### Nuprl Lemma : fps-elim-div

`∀[X:Type]`
`  ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)]. ∀[z:|r|]. ∀[x:X].`
`    (fps-elim(x) (f÷g)) = (fps-elim(x) f÷fps-elim(x) g) ∈ PowerSeries(X;r) `
`    supposing (¬((fps-elim(x) f) = 0 ∈ PowerSeries(X;r))) ∧ ((g[{}] * z) = 1 ∈ |r|) `
`  supposing valueall-type(X)`

Proof

Definitions occuring in Statement :  fps-elim: `fps-elim(x)` fps-div: `(f÷g)` fps-zero: `0` 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` not: `¬A` and: `P ∧ Q` apply: `f a` 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` and: `P ∧ Q` squash: `↓T` prop: `ℙ` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` fun_thru_2op: `FunThru2op(A;B;opa;opb;f)` infix_ap: `x f y` all: `∀x:A. B[x]` cand: `A c∧ B` fps-coeff: `f[b]` fps-elim: `fps-elim(x)` ifthenelse: `if b then t else f fi ` bag-deq-member: `bag-deq-member(eq;x;b)` deq-member: `x ∈b L` reduce: `reduce(f;k;as)` list_ind: list_ind empty-bag: `{}` nil: `[]` it: `⋅` bfalse: `ff` crng: `CRng` rng: `Rng`
Lemmas referenced :  fps-div-property equal_wf squash_wf true_wf power-series_wf fps-elim_wf iff_weakening_equal fps-elim-hom fps-div_wf fps-div-unique not_wf fps-zero_wf rng_car_wf rng_times_wf fps-coeff_wf empty-bag_wf rng_one_wf crng_wf deq_wf valueall-type_wf
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination productElimination applyEquality lambdaEquality imageElimination equalityTransitivity equalitySymmetry universeEquality cumulativity because_Cache natural_numberEquality sqequalRule imageMemberEquality baseClosed independent_functionElimination dependent_functionElimination independent_pairFormation productEquality setElimination rename isect_memberEquality axiomEquality

Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[f,g:PowerSeries(X;r)].  \mforall{}[z:|r|].  \mforall{}[x:X].
(fps-elim(x)  (f\mdiv{}g))  =  (fps-elim(x)  f\mdiv{}fps-elim(x)  g)
supposing  (\mneg{}((fps-elim(x)  f)  =  0))  \mwedge{}  ((g[\{\}]  *  z)  =  1)
supposing  valueall-type(X)

Date html generated: 2018_05_21-PM-09_59_12
Last ObjectModification: 2017_07_26-PM-06_33_43

Theory : power!series

Home Index