Nuprl Lemma : sum-in-vs-split
∀[n,m,k:ℤ]. ∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[f:{n..m + 1-} ⟶ Point(vs)].
Σ{f[i] | n≤i≤m} = Σ{f[i] | n≤i≤k} + Σ{f[i] | k + 1≤i≤m} ∈ Point(vs) supposing (n ≤ k) ∧ (k ≤ m)
Proof
Definitions occuring in Statement :
sum-in-vs: Σ{f[i] | n≤i≤m},
vs-add: x + y,
vector-space: VectorSpace(K),
vs-point: Point(vs),
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
and: P ∧ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
sum-in-vs: Σ{f[i] | n≤i≤m},
bag-append: as + bs,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
true: True,
rng: Rng
Lemmas referenced :
from-upto-split,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
from-upto_wf,
list-subtype-bag,
le_wf,
less_than_wf,
int_seg_wf,
subtype_rel_sets_simple,
lelt_wf,
decidable__lt,
istype-false,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
less-iff-le,
add-associates,
zero-add,
add_functionality_wrt_le,
le-add-cancel2,
istype-le,
istype-less_than,
not-le-2,
le-add-cancel,
vs-bag-add-append,
vs-point_wf,
vector-space_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
addEquality,
natural_numberEquality,
independent_isectElimination,
dependent_functionElimination,
because_Cache,
hypothesis,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
applyEquality,
setEquality,
intEquality,
productEquality,
inhabitedIsType,
lambdaFormation_alt,
imageElimination,
minusEquality,
productIsType,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
isectIsTypeImplies,
functionIsType,
setElimination,
rename
Latex:
\mforall{}[n,m,k:\mBbbZ{}]. \mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[f:\{n..m + 1\msupminus{}\} {}\mrightarrow{} Point(vs)].
\mSigma{}\{f[i] | n\mleq{}i\mleq{}m\} = \mSigma{}\{f[i] | n\mleq{}i\mleq{}k\} + \mSigma{}\{f[i] | k + 1\mleq{}i\mleq{}m\} supposing (n \mleq{} k) \mwedge{} (k \mleq{} m)
Date html generated:
2019_10_31-AM-06_26_14
Last ObjectModification:
2019_08_08-PM-00_12_44
Theory : linear!algebra
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