Nuprl Lemma : bag-summation-minus
∀[T:Type]. ∀[r:Rng]. ∀[b:bag(T)]. ∀[f:T ⟶ |r|]. (Σ(x∈b). -r f[x] = (-r 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],
so_apply: x[s],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
rng_minus: -r,
rng_zero: 0,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rng: Rng,
comm: Comm(T;op),
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
exists: ∃x:A. B[x],
rng_sig: RngSig,
prop: ℙ,
ring_p: IsRing(T;plus;zero;neg;times;one),
all: ∀x:A. B[x],
squash: ↓T,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
rng_car_wf,
bag_wf,
rng_wf,
rng_plus_comm,
bag-summation-linear1,
rng_plus_wf,
rng_times_wf,
rng_zero_wf,
rng_minus_wf,
rng_properties,
group_p_wf,
rng_all_properties,
rng_one_wf,
equal_wf,
squash_wf,
true_wf,
bag-summation_wf,
assoc_wf,
comm_wf,
rng_times_over_minus,
rng_times_one,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
functionEquality,
cumulativity,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
universeEquality,
independent_isectElimination,
dependent_pairFormation,
productElimination,
functionExtensionality,
applyEquality,
independent_pairFormation,
dependent_functionElimination,
hyp_replacement,
equalitySymmetry,
lambdaEquality,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productEquality,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[r:Rng]. \mforall{}[b:bag(T)]. \mforall{}[f:T {}\mrightarrow{} |r|]. (\mSigma{}(x\mmember{}b). -r f[x] = (-r \mSigma{}(x\mmember{}b). f[x]))
Date html generated:
2017_10_01-AM-08_51_01
Last ObjectModification:
2017_07_26-PM-04_33_04
Theory : bags
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