Nuprl Lemma : divisibility-by-3-rule

n:ℕ. ∀a:ℕn ⟶ ℤ.  (3 | Σi<n.a[i]*10^i ⇐⇒ | Σ(a[i] i < n))

This theorem is one of freek's list of 100 theorems


Definitions occuring in Statement :  power-sum: Σi<n.a[i]*x^i divides: a sum: Σ(f[x] x < k) int_seg: {i..j-} nat: so_apply: x[s] all: x:A. B[x] iff: ⇐⇒ Q function: x:A ⟶ B[x] natural_number: $n int:
Definitions unfolded in proof :  all: x:A. B[x] member: t ∈ T uall: [x:A]. B[x] nat: eqmod: a ≡ mod m divides: a exists: x:A. B[x] ge: i ≥  decidable: Dec(P) or: P ∨ Q uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) top: Top subtract: m prop: false: False subtype_rel: A ⊆B power-sum: Σi<n.a[i]*x^i iff: ⇐⇒ Q and: P ∧ Q implies:  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:  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

\mforall{}n:\mBbbN{}.  \mforall{}a:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}.    (3  |  \mSigma{}i<n.a[i]*10\^{}i  \mLeftarrow{}{}\mRightarrow{}  3  |  \mSigma{}(a[i]  |  i  <  n))

Date html generated: 2018_05_21-PM-08_31_15
Last ObjectModification: 2017_07_26-PM-05_57_56

Theory : general

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