Nuprl Lemma : bag-summation-ring-linear1
∀[T:Type]. ∀[r:Rng]. ∀[b:bag(T)]. ∀[f:T ⟶ |r|].
∀a:|r|. ((Σ(x∈b). f[x] * a = (Σ(x∈b). f[x] * a) ∈ |r|) ∧ (Σ(x∈b). a * f[x] = (a * Σ(x∈b). f[x]) ∈ |r|))
Proof
Definitions occuring in Statement :
bag-summation: Σ(x∈b). f[x],
bag: bag(T),
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
rng_times: *,
rng_zero: 0,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
comm: Comm(T;op),
exists: ∃x:A. B[x],
rng: Rng,
rng_sig: RngSig,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
ring_p: IsRing(T;plus;zero;neg;times;one)
Lemmas referenced :
rng_plus_comm,
rng_all_properties,
rng_minus_wf,
rng_properties,
group_p_wf,
rng_car_wf,
rng_plus_wf,
rng_zero_wf,
bag-summation-linear1-right,
rng_times_wf,
bag-summation-linear1,
bag_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
hypothesis,
independent_pairFormation,
dependent_pairFormation,
setElimination,
rename,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
independent_isectElimination,
dependent_functionElimination,
independent_pairEquality,
axiomEquality,
functionEquality,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[r:Rng]. \mforall{}[b:bag(T)]. \mforall{}[f:T {}\mrightarrow{} |r|].
\mforall{}a:|r|. ((\mSigma{}(x\mmember{}b). f[x] * a = (\mSigma{}(x\mmember{}b). f[x] * a)) \mwedge{} (\mSigma{}(x\mmember{}b). a * f[x] = (a * \mSigma{}(x\mmember{}b). f[x])))
Date html generated:
2016_05_15-PM-02_32_21
Last ObjectModification:
2015_12_27-AM-09_47_51
Theory : bags
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