### 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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