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-div: (f÷g),
fps-one: 1,
power-series: PowerSeries(X;r),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_one: 1
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
crng: CRng,
rng: Rng,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
true: True,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
fps-coeff: f[b],
fps-one: 1,
ifthenelse: if b then t else f fi ,
bag-null: bag-null(bs),
null: null(as),
empty-bag: {},
nil: [],
it: ⋅,
btrue: tt,
rng_one: 1,
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
power-series_wf,
crng_wf,
deq_wf,
valueall-type_wf,
fps-one_wf,
rng_one_wf,
fps-div-unique,
equal_wf,
squash_wf,
true_wf,
mul_one_fps,
iff_weakening_equal,
rng_car_wf,
rng_times_one,
bag_null_empty_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
equalitySymmetry,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
universeEquality,
setElimination,
rename,
independent_pairFormation,
independent_isectElimination,
dependent_functionElimination,
natural_numberEquality,
applyEquality,
lambdaEquality,
imageElimination,
productElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f:PowerSeries(X;r)]. ((f\mdiv{}1) = f) supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_57_51
Last ObjectModification:
2017_07_26-PM-06_33_29
Theory : power!series
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