### Nuprl Lemma : fps-deriv-mul

`∀[X:Type]`
`  ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)]. ∀[x:X].`
`    (d(f*g)/dx = ((f*dg/dx)+(df/dx*g)) ∈ PowerSeries(X;r)) `
`  supposing valueall-type(X)`

Proof

Definitions occuring in Statement :  fps-deriv: `df/dx` fps-mul: `(f*g)` fps-add: `(f+g)` power-series: `PowerSeries(X;r)` deq: `EqDecider(T)` valueall-type: `valueall-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` universe: `Type` equal: `s = t ∈ T` crng: `CRng`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` and: `P ∧ Q` cand: `A c∧ B` all: `∀x:A. B[x]` crng: `CRng` rng: `Rng` implies: `P `` Q` compose: `f o g` squash: `↓T` prop: `ℙ` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` nat: `ℕ` 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)` bnot: `¬bb` assert: `↑b` false: `False` top: `Top` decidable: `Dec(P)` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` nequal: `a ≠ b ∈ T ` infix_ap: `x f y` ge: `i ≥ j ` le: `A ≤ B` fps-deriv: `df/dx` fps-add: `(f+g)`
Lemmas referenced :  fps-linear-ucont-equal fps-deriv_wf fps-mul_wf power-series_wf fps-add_wf rng_car_wf bag_wf crng_wf deq_wf valueall-type_wf fps-ucont-composition fps-deriv-ucont fps-mul-ucont fps-add-ucont-general equal_wf squash_wf true_wf fps-deriv-add subtype_rel_self iff_weakening_equal mul_over_plus_fps mul_comm_fps mon_assoc_fps abmonoid_ac_1_fps fps-scalar-mul_wf fps-deriv-scalar-mul fps-scalar-mul-mul fps-scalar-mul-add fps-single_wf fps-ucont_wf fps-mul-comm abmonoid_comm_fps bag-append_wf int-to-ring_wf bag-count_wf nat_wf bag-drop_wf fps-mul-single fps-deriv-single eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int bag-drop-append int_subtype_base int-to-ring-zero mon_ident_fps rng_wf fps-scalar-mul-zero bag-count-append decidable__equal_int full-omega-unsat intformand_wf intformnot_wf intformeq_wf itermAdd_wf itermVar_wf itermConstant_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_add_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_wf decidable__lt assert_wf bnot_wf not_wf equal-wf-T-base intformless_wf int_formula_prop_less_lemma bool_cases iff_transitivity iff_weakening_uiff assert_of_bnot bag-append-comm rng_plus_wf fps-scalar-mul-rng-add int-to-ring-add set_subtype_base le_wf decidable__le nat_properties intformle_wf int_formula_prop_le_lemma fps-zero_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache independent_isectElimination hypothesis hypothesisEquality sqequalRule lambdaEquality independent_pairFormation lambdaFormation setElimination rename isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry universeEquality dependent_functionElimination independent_functionElimination applyEquality imageElimination natural_numberEquality imageMemberEquality baseClosed instantiate productElimination hyp_replacement functionEquality unionElimination equalityElimination dependent_pairFormation promote_hyp cumulativity voidElimination intEquality voidEquality approximateComputation int_eqEquality applyLambdaEquality impliesFunctionality dependent_set_memberEquality addEquality

Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[f,g:PowerSeries(X;r)].  \mforall{}[x:X].
(d(f*g)/dx  =  ((f*dg/dx)+(df/dx*g)))
supposing  valueall-type(X)

Date html generated: 2018_05_21-PM-10_16_51
Last ObjectModification: 2018_05_19-PM-04_18_58

Theory : power!series

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