### Nuprl Lemma : fps-mul-comm

`∀[X:Type]. ∀[eq:EqDecider(X)].`
`  ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)].  ((f*g) = (g*f) ∈ PowerSeries(X;r)) supposing valueall-type(X)`

Proof

Definitions occuring in Statement :  fps-mul: `(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` power-series: `PowerSeries(X;r)` fps-mul: `(f*g)` fps-coeff: `f[b]` infix_ap: `x f y` crng: `CRng` comm: `Comm(T;op)` and: `P ∧ Q` cand: `A c∧ B` rng: `Rng` so_lambda: `λ2x.t[x]` pi1: `fst(t)` pi2: `snd(t)` so_apply: `x[s]` true: `True` squash: `↓T` prop: `ℙ` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` top: `Top` all: `∀x:A. B[x]`
Lemmas referenced :  rng_plus_comm rng_all_properties bag_wf power-series_wf valueall-type_wf rng_car_wf bag-summation_wf rng_plus_wf rng_zero_wf infix_ap_wf rng_times_wf fps-coeff_wf bag-partitions_wf equal_wf squash_wf true_wf bag-partitions-symmetry iff_weakening_equal bag-summation-map bag-subtype-list assoc_wf comm_wf crng_times_comm pi1_wf_top pi2_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule lambdaEquality extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis because_Cache productElimination independent_pairFormation cumulativity isect_memberEquality axiomEquality equalityTransitivity equalitySymmetry productEquality independent_isectElimination natural_numberEquality applyEquality imageElimination imageMemberEquality baseClosed universeEquality independent_functionElimination voidElimination voidEquality dependent_functionElimination functionExtensionality functionEquality independent_pairEquality

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

Date html generated: 2018_05_21-PM-09_54_58
Last ObjectModification: 2017_07_26-PM-06_32_35

Theory : power!series

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