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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