Nuprl Lemma : rng_mssum_when_swap
∀s:DSet. ∀r:Rng. ∀f:|s| ⟶ |r|. ∀b:𝔹. ∀a:MSet{s}. ((Σx ∈ a. (when b. f[x])) = (when b. (Σx ∈ a. f[x])) ∈ |r|)
Proof
Definitions occuring in Statement :
rng_mssum: rng_mssum,
mset: MSet{s},
bool: 𝔹,
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T,
rng_when: rng_when,
rng: Rng,
rng_car: |r|,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
rng: Rng,
dset: DSet,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
rng_when: rng_when,
prop: ℙ
Lemmas referenced :
rng_mssum_elim_lemma,
all_mset_elim,
equal_wf,
rng_car_wf,
rng_mssum_wf,
rng_when_wf,
set_car_wf,
mset_wf,
sq_stable__equal,
all_wf,
list_wf,
rng_lsum_wf,
bool_wf,
rng_wf,
dset_wf,
rng_lsum_when_swap
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
addLevel,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
lambdaEquality,
setElimination,
rename,
applyEquality,
independent_functionElimination,
productElimination,
levelHypothesis,
functionEquality
Latex:
\mforall{}s:DSet. \mforall{}r:Rng. \mforall{}f:|s| {}\mrightarrow{} |r|. \mforall{}b:\mBbbB{}. \mforall{}a:MSet\{s\}.
((\mSigma{}x \mmember{} a. (when b. f[x])) = (when b. (\mSigma{}x \mmember{} a. f[x])))
Date html generated:
2018_05_22-AM-07_46_09
Last ObjectModification:
2018_05_19-AM-08_30_36
Theory : list_3
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