Nuprl Lemma : empty-sum-in-vs
∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[n,m:ℤ]. ∀[f:Top]. Σ{f[i] | n≤i≤m} = 0 ∈ Point(vs) supposing m < n
Proof
Definitions occuring in Statement :
sum-in-vs: Σ{f[i] | n≤i≤m},
vs-0: 0,
vector-space: VectorSpace(K),
vs-point: Point(vs),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
int: ℤ,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sum-in-vs: Σ{f[i] | n≤i≤m},
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
from-upto: [n, m),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
empty-bag: {},
nil: [],
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),
list_accum: list_accum,
vs-0: 0,
record-select: r.x,
rng: Rng
Lemmas referenced :
subtype_base_sq,
list_wf,
le_wf,
less_than_wf,
list_subtype_base,
set_subtype_base,
int_subtype_base,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
istype-less_than,
nil_wf,
vs-0_wf,
istype-top,
vector-space_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
setEquality,
intEquality,
productEquality,
hypothesisEquality,
hypothesis,
addEquality,
natural_numberEquality,
independent_isectElimination,
sqequalRule,
lambdaEquality_alt,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
because_Cache,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
equalityIstype,
promote_hyp,
setElimination,
rename,
axiomEquality,
isectIsTypeImplies
Latex:
\mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[n,m:\mBbbZ{}]. \mforall{}[f:Top]. \mSigma{}\{f[i] | n\mleq{}i\mleq{}m\} = 0 supposing m < n
Date html generated:
2019_10_31-AM-06_26_03
Last ObjectModification:
2019_08_13-PM-05_02_22
Theory : linear!algebra
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