Nuprl Lemma : divisibility-by-9-rule
∀n:ℕ. ∀a:ℕn ⟶ ℤ. (9 | Σi<n.a[i]*10^i ⇐⇒ 9 | Σ(a[i] | i < n))
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
power-sum: Σi<n.a[i]*x^i,
divides: b | a,
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
eqmod: a ≡ b mod m,
divides: b | a,
exists: ∃x:A. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
subtract: n - m,
prop: ℙ,
false: False,
subtype_rel: A ⊆r B,
power-sum: Σi<n.a[i]*x^i,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
rev_implies: P ⇐ Q,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
int_seg_wf,
nat_wf,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermConstant_wf,
itermMultiply_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
eqmod_wf,
squash_wf,
true_wf,
sum_wf,
exp_wf2,
int_seg_subtype_nat,
false_wf,
int_seg_properties,
itermVar_wf,
int_term_value_var_lemma,
power-sum_wf,
iff_wf,
eqmod_functionality_wrt_eqmod,
power-sum_functionality_wrt_eqmod,
eqmod_weakening,
equal_wf,
exp-one,
iff_weakening_equal,
minus-zero,
add-zero,
divides_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
functionEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
intEquality,
dependent_pairFormation,
dependent_functionElimination,
because_Cache,
unionElimination,
independent_isectElimination,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
baseClosed,
baseApply,
closedConclusion,
applyEquality,
independent_pairFormation,
addLevel,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
imageMemberEquality,
levelHypothesis,
multiplyEquality,
functionExtensionality,
productElimination,
dependent_set_memberEquality,
int_eqEquality,
impliesFunctionality,
independent_functionElimination,
universeEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}. (9 | \mSigma{}i<n.a[i]*10\^{}i \mLeftarrow{}{}\mRightarrow{} 9 | \mSigma{}(a[i] | i < n))
Date html generated:
2018_05_21-PM-08_31_29
Last ObjectModification:
2017_07_26-PM-05_58_08
Theory : general
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