### Nuprl Lemma : fps-compose-neg

`∀[X:Type]`
`  ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x:X]. ∀[g,f:PowerSeries(X;r)].  (-(g)(x:=f) = -(g(x:=f)) ∈ PowerSeries(X;r)) `
`  supposing valueall-type(X)`

Proof

Definitions occuring in Statement :  fps-compose: `g(x:=f)` fps-neg: `-(f)` power-series: `PowerSeries(X;r)` deq: `EqDecider(T)` valueall-type: `valueall-type(T)` uimplies: `b supposing a` 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` uimplies: `b supposing a` fps-compose: `g(x:=f)` fps-neg: `-(f)` power-series: `PowerSeries(X;r)` fps-coeff: `f[b]` all: `∀x:A. B[x]` implies: `P `` Q` crng: `CRng` comm: `Comm(T;op)` and: `P ∧ Q` cand: `A c∧ B` rng: `Rng` listp: `A List+` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` squash: `↓T` prop: `ℙ` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  bag-parts'_wf bag_wf listp_wf rng_plus_comm crng_properties rng_all_properties listp_properties bag-product_wf rng_car_wf rng_times_wf rng_one_wf tl_wf list-subtype-bag bag-summation-minus equal_wf squash_wf true_wf infix_ap_wf bag-append_wf hd_wf iff_weakening_equal bag-summation_wf assoc_wf comm_wf rng_times_over_minus power-series_wf crng_wf deq_wf valueall-type_wf rng_plus_wf rng_zero_wf rng_minus_wf bag-rep_wf length_wf_nat
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule lambdaEquality extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality independent_isectElimination hypothesis lambdaFormation setElimination rename productElimination because_Cache independent_pairFormation applyEquality imageElimination equalityTransitivity equalitySymmetry natural_numberEquality imageMemberEquality baseClosed universeEquality independent_functionElimination productEquality functionExtensionality functionEquality dependent_functionElimination isect_memberEquality axiomEquality

Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[x:X].  \mforall{}[g,f:PowerSeries(X;r)].    (-(g)(x:=f)  =  -(g(x:=f)))
supposing  valueall-type(X)

Date html generated: 2018_05_21-PM-09_59_40
Last ObjectModification: 2017_07_26-PM-06_33_58

Theory : power!series

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