Nuprl Lemma : fps-exp-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-exp: (f)^(n),
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,
squash: ↓T,
prop: ℙ,
nat_plus: ℕ+,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
subtract: n - m
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
fps-exp-unroll,
less_than_wf,
iff_weakening_equal,
power-series_wf,
crng_wf,
deq_wf,
valueall-type_wf,
fps-mul_wf,
fps-exp-zero,
mul_one_fps
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
independent_isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination,
cumulativity,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f:PowerSeries(X;r)]. ((f)\^{}(1) = f) supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_58_33
Last ObjectModification:
2017_07_26-PM-06_33_36
Theory : power!series
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