Nuprl Lemma : mset_for_of_op
∀g:IAbMonoid. ∀s:DSet. ∀e,f:|s| ⟶ |g|. ∀a:MSet{s}.
((msFor{g} x ∈ a. (e[x] * f[x])) = ((msFor{g} x ∈ a. e[x]) * (msFor{g} x ∈ a. f[x])) ∈ |g|)
Proof
Definitions occuring in Statement :
mset_for: mset_for,
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,
iabmonoid: IAbMonoid,
grp_op: *,
grp_car: |g|,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
top: Top,
so_apply: x[s],
uall: ∀[x:A]. B[x],
iabmonoid: IAbMonoid,
imon: IMonoid,
dset: DSet,
implies: P ⇒ Q,
infix_ap: x f y,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ
Lemmas referenced :
mset_for_elim_lemma,
all_mset_elim,
equal_wf,
grp_car_wf,
mset_for_wf,
infix_ap_wf,
grp_op_wf,
set_car_wf,
mset_wf,
sq_stable__equal,
all_wf,
list_wf,
mon_for_wf,
dset_wf,
iabmonoid_wf,
mon_for_of_op
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
addLevel,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
hypothesisEquality,
lambdaEquality,
isectElimination,
setElimination,
rename,
because_Cache,
applyEquality,
independent_functionElimination,
productElimination,
levelHypothesis,
functionEquality
Latex:
\mforall{}g:IAbMonoid. \mforall{}s:DSet. \mforall{}e,f:|s| {}\mrightarrow{} |g|. \mforall{}a:MSet\{s\}.
((msFor\{g\} x \mmember{} a. (e[x] * f[x])) = ((msFor\{g\} x \mmember{} a. e[x]) * (msFor\{g\} x \mmember{} a. f[x])))
Date html generated:
2016_05_16-AM-07_47_52
Last ObjectModification:
2015_12_28-PM-06_03_10
Theory : mset
Home
Index