Nuprl Lemma : bag-summation-linear1

`∀[T,R:Type]. ∀[add,mul:R ⟶ R ⟶ R]. ∀[zero:R]. ∀[b:bag(T)]. ∀[f:T ⟶ R].`
`  ∀a:R. (Σ(x∈b). a mul f[x] = (a mul Σ(x∈b). f[x]) ∈ R) `
`  supposing (∃minus:R ⟶ R. IsGroup(R;add;zero;minus)) ∧ Comm(R;add) ∧ BiLinear(R;add;mul)`

Proof

Definitions occuring in Statement :  bag-summation: `Σ(x∈b). f[x]` bag: `bag(T)` comm: `Comm(T;op)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` infix_ap: `x f y` so_apply: `x[s]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` and: `P ∧ Q` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T` group_p: `IsGroup(T;op;id;inv)` bilinear: `BiLinear(T;pl;tm)`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` and: `P ∧ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` cand: `A c∧ B` prop: `ℙ` squash: `↓T` exists: `∃x:A. B[x]` infix_ap: `x f y` true: `True` group_p: `IsGroup(T;op;id;inv)` monoid_p: `IsMonoid(T;op;id)` guard: `{T}` ident: `Ident(T;op;id)` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` assoc: `Assoc(T;op)`
Lemmas referenced :  bag-summation-linear equal_wf squash_wf true_wf bag-summation_wf exists_wf group_p_wf comm_wf bilinear_wf bag_wf bag-summation-zero iff_weakening_equal
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality lambdaFormation productElimination sqequalRule lambdaEquality cumulativity independent_isectElimination independent_pairFormation dependent_functionElimination hyp_replacement equalitySymmetry applyEquality imageElimination equalityTransitivity universeEquality because_Cache imageMemberEquality baseClosed natural_numberEquality axiomEquality productEquality functionEquality functionExtensionality independent_functionElimination

Latex:
\mforall{}[T,R:Type].  \mforall{}[add,mul:R  {}\mrightarrow{}  R  {}\mrightarrow{}  R].  \mforall{}[zero:R].  \mforall{}[b:bag(T)].  \mforall{}[f:T  {}\mrightarrow{}  R].
\mforall{}a:R.  (\mSigma{}(x\mmember{}b).  a  mul  f[x]  =  (a  mul  \mSigma{}(x\mmember{}b).  f[x]))

Date html generated: 2017_10_01-AM-08_50_53
Last ObjectModification: 2017_07_26-PM-04_32_58

Theory : bags

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