### 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: `s = t ∈ T` crng: `CRng`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b 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: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` squash: `↓T` prop: `ℙ` crng: `CRng` rng: `Rng` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` 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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