### Nuprl Lemma : fps-scalar-mul-slice

`∀[X:Type]. ∀[r:CRng]. ∀[c:|r|]. ∀[n:ℤ]. ∀[g:PowerSeries(X;r)].  ([(c)*g]_n = (c)*[g]_n ∈ PowerSeries(X;r))`

Proof

Definitions occuring in Statement :  fps-scalar-mul: `(c)*f` fps-slice: `[f]_n` power-series: `PowerSeries(X;r)` uall: `∀[x:A]. B[x]` int: `ℤ` universe: `Type` equal: `s = t ∈ T` crng: `CRng` rng_car: `|r|`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` all: `∀x:A. B[x]` fps-slice: `[f]_n` fps-scalar-mul: `(c)*f` fps-coeff: `f[b]` subtype_rel: `A ⊆r B` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` nat: `ℕ` ifthenelse: `if b then t else f fi ` crng: `CRng` rng: `Rng` power-series: `PowerSeries(X;r)` bfalse: `ff` exists: `∃x:A. B[x]` prop: `ℙ` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False`
Lemmas referenced :  fps-ext fps-slice_wf fps-scalar-mul_wf eq_int_wf bag-size_wf bool_wf eqtt_to_assert assert_of_eq_int nat_wf infix_ap_wf rng_car_wf rng_times_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int rng_times_zero bag_wf power-series_wf crng_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality cumulativity hypothesis productElimination independent_isectElimination lambdaFormation sqequalRule applyEquality because_Cache unionElimination equalityElimination equalityTransitivity equalitySymmetry lambdaEquality setElimination rename dependent_pairFormation promote_hyp dependent_functionElimination instantiate independent_functionElimination voidElimination isect_memberEquality axiomEquality intEquality universeEquality

Latex:
\mforall{}[X:Type].  \mforall{}[r:CRng].  \mforall{}[c:|r|].  \mforall{}[n:\mBbbZ{}].  \mforall{}[g:PowerSeries(X;r)].    ([(c)*g]\_n  =  (c)*[g]\_n)

Date html generated: 2018_05_21-PM-09_57_48
Last ObjectModification: 2017_07_26-PM-06_33_27

Theory : power!series

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