Nuprl Lemma : mset_for_mset_sum
∀s:DSet. ∀g:IAbMonoid. ∀f:|s| ⟶ |g|. ∀a,b:MSet{s}.
((msFor{g} x ∈ a + b. f[x]) = ((msFor{g} x ∈ a. f[x]) * (msFor{g} x ∈ b. f[x])) ∈ |g|)
Proof
Definitions occuring in Statement :
mset_for: mset_for,
mset_sum: a + b,
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],
member: t ∈ T,
uall: ∀[x:A]. B[x],
dset: DSet,
iabmonoid: IAbMonoid,
imon: IMonoid,
mset: MSet{s},
quotient: x,y:A//B[x; y],
and: P ∧ Q,
implies: P ⇒ Q,
mset_for: mset_for,
mset_sum: a + b,
prop: ℙ,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
mset_wf,
set_car_wf,
grp_car_wf,
iabmonoid_wf,
dset_wf,
list_wf,
permr_wf,
equal_wf,
equal-wf-base,
squash_wf,
true_wf,
mon_for_functionality_wrt_permr,
append_wf,
append_functionality_wrt_permr,
mem_f_wf,
infix_ap_wf,
grp_op_wf,
mon_for_wf,
iff_weakening_equal,
mon_for_append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
functionEquality,
isectElimination,
setElimination,
rename,
pointwiseFunctionalityForEquality,
sqequalRule,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
independent_functionElimination,
productEquality,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
functionExtensionality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}s:DSet. \mforall{}g:IAbMonoid. \mforall{}f:|s| {}\mrightarrow{} |g|. \mforall{}a,b:MSet\{s\}.
((msFor\{g\} x \mmember{} a + b. f[x]) = ((msFor\{g\} x \mmember{} a. f[x]) * (msFor\{g\} x \mmember{} b. f[x])))
Date html generated:
2017_10_01-AM-09_59_19
Last ObjectModification:
2017_03_03-PM-01_00_12
Theory : mset
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