### Nuprl Lemma : bag-summation-equal

`∀[T,R:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R]. ∀[b:bag(T)]. ∀[f,g:T ⟶ R].`
`  Σ(x∈b). f[x] = Σ(x∈b). g[x] ∈ R supposing (∀x:T. (x ↓∈ b `` (f[x] = g[x] ∈ 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` prop: `ℙ` so_lambda: `λ2x.t[x]` implies: `P `` Q` so_apply: `x[s]` all: `∀x:A. B[x]` squash: `↓T` cand: `A c∧ B` sq_stable: `SqStable(P)` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` monoid_p: `IsMonoid(T;op;id)`
Lemmas referenced :  all_wf bag-member_wf equal_wf monoid_p_wf comm_wf bag_wf bag-subtype bag-summation_wf squash_wf assoc_wf set_wf true_wf sq_stable__bag-member iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution productElimination thin hypothesis productEquality extract_by_obid isectElimination cumulativity hypothesisEquality sqequalRule lambdaEquality functionEquality applyEquality functionExtensionality isect_memberEquality axiomEquality because_Cache equalityTransitivity equalitySymmetry dependent_functionElimination setEquality lambdaFormation independent_functionElimination imageElimination independent_isectElimination independent_pairFormation universeEquality setElimination rename imageMemberEquality baseClosed natural_numberEquality

Latex:
\mforall{}[T,R:Type].  \mforall{}[add:R  {}\mrightarrow{}  R  {}\mrightarrow{}  R].  \mforall{}[zero:R].  \mforall{}[b:bag(T)].  \mforall{}[f,g:T  {}\mrightarrow{}  R].
\mSigma{}(x\mmember{}b).  f[x]  =  \mSigma{}(x\mmember{}b).  g[x]
supposing  (\mforall{}x:T.  (x  \mdownarrow{}\mmember{}  b  {}\mRightarrow{}  (f[x]  =  g[x])))  \mwedge{}  IsMonoid(R;add;zero)  \mwedge{}  Comm(R;add)

Date html generated: 2017_10_01-AM-09_01_30
Last ObjectModification: 2017_07_26-PM-04_42_54

Theory : bags

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