Nuprl Lemma : fps-ext
∀[X:Type]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)]. uiff(f = g ∈ PowerSeries(X;r);∀b:bag(X). (f[b] = g[b] ∈ |r|))
Proof
Definitions occuring in Statement :
fps-coeff: f[b],
power-series: PowerSeries(X;r),
bag: bag(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀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],
prop: ℙ,
power-series: PowerSeries(X;r),
fps-coeff: f[b],
squash: ↓T,
crng: CRng,
rng: Rng,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
and_wf,
equal_wf,
power-series_wf,
fps-coeff_wf,
squash_wf,
true_wf,
rng_car_wf,
iff_weakening_equal,
all_wf,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
equalitySymmetry,
dependent_set_memberEquality,
hypothesis,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyLambdaEquality,
setElimination,
rename,
productElimination,
because_Cache,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
cumulativity,
functionExtensionality,
applyEquality,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
independent_functionElimination,
independent_pairEquality,
isect_memberEquality
Latex:
\mforall{}[X:Type]. \mforall{}[r:CRng]. \mforall{}[f,g:PowerSeries(X;r)]. uiff(f = g;\mforall{}b:bag(X). (f[b] = g[b]))
Date html generated:
2018_05_21-PM-09_54_44
Last ObjectModification:
2017_07_26-PM-06_32_32
Theory : power!series
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