Nuprl Lemma : fps-elim-x-neg

[X:Type]. ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x:X]. ∀[g:PowerSeries(X;r)].  (-(g)(x:=0) -(g(x:=0)) ∈ PowerSeries(X;r))


Proof




Definitions occuring in Statement :  fps-elim-x: f(x:=0) fps-neg: -(f) power-series: PowerSeries(X;r) deq: EqDecider(T) uall: [x:A]. B[x] universe: Type equal: t ∈ T crng: CRng
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uiff: uiff(P;Q) and: P ∧ Q uimplies: supposing a all: x:A. B[x] fps-elim-x: f(x:=0) fps-neg: -(f) fps-coeff: f[b] fps-elim: fps-elim(x) implies:  Q bool: 𝔹 unit: Unit it: btrue: tt ifthenelse: if then else fi  squash: T prop: crng: CRng rng: Rng true: True subtype_rel: A ⊆B guard: {T} iff: ⇐⇒ Q rev_implies:  Q bfalse: ff exists: x:A. B[x] or: P ∨ Q sq_type: SQType(T) bnot: ¬bb assert: b false: False not: ¬A power-series: PowerSeries(X;r)
Lemmas referenced :  fps-ext fps-elim-x_wf fps-neg_wf bag-deq-member_wf bool_wf eqtt_to_assert assert-bag-deq-member equal_wf squash_wf true_wf rng_car_wf rng_zero_wf rng_minus_zero iff_weakening_equal eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot bag-member_wf rng_minus_wf bag_wf power-series_wf crng_wf deq_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality cumulativity hypothesis productElimination independent_isectElimination lambdaFormation sqequalRule unionElimination equalityElimination equalityTransitivity equalitySymmetry applyEquality lambdaEquality imageElimination setElimination rename natural_numberEquality imageMemberEquality baseClosed universeEquality independent_functionElimination dependent_pairFormation promote_hyp dependent_functionElimination instantiate voidElimination isect_memberEquality axiomEquality

Latex:
\mforall{}[X:Type].  \mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[x:X].  \mforall{}[g:PowerSeries(X;r)].    (-(g)(x:=0)  =  -(g(x:=0)))



Date html generated: 2018_05_21-PM-09_59_15
Last ObjectModification: 2017_07_26-PM-06_33_45

Theory : power!series


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