### Nuprl Lemma : bag-summation-single-non-zero-no-repeats

`∀[T,R:Type]. ∀[eq:EqDecider(T)]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R]. ∀[b:bag(T)]. ∀[f:T ⟶ R].`
`  ∀z:T`
`    (Σ(x∈b). f[x] = f[z] ∈ R) supposing `
`       ((bag-no-repeats(T;b) ∧ z ↓∈ b) and `
`       (∀x:T. (x ↓∈ b `` ((x = z ∈ T) ∨ (f[x] = zero ∈ R))))) `
`  supposing IsMonoid(R;add;zero) ∧ Comm(R;add)`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag-no-repeats: `bag-no-repeats(T;bs)` bag-summation: `Σ(x∈b). f[x]` bag: `bag(T)` deq: `EqDecider(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` or: `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` all: `∀x:A. B[x]` and: `P ∧ Q` squash: `↓T` prop: `ℙ` so_apply: `x[s]` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` so_lambda: `λ2x.t[x]` deq: `EqDecider(T)` cand: `A c∧ B` uiff: `uiff(P;Q)` bag-member: `x ↓∈ bs` or: `P ∨ Q` monoid_p: `IsMonoid(T;op;id)` eqof: `eqof(d)` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  bag-summation-single-non-zero equal_wf squash_wf true_wf iff_weakening_equal bag-extensionality-no-repeats decidable-equal-deq bag-filter_wf subtype_rel_bag assert_wf single-bag_wf bag-single-no-repeats bag-member-single bag-member_wf bag-summation-single bag-summation_wf bag-no-repeats_wf all_wf or_wf monoid_p_wf comm_wf bag_wf deq_wf bag-filter-no-repeats bag-member-filter safe-assert-deq and_wf
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination lambdaFormation dependent_functionElimination productElimination applyEquality lambdaEquality imageElimination equalityTransitivity equalitySymmetry universeEquality functionExtensionality cumulativity natural_numberEquality sqequalRule imageMemberEquality baseClosed independent_functionElimination because_Cache setElimination rename setEquality independent_pairFormation hyp_replacement applyLambdaEquality productEquality isect_memberEquality axiomEquality functionEquality dependent_set_memberEquality

Latex:
\mforall{}[T,R:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[add:R  {}\mrightarrow{}  R  {}\mrightarrow{}  R].  \mforall{}[zero:R].  \mforall{}[b:bag(T)].  \mforall{}[f:T  {}\mrightarrow{}  R].
\mforall{}z:T
(\mSigma{}(x\mmember{}b).  f[x]  =  f[z])  supposing
((bag-no-repeats(T;b)  \mwedge{}  z  \mdownarrow{}\mmember{}  b)  and
(\mforall{}x:T.  (x  \mdownarrow{}\mmember{}  b  {}\mRightarrow{}  ((x  =  z)  \mvee{}  (f[x]  =  zero)))))