### Nuprl Lemma : fps-geometric-slice1

`∀[X:Type]`
`  ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[m:ℕ]. ∀[g:PowerSeries(X;r)].  ([(1÷(1-[g]_1))]_m = ([g]_1)^(m) ∈ PowerSeries(X;r)) `
`  supposing valueall-type(X)`

Proof

Definitions occuring in Statement :  fps-exp: `(f)^(n)` fps-slice: `[f]_n` fps-div: `(f÷g)` fps-sub: `(f-g)` fps-one: `1` power-series: `PowerSeries(X;r)` deq: `EqDecider(T)` nat: `ℕ` valueall-type: `valueall-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T` crng: `CRng` rng_one: `1`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` and: `P ∧ Q` prop: `ℙ` nat: `ℕ` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` nequal: `a ≠ b ∈ T ` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  fps-geometric-slice less_than_wf power-series_wf nat_wf crng_wf deq_wf valueall-type_wf fps-slice_wf fps-exp_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int nat_properties satisfiable-full-omega-tt intformnot_wf intformeq_wf itermConstant_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_constant_lemma int_formula_prop_wf squash_wf true_wf rem-one div-one iff_weakening_equal fps-slice-slice
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination dependent_set_memberEquality natural_numberEquality sqequalRule independent_pairFormation imageMemberEquality baseClosed cumulativity isect_memberEquality axiomEquality because_Cache universeEquality setElimination rename lambdaFormation unionElimination equalityElimination productElimination dependent_pairFormation equalityTransitivity equalitySymmetry promote_hyp dependent_functionElimination instantiate independent_functionElimination voidElimination lambdaEquality intEquality voidEquality computeAll applyEquality imageElimination

Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[m:\mBbbN{}].  \mforall{}[g:PowerSeries(X;r)].    ([(1\mdiv{}(1-[g]\_1))]\_m  =  ([g]\_1)\^{}(m))
supposing  valueall-type(X)

Date html generated: 2018_05_21-PM-09_59_03
Last ObjectModification: 2017_07_26-PM-06_33_40

Theory : power!series

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