### Nuprl Lemma : fps-div-coeff-property

`∀[X:Type]`
`  ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)]. ∀[x:|r|].`
`    (g*λb.fps-div-coeff(eq;r;f;g;x;b)) = f ∈ PowerSeries(X;r) supposing (g[{}] * x) = 1 ∈ |r| `
`  supposing valueall-type(X)`

Proof

Definitions occuring in Statement :  fps-div-coeff: `fps-div-coeff(eq;r;f;g;x;b)` 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` lambda: `λx.A[x]` 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` crng: `CRng` comm: `Comm(T;op)` rng: `Rng` prop: `ℙ` and: `P ∧ Q` fps-mul: `(f*g)` power-series: `PowerSeries(X;r)` fps-coeff: `f[b]` infix_ap: `x f y` fps-div-coeff: `fps-div-coeff(eq;r;f;g;x;b)` so_lambda: `λ2x.t[x]` pi1: `fst(t)` pi2: `snd(t)` so_apply: `x[s]` cand: `A c∧ B` all: `∀x:A. B[x]` top: `Top` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` ring_p: `IsRing(T;plus;zero;neg;times;one)` group_p: `IsGroup(T;op;id;inv)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` bfalse: `ff` assert: `↑b` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` false: `False` not: `¬A`
Lemmas referenced :  rng_plus_comm crng_properties rng_properties rng_all_properties ring_p_wf rng_car_wf rng_plus_wf rng_zero_wf rng_minus_wf rng_times_wf rng_one_wf bag_wf equal_wf fps-coeff_wf empty-bag_wf power-series_wf crng_wf deq_wf valueall-type_wf fps-div-coeff_wf bag-summation_wf bag-partitions_wf infix_ap_wf assert_wf bnot_wf bag-null_wf pi1_wf_top bag-filter_wf bag-summation-single squash_wf true_wf pi2_wf iff_weakening_equal and_wf bag-summation-append subtype_rel_bag single-bag_wf bag-split bag-append_wf empty_bag_append_lemma bag-partitions-with-one-given bag-eq_wf bool_wf eqtt_to_assert assert-bag-null iff_imp_equal_bool btrue_wf equal-wf-T-base assert-bag-eq iff_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot bfalse_wf false_wf rng_times_assoc rng_times_over_plus rng_times_over_minus rng_times_one rng_plus_ac_1 rng_plus_inv rng_plus_zero
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis dependent_set_memberEquality productElimination sqequalRule functionExtensionality applyEquality because_Cache cumulativity isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry universeEquality independent_isectElimination hyp_replacement applyLambdaEquality productEquality lambdaEquality independent_pairFormation setEquality dependent_functionElimination lambdaFormation independent_pairEquality voidElimination voidEquality imageElimination natural_numberEquality imageMemberEquality baseClosed independent_functionElimination instantiate isectEquality unionElimination equalityElimination addLevel impliesFunctionality dependent_pairFormation promote_hyp

Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[f,g:PowerSeries(X;r)].  \mforall{}[x:|r|].
(g*\mlambda{}b.fps-div-coeff(eq;r;f;g;x;b))  =  f  supposing  (g[\{\}]  *  x)  =  1
supposing  valueall-type(X)

Date html generated: 2018_05_21-PM-09_55_37
Last ObjectModification: 2017_07_26-PM-06_32_44

Theory : power!series

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