Nuprl Lemma : bag-summation-is-zero
∀[T,R:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R]. ∀[b:bag(T)]. ∀[f:T ⟶ R].
  Σ(x∈b). f[x] = zero ∈ R supposing (∀x:T. (x ↓∈ b 
⇒ (f[x] = zero ∈ R))) ∧ IsMonoid(R;add;zero) ∧ Comm(R;add)
Proof
Definitions occuring in Statement : 
bag-member: x ↓∈ bs
, 
bag-summation: Σ(x∈b). f[x]
, 
bag: bag(T)
, 
comm: Comm(T;op)
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
, 
monoid_p: IsMonoid(T;op;id)
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
and: P ∧ Q
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
cand: A c∧ B
, 
squash: ↓T
, 
prop: ℙ
, 
true: True
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
Lemmas referenced : 
bag-summation-equal, 
equal_wf, 
squash_wf, 
true_wf, 
bag-summation-zero, 
iff_weakening_equal, 
all_wf, 
bag-member_wf, 
monoid_p_wf, 
comm_wf, 
bag_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
sqequalHypSubstitution, 
productElimination, 
thin, 
extract_by_obid, 
isectElimination, 
hypothesisEquality, 
because_Cache, 
hypothesis, 
sqequalRule, 
lambdaEquality, 
applyEquality, 
functionExtensionality, 
cumulativity, 
independent_isectElimination, 
independent_pairFormation, 
imageElimination, 
equalityTransitivity, 
equalitySymmetry, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
universeEquality, 
independent_functionElimination, 
productEquality, 
functionEquality, 
isect_memberEquality, 
axiomEquality
Latex:
\mforall{}[T,R:Type].  \mforall{}[add:R  {}\mrightarrow{}  R  {}\mrightarrow{}  R].  \mforall{}[zero:R].  \mforall{}[b:bag(T)].  \mforall{}[f:T  {}\mrightarrow{}  R].
    \mSigma{}(x\mmember{}b).  f[x]  =  zero 
    supposing  (\mforall{}x:T.  (x  \mdownarrow{}\mmember{}  b  {}\mRightarrow{}  (f[x]  =  zero)))  \mwedge{}  IsMonoid(R;add;zero)  \mwedge{}  Comm(R;add)
Date html generated:
2017_10_01-AM-09_01_43
Last ObjectModification:
2017_07_26-PM-04_43_05
Theory : bags
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