Nuprl Lemma : sum-in-vs-const
∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[n,m:ℤ]. ∀[f:{n..m + 1-} ⟶ |K|]. ∀[a:Point(vs)].
(Σ{f[i] * a | n≤i≤m} = Σ(K) n ≤ i < m + 1. f[i] * a ∈ Point(vs))
Proof
Definitions occuring in Statement :
sum-in-vs: Σ{f[i] | n≤i≤m},
vs-mul: a * x,
vector-space: VectorSpace(K),
vs-point: Point(vs),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
rng_sum: rng_sum,
rng: Rng,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rng: Rng,
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),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y,
cand: A c∧ B,
less_than: a < b,
subtract: n - m
Lemmas referenced :
vs-point_wf,
int_seg_wf,
rng_car_wf,
istype-int,
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,
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,
int_seg_properties,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
vs-mul-zero,
equal_wf,
squash_wf,
true_wf,
istype-universe,
empty-sum-in-vs,
vs-mul_wf,
rng_sig_wf,
rng_sum_unroll_empty,
iff_weakening_equal,
sum-in-vs_wf,
rng_sum_unroll_hi,
add-subtract-cancel,
subtract-add-cancel,
rng_plus_wf,
rng_sum_wf,
vs-mul-add,
vs-add_wf,
sum-in-vs-split,
add-associates,
add-commutes,
add-swap,
zero-add,
vs-add-comm-nu,
sum-in-vs-single,
vs-0_wf,
vs-zero-add,
rng_zero_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
addEquality,
natural_numberEquality,
dependent_functionElimination,
lambdaFormation_alt,
intWeakElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
functionIsTypeImplies,
productElimination,
unionElimination,
applyEquality,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
because_Cache,
productIsType,
hypothesis_subsumption,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
minusEquality
Latex:
\mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[n,m:\mBbbZ{}]. \mforall{}[f:\{n..m + 1\msupminus{}\} {}\mrightarrow{} |K|]. \mforall{}[a:Point(vs)].
(\mSigma{}\{f[i] * a | n\mleq{}i\mleq{}m\} = \mSigma{}(K) n \mleq{} i < m + 1. f[i] * a)
Date html generated:
2019_10_31-AM-06_26_28
Last ObjectModification:
2019_08_19-AM-10_33_16
Theory : linear!algebra
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