### Nuprl Lemma : fps-mul-coeff-bag-rep-simple

`∀[X:Type]`
`  ∀[eq:EqDecider(X)]. ∀[n:ℕ]. ∀[k:ℕn + 1]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)]. ∀[x:X].`
`    (f*g)[bag-rep(n;x)] = (* f[bag-rep(k;x)] g[bag-rep(n - k;x)]) ∈ |r| `
`    supposing ∀i:ℕn + 1. ((¬(i = k ∈ ℤ)) `` (f[bag-rep(i;x)] = 0 ∈ |r|)) `
`  supposing valueall-type(X)`

Proof

Definitions occuring in Statement :  fps-mul: `(f*g)` fps-coeff: `f[b]` power-series: `PowerSeries(X;r)` bag-rep: `bag-rep(n;x)` deq: `EqDecider(T)` int_seg: `{i..j-}` nat: `ℕ` valueall-type: `valueall-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` apply: `f a` subtract: `n - m` add: `n + m` natural_number: `\$n` int: `ℤ` universe: `Type` equal: `s = t ∈ T` crng: `CRng` rng_times: `*` rng_zero: `0` rng_car: `|r|`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` fps-coeff: `f[b]` fps-mul: `(f*g)` crng: `CRng` rng: `Rng` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` top: `Top` so_apply: `x[s]` and: `P ∧ Q` cand: `A c∧ B` monoid_p: `IsMonoid(T;op;id)` all: `∀x:A. B[x]` nat: `ℕ` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` int_seg: `{i..j-}` guard: `{T}` ge: `i ≥ j ` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` uiff: `uiff(P;Q)` pi1: `fst(t)` pi2: `snd(t)` squash: `↓T` label: `...\$L... t` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_type: `SQType(T)` infix_ap: `x f y` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  bag-summation-single-non-zero-no-repeats bag_wf rng_car_wf product-deq_wf bag-deq_wf rng_plus_wf rng_zero_wf bag-partitions_wf bag-rep_wf list-subtype-bag rng_times_wf fps-coeff_wf pi1_wf_top pi2_wf rng_all_properties rng_plus_comm2 int_seg_subtype_nat false_wf subtract_wf int_seg_properties nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_term_value_add_lemma int_formula_prop_wf le_wf bag-member_wf all_wf int_seg_wf not_wf equal_wf power-series_wf crng_wf nat_wf deq_wf valueall-type_wf bag-member-partitions bag-append-equal-bag-rep decidable__equal_int bag-size_wf squash_wf true_wf intformeq_wf int_formula_prop_eq_lemma iff_weakening_equal subtype_base_sq int_subtype_base add-is-int-iff decidable__lt lelt_wf and_wf rng_times_zero no-repeats-bag-partitions bag-size-rep
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin productEquality cumulativity hypothesisEquality hypothesis setElimination rename because_Cache independent_isectElimination applyEquality lambdaEquality productElimination independent_pairEquality isect_memberEquality voidElimination voidEquality independent_pairFormation dependent_functionElimination natural_numberEquality addEquality lambdaFormation dependent_set_memberEquality unionElimination dependent_pairFormation int_eqEquality intEquality computeAll functionEquality axiomEquality equalityTransitivity equalitySymmetry universeEquality inlFormation imageElimination imageMemberEquality baseClosed independent_functionElimination instantiate pointwiseFunctionality promote_hyp baseApply closedConclusion inrFormation hyp_replacement applyLambdaEquality

Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)].  \mforall{}[n:\mBbbN{}].  \mforall{}[k:\mBbbN{}n  +  1].  \mforall{}[r:CRng].  \mforall{}[f,g:PowerSeries(X;r)].  \mforall{}[x:X].
(f*g)[bag-rep(n;x)]  =  (*  f[bag-rep(k;x)]  g[bag-rep(n  -  k;x)])
supposing  \mforall{}i:\mBbbN{}n  +  1.  ((\mneg{}(i  =  k))  {}\mRightarrow{}  (f[bag-rep(i;x)]  =  0))
supposing  valueall-type(X)

Date html generated: 2018_05_21-PM-09_54_53
Last ObjectModification: 2017_07_26-PM-06_32_34

Theory : power!series

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