### Nuprl Lemma : fps-div-unique

`∀[X:Type]`
`  ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)]. ∀[x:|r|].`
`    ∀q:PowerSeries(X;r)`
`      q = (f÷g) ∈ PowerSeries(X;r) supposing ((g*q) = f ∈ PowerSeries(X;r)) ∧ ((g[{}] * x) = 1 ∈ |r|) `
`  supposing valueall-type(X)`

Proof

Definitions occuring in Statement :  fps-div: `(f÷g)` fps-mul: `(f*g)` fps-coeff: `f[b]` power-series: `PowerSeries(X;r)` empty-bag: `{}` deq: `EqDecider(T)` valueall-type: `valueall-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` infix_ap: `x f y` all: `∀x:A. B[x]` and: `P ∧ Q` universe: `Type` equal: `s = t ∈ T` crng: `CRng` rng_one: `1` rng_times: `*` rng_car: `|r|`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` and: `P ∧ Q` prop: `ℙ` crng: `CRng` rng: `Rng` infix_ap: `x f y` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` fps-rng: `fps-rng(r)` rng_car: `|r|` pi1: `fst(t)` rng_plus: `+r` pi2: `snd(t)` rng_zero: `0` rng_minus: `-r` rng_times: `*` rng_one: `1` ring_p: `IsRing(T;plus;zero;neg;times;one)` monoid_p: `IsMonoid(T;op;id)` ident: `Ident(T;op;id)`
Lemmas referenced :  fps-div-property equal_wf power-series_wf fps-mul_wf rng_car_wf rng_times_wf fps-coeff_wf empty-bag_wf rng_one_wf crng_wf deq_wf valueall-type_wf squash_wf true_wf fps-div_wf fps-one_wf iff_weakening_equal fps-mul-assoc fps-mul-comm fps-rng_wf crng_properties rng_properties
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination lambdaFormation productElimination equalityTransitivity equalitySymmetry productEquality cumulativity setElimination rename applyEquality because_Cache sqequalRule isect_memberEquality lambdaEquality dependent_functionElimination axiomEquality universeEquality imageElimination imageMemberEquality baseClosed natural_numberEquality independent_functionElimination hyp_replacement applyLambdaEquality

Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[f,g:PowerSeries(X;r)].  \mforall{}[x:|r|].
\mforall{}q:PowerSeries(X;r).  q  =  (f\mdiv{}g)  supposing  ((g*q)  =  f)  \mwedge{}  ((g[\{\}]  *  x)  =  1)
supposing  valueall-type(X)

Date html generated: 2018_05_21-PM-09_55_41
Last ObjectModification: 2017_07_26-PM-06_32_46

Theory : power!series

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