Nuprl Lemma : fps-scalar-mul-property
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng].
((IsAction(|r|;*;1;PowerSeries(X;r);λc,f. (c)*f)
∧ IsBilinear(|r|;PowerSeries(X;r);PowerSeries(X;r);+r;λf,g. (f+g);λf,g. (f+g);λc,f. (c)*f))
∧ (∀c:|r|. Dist1op2opLR(PowerSeries(X;r);λf.(c)*f;λf,g. (f*g))))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-scalar-mul: (c)*f,
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],
all: ∀x:A. B[x],
and: P ∧ Q,
lambda: λx.A[x],
universe: Type,
crng: CRng,
rng_one: 1,
rng_times: *,
rng_plus: +r,
rng_car: |r|,
dist_1op_2op_lr: Dist1op2opLR(A;1op;2op),
action_p: IsAction(A;x;e;S;f),
bilinear_p: IsBilinear(A;B;C;+a;+b;+c;f)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
crng: CRng,
comm: Comm(T;op),
rng: Rng,
prop: ℙ,
and: P ∧ Q,
cand: A c∧ B,
exists: ∃x:A. B[x],
action_p: IsAction(A;x;e;S;f),
all: ∀x:A. B[x],
bilinear_p: IsBilinear(A;B;C;+a;+b;+c;f),
dist_1op_2op_lr: Dist1op2opLR(A;1op;2op),
ring_p: IsRing(T;plus;zero;neg;times;one),
group_p: IsGroup(T;op;id;inv),
infix_ap: x f y,
uiff: uiff(P;Q),
fps-scalar-mul: (c)*f,
fps-coeff: f[b],
fps-add: (f+g),
fps-mul: (f*g),
power-series: PowerSeries(X;r),
true: True,
so_lambda: λ2x.t[x],
pi1: fst(t),
pi2: snd(t),
so_apply: x[s],
top: Top,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
rng_plus_comm,
crng_properties,
rng_properties,
rng_all_properties,
ring_p_wf,
rng_car_wf,
rng_plus_wf,
rng_zero_wf,
rng_minus_wf,
rng_times_wf,
rng_one_wf,
group_p_wf,
crng_wf,
deq_wf,
valueall-type_wf,
power-series_wf,
fps-ext,
fps-scalar-mul_wf,
bag_wf,
fps-add_wf,
fps-mul_wf,
infix_ap_wf,
bag-summation_wf,
bag-partitions_wf,
pi1_wf_top,
pi2_wf,
equal_wf,
bag-summation-linear1-right,
iff_weakening_equal,
bag-summation-equal,
rng_times_assoc,
bag-member_wf,
squash_wf,
true_wf,
crng_times_comm,
crng_times_ac_1,
rng_times_one,
rng_times_over_plus,
assoc_wf,
comm_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
productElimination,
because_Cache,
independent_pairFormation,
dependent_pairFormation,
functionExtensionality,
applyEquality,
sqequalRule,
independent_pairEquality,
isect_memberEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
universeEquality,
lambdaFormation,
independent_isectElimination,
natural_numberEquality,
productEquality,
voidElimination,
voidEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination,
functionEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng].
((IsAction(|r|;*;1;PowerSeries(X;r);\mlambda{}c,f. (c)*f)
\mwedge{} IsBilinear(|r|;PowerSeries(X;r);PowerSeries(X;r);+r;\mlambda{}f,g. (f+g);\mlambda{}f,g. (f+g);\mlambda{}c,f. (c)*f))
\mwedge{} (\mforall{}c:|r|. Dist1op2opLR(PowerSeries(X;r);\mlambda{}f.(c)*f;\mlambda{}f,g. (f*g))))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_57_26
Last ObjectModification:
2017_07_26-PM-06_33_21
Theory : power!series
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