Nuprl Lemma : neg_assoc_fps
∀[X:Type]. ∀[r:CRng]. ∀[a,b:PowerSeries(X;r)].
(((a+(-(a)+b)) = b ∈ PowerSeries(X;r)) ∧ ((-(a)+(a+b)) = b ∈ PowerSeries(X;r)))
Proof
Definitions occuring in Statement :
fps-neg: -(f),
fps-add: (f+g),
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,
fps-neg: -(f),
fps-add: (f+g),
power-series: PowerSeries(X;r),
fps-coeff: f[b],
crng: CRng,
rng: Rng,
true: True,
squash: ↓T,
prop: ℙ,
infix_ap: x f y,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
bag_wf,
power-series_wf,
crng_wf,
rng_car_wf,
rng_minus_wf,
equal_wf,
squash_wf,
true_wf,
rng_plus_inv_assoc,
iff_weakening_equal,
rng_plus_ac_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
functionExtensionality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_pairFormation,
because_Cache,
productElimination,
independent_pairEquality,
axiomEquality,
cumulativity,
isect_memberEquality,
universeEquality,
setElimination,
rename,
applyEquality,
natural_numberEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[X:Type]. \mforall{}[r:CRng]. \mforall{}[a,b:PowerSeries(X;r)]. (((a+(-(a)+b)) = b) \mwedge{} ((-(a)+(a+b)) = b))
Date html generated:
2018_05_21-PM-09_56_44
Last ObjectModification:
2017_07_26-PM-06_33_04
Theory : power!series
Home
Index