### Nuprl Lemma : divisibility-by-3-rule

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