Nuprl Lemma : negerse_fps
∀[X:Type]. ∀[r:CRng]. ∀[a:PowerSeries(X;r)]. (((a+-(a)) = 0 ∈ PowerSeries(X;r)) ∧ ((-(a)+a) = 0 ∈ PowerSeries(X;r)))
Proof
Definitions occuring in Statement :
fps-neg: -(f),
fps-add: (f+g),
fps-zero: 0,
power-series: PowerSeries(X;r),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
fps-zero: 0,
fps-neg: -(f),
fps-add: (f+g),
power-series: PowerSeries(X;r),
fps-coeff: f[b],
crng: CRng,
rng: Rng,
infix_ap: x f y
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
power-series_wf,
fps-add-comm,
fps-neg_wf,
iff_weakening_equal,
crng_wf,
bag_wf,
rng_car_wf,
rng_zero_wf,
rng_plus_inv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
cumulativity,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
because_Cache,
functionExtensionality,
setElimination,
rename
Latex:
\mforall{}[X:Type]. \mforall{}[r:CRng]. \mforall{}[a:PowerSeries(X;r)]. (((a+-(a)) = 0) \mwedge{} ((-(a)+a) = 0))
Date html generated:
2018_05_21-PM-09_56_35
Last ObjectModification:
2017_07_26-PM-06_32_59
Theory : power!series
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