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]))
supposing (\mexists{}minus:R {}\mrightarrow{} R. IsGroup(R;add;zero;minus)) \mwedge{} Comm(R;add) \mwedge{} BiLinear(R;add;mul)
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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