Nuprl Lemma : vs-bag-add-add
∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[S:Type]. ∀[f,g:S ⟶ Point(vs)]. ∀[bs:bag(S)].
(Σ{f[b] + g[b] | b ∈ bs} = Σ{f[b] | b ∈ bs} + Σ{g[b] | b ∈ bs} ∈ Point(vs))
Proof
Definitions occuring in Statement :
vs-bag-add: Σ{f[b] | b ∈ bs},
vs-add: x + y,
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,
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bag: bag(T),
rng: Rng,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
prop: ℙ,
all: ∀x:A. B[x],
vs-bag-add: Σ{f[b] | b ∈ bs},
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
empty-bag: {},
single-bag: {x},
bag-append: as + bs,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cand: A c∧ B,
monoid_p: IsMonoid(T;op;id),
assoc: Assoc(T;op),
infix_ap: x f y,
ident: Ident(T;op;id),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
comm: Comm(T;op)
Lemmas referenced :
vs-point_wf,
equal-wf-base,
list_wf,
permutation_wf,
bag_wf,
vector-space_wf,
rng_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
int_subtype_base,
list-cases,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
decidable__equal_int,
vs-zero-add,
vs-0_wf,
bag-summation-empty,
list_ind_cons_lemma,
list_ind_nil_lemma,
vs-add_wf,
vs-mon_assoc,
vs-mon_ident,
vs-add-comm-nu,
iff_weakening_equal,
single-bag_wf,
list-subtype-bag,
squash_wf,
true_wf,
bag-summation-append,
rng_sig_wf,
subtype_rel_self,
bag-summation_wf,
bag-summation-single,
vs-add-assoc,
vs-ac_1,
vs-bag-add_wf,
quotient-member-eq,
permutation-equiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
sqequalRule,
pertypeElimination,
productElimination,
productEquality,
isect_memberEquality,
axiomEquality,
functionEquality,
universeEquality,
dependent_functionElimination,
lambdaFormation,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
applyEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
cumulativity,
imageElimination,
imageMemberEquality,
independent_pairEquality,
functionExtensionality,
hyp_replacement
Latex:
\mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[S:Type]. \mforall{}[f,g:S {}\mrightarrow{} Point(vs)]. \mforall{}[bs:bag(S)].
(\mSigma{}\{f[b] + g[b] | b \mmember{} bs\} = \mSigma{}\{f[b] | b \mmember{} bs\} + \mSigma{}\{g[b] | b \mmember{} bs\})
Date html generated:
2018_05_22-PM-09_41_42
Last ObjectModification:
2018_05_20-PM-10_41_57
Theory : linear!algebra
Home
Index