Nuprl Lemma : power-sum-linear
∀[n:ℕ]. ∀[x:ℤ]. ∀[a,b:ℕn ⟶ ℤ]. ∀[c,d:ℤ].
(Σi<n.(c * a[i]) + (d * b[i])*x^i = ((c * Σi<n.a[i]*x^i) + (d * Σi<n.b[i]*x^i)) ∈ ℤ)
Proof
Definitions occuring in Statement :
power-sum: Σi<n.a[i]*x^i,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
power-sum: Σi<n.a[i]*x^i,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
int_seg_wf,
nat_wf,
sum_wf,
exp_wf2,
int_seg_subtype_nat,
false_wf,
equal_wf,
squash_wf,
true_wf,
add_functionality_wrt_eq,
sum_scalar_mult,
iff_weakening_equal,
sum_linear,
int_seg_properties,
nat_properties,
decidable__equal_int,
le_wf,
lelt_wf,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
intEquality,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
because_Cache,
functionEquality,
extract_by_obid,
natural_numberEquality,
setElimination,
rename,
lambdaEquality,
multiplyEquality,
addEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
dependent_set_memberEquality,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbZ{}]. \mforall{}[a,b:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. \mforall{}[c,d:\mBbbZ{}].
(\mSigma{}i<n.(c * a[i]) + (d * b[i])*x\^{}i = ((c * \mSigma{}i<n.a[i]*x\^{}i) + (d * \mSigma{}i<n.b[i]*x\^{}i)))
Date html generated:
2018_05_21-PM-08_30_16
Last ObjectModification:
2017_07_26-PM-05_57_20
Theory : general
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