Nuprl Lemma : vs-map-bag-add
∀[K:Rng]. ∀[vs,ws:VectorSpace(K)]. ∀[g:vs ⟶ ws]. ∀[S:Type]. ∀[f:S ⟶ Point(vs)]. ∀[bs:bag(S)].
((g Σ{f[b] | b ∈ bs}) = Σ{g f[b] | b ∈ bs} ∈ Point(ws))
Proof
Definitions occuring in Statement :
vs-map: A ⟶ B,
vs-bag-add: Σ{f[b] | b ∈ bs},
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
so_apply: x[s],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
and: P ∧ Q,
prop: ℙ,
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
subtype_rel: A ⊆r B,
rng: Rng,
vs-map: A ⟶ B,
vs-bag-add: Σ{f[b] | b ∈ bs},
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
cand: A c∧ B,
bag-append: as + bs,
append: as @ bs,
list_ind: list_ind,
single-bag: {x},
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
monoid_p: IsMonoid(T;op;id),
assoc: Assoc(T;op),
infix_ap: x f y,
ident: Ident(T;op;id),
comm: Comm(T;op)
Lemmas referenced :
bag_to_squash_list,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-le,
list_wf,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
istype-nat,
equal_wf,
vs-point_wf,
vs-bag-add_wf,
bag_wf,
istype-universe,
vs-map_wf,
vector-space_wf,
rng_wf,
list_accum_nil_lemma,
vs-map-0,
vs-add_wf,
rng_properties,
rng_car_wf,
vs-mul_wf,
single-bag_wf,
list-subtype-bag,
squash_wf,
true_wf,
vs-bag-add-append,
subtype_rel_self,
iff_weakening_equal,
rng_sig_wf,
vs-0_wf,
vs-mon_assoc,
vs-mon_ident,
vs-add-comm-nu,
bag-summation-single
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
rename,
lambdaFormation_alt,
setElimination,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
unionElimination,
hypothesis_subsumption,
equalityIstype,
because_Cache,
dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
sqequalBase,
hyp_replacement,
isectIsTypeImplies,
functionIsType,
universeEquality,
productIsType,
imageMemberEquality,
independent_pairEquality
Latex:
\mforall{}[K:Rng]. \mforall{}[vs,ws:VectorSpace(K)]. \mforall{}[g:vs {}\mrightarrow{} ws]. \mforall{}[S:Type]. \mforall{}[f:S {}\mrightarrow{} Point(vs)]. \mforall{}[bs:bag(S)].
((g \mSigma{}\{f[b] | b \mmember{} bs\}) = \mSigma{}\{g f[b] | b \mmember{} bs\})
Date html generated:
2019_10_31-AM-06_27_07
Last ObjectModification:
2019_08_07-AM-11_26_39
Theory : linear!algebra
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