Nuprl Lemma : vs-bag-add-mul
∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[S:Type]. ∀[f:S ⟶ Point(vs)]. ∀[bs:bag(S)]. ∀[k:|K|].
(k * Σ{f[b] | b ∈ bs} = Σ{k * f[b] | b ∈ bs} ∈ Point(vs))
Proof
Definitions occuring in Statement :
vs-bag-add: Σ{f[b] | b ∈ bs},
vs-mul: a * x,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
rng_car: |r|,
bag: bag(T)
Definitions unfolded in proof :
all: ∀x:A. B[x],
prop: ℙ,
and: P ∧ Q,
quotient: x,y:A//B[x; y],
rng: Rng,
bag: bag(T),
member: t ∈ T,
uall: ∀[x:A]. B[x],
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
it: ⋅,
nil: [],
decidable: Dec(P),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
colength: colength(L),
cons: [a / b],
or: P ∨ Q,
guard: {T},
subtype_rel: A ⊆r B,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
vs-bag-add: Σ{f[b] | b ∈ bs},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
empty-bag: {},
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
bag-append: as + bs,
single-bag: {x},
comm: Comm(T;op),
ident: Ident(T;op;id),
infix_ap: x f y,
assoc: Assoc(T;op),
monoid_p: IsMonoid(T;op;id),
cand: A c∧ B
Lemmas referenced :
rng_wf,
vector-space_wf,
bag_wf,
rng_car_wf,
permutation_wf,
list_wf,
equal-wf-base,
vs-point_wf,
decidable__equal_int,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
equal_wf,
le_wf,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__le,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
spread_cons_lemma,
product_subtype_list,
list-cases,
less_than_irreflexivity,
less_than_transitivity1,
colength_wf_list,
nat_wf,
equal-wf-T-base,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
iff_weakening_equal,
vs-0_wf,
vs-zero-mul,
true_wf,
squash_wf,
bag-summation-empty,
list_ind_nil_lemma,
list_ind_cons_lemma,
vs-mul_wf,
list-subtype-bag,
single-bag_wf,
vs-add-comm,
vs-mon_ident,
vs-mon_assoc,
vs-add_wf,
rng_sig_wf,
bag-summation-append,
bag-summation_wf,
vs-mul-linear,
bag-summation-single,
vs-bag-add_wf,
permutation-equiv,
quotient-member-eq
Rules used in proof :
dependent_functionElimination,
universeEquality,
functionEquality,
axiomEquality,
isect_memberEquality,
cumulativity,
productEquality,
productElimination,
pertypeElimination,
sqequalRule,
hypothesisEquality,
hypothesis,
because_Cache,
rename,
setElimination,
thin,
isectElimination,
extract_by_obid,
pointwiseFunctionalityForEquality,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
imageElimination,
instantiate,
baseClosed,
addEquality,
dependent_set_memberEquality,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
hypothesis_subsumption,
promote_hyp,
unionElimination,
applyEquality,
independent_pairFormation,
voidEquality,
voidElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
lambdaFormation,
imageMemberEquality,
functionExtensionality,
independent_pairEquality,
hyp_replacement
Latex:
\mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[S:Type]. \mforall{}[f:S {}\mrightarrow{} Point(vs)]. \mforall{}[bs:bag(S)]. \mforall{}[k:|K|].
(k * \mSigma{}\{f[b] | b \mmember{} bs\} = \mSigma{}\{k * f[b] | b \mmember{} bs\})
Date html generated:
2018_05_22-PM-09_41_35
Last ObjectModification:
2018_01_09-PM-01_03_52
Theory : linear!algebra
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