Nuprl Lemma : fps-exp-zero
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f:PowerSeries(X;r)]. ((f)^(0) = 1 ∈ PowerSeries(X;r)) supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-exp: (f)^(n),
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],
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
fps-rng: fps-rng(r),
rng_car: |r|,
pi1: fst(t),
rng_one: 1,
pi2: snd(t),
fps-exp: (f)^(n)
Lemmas referenced :
rng_nexp_zero,
fps-rng_wf,
crng_subtype_rng,
crng_wf,
deq_wf,
valueall-type_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
applyEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
universeIsType,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f:PowerSeries(X;r)]. ((f)\^{}(0) = 1) supposing valueall-type(X)
Date html generated:
2020_05_20-AM-09_05_43
Last ObjectModification:
2020_01_25-AM-11_14_49
Theory : power!series
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