Nuprl Lemma : mset_for_swap
∀g:IAbMonoid. ∀s,s':DSet. ∀f:|s| ⟶ |s'| ⟶ |g|. ∀a:MSet{s}. ∀b:MSet{s'}.
((msFor{g} x ∈ a. msFor{g} y ∈ b. f[x;y]) = (msFor{g} y ∈ b. msFor{g} x ∈ a. f[x;y]) ∈ |g|)
Proof
Definitions occuring in Statement :
mset_for: mset_for,
mset: MSet{s},
so_apply: x[s1;s2],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T,
iabmonoid: IAbMonoid,
grp_car: |g|,
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 y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
iabmonoid: IAbMonoid,
imon: IMonoid,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ
Lemmas referenced :
mon_for_swap,
set_car_wf,
list_wf,
mset_for_elim_lemma,
all_mset_elim,
equal_wf,
grp_car_wf,
mset_for_wf,
mk_mset_wf,
mset_wf,
sq_stable__equal,
all_wf,
sq_stable__all,
mon_for_wf,
dset_wf,
iabmonoid_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
setElimination,
rename,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
addLevel,
isect_memberEquality,
voidElimination,
voidEquality,
allFunctionality,
because_Cache,
independent_functionElimination,
productElimination,
levelHypothesis,
allLevelFunctionality,
functionEquality
Latex:
\mforall{}g:IAbMonoid. \mforall{}s,s':DSet. \mforall{}f:|s| {}\mrightarrow{} |s'| {}\mrightarrow{} |g|. \mforall{}a:MSet\{s\}. \mforall{}b:MSet\{s'\}.
((msFor\{g\} x \mmember{} a. msFor\{g\} y \mmember{} b. f[x;y]) = (msFor\{g\} y \mmember{} b. msFor\{g\} x \mmember{} a. f[x;y]))
Date html generated:
2016_05_16-AM-07_47_57
Last ObjectModification:
2015_12_28-PM-06_03_24
Theory : mset
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