Nuprl Lemma : fps-compose-zero
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x:X]. ∀[f:PowerSeries(X;r)]. (0(x:=f) = 0 ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-compose: g(x:=f),
fps-zero: 0,
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,
fps-zero: 0,
fps-compose: g(x:=f),
power-series: PowerSeries(X;r),
fps-coeff: f[b],
crng: CRng,
rng: Rng,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
ring_p: IsRing(T;plus;zero;neg;times;one),
group_p: IsGroup(T;op;id;inv)
Lemmas referenced :
bag-summation-is-zero,
list_wf,
bag_wf,
rng_car_wf,
rng_plus_wf,
rng_zero_wf,
bag-parts'_wf2,
infix_ap_wf,
bag-product_wf,
rng_all_properties,
crng_all_properties,
rng_times_zero,
bag-member_wf,
crng_properties,
rng_properties,
rng_plus_comm2,
power-series_wf,
crng_wf,
deq_wf,
valueall-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lambdaEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
independent_isectElimination,
applyEquality,
because_Cache,
productElimination,
independent_pairFormation,
lambdaFormation,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[x:X]. \mforall{}[f:PowerSeries(X;r)]. (0(x:=f) = 0)
supposing valueall-type(X)
Date html generated:
2016_05_15-PM-09_54_10
Last ObjectModification:
2015_12_27-PM-04_36_49
Theory : power!series
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