Nuprl Lemma : rng_times_mssum_r
∀s:DSet. ∀r:Rng. ∀f:|s| ⟶ |r|. ∀u:|r|. ∀a:MSet{s}. (((Σx ∈ a. f[x]) * u) = (Σx ∈ a. (f[x] * u)) ∈ |r|)
Proof
Definitions occuring in Statement :
rng_mssum: rng_mssum,
mset: MSet{s},
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T,
rng: Rng,
rng_times: *,
rng_car: |r|,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
dset: DSet,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
rng: Rng,
infix_ap: x f y,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ
Lemmas referenced :
rng_times_lsum_r,
set_car_wf,
list_wf,
rng_mssum_elim_lemma,
all_mset_elim,
equal_wf,
rng_car_wf,
rng_times_wf,
rng_mssum_wf,
mset_wf,
sq_stable__equal,
all_wf,
rng_lsum_wf,
rng_wf,
dset_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
setElimination,
rename,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
addLevel,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination,
productElimination,
levelHypothesis,
functionEquality
Latex:
\mforall{}s:DSet. \mforall{}r:Rng. \mforall{}f:|s| {}\mrightarrow{} |r|. \mforall{}u:|r|. \mforall{}a:MSet\{s\}. (((\mSigma{}x \mmember{} a. f[x]) * u) = (\mSigma{}x \mmember{} a. (f[x] * u)))
Date html generated:
2018_05_22-AM-07_46_07
Last ObjectModification:
2018_05_19-AM-08_30_32
Theory : list_3
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