### Nuprl Lemma : divisibility-by-2-rule

`∀n:ℕ+. ∀a:ℕn ⟶ ℤ.  (2 | Σi<n.a[i]*10^i `⇐⇒` 2 | a[0])`

Proof

Definitions occuring in Statement :  power-sum: `Σi<n.a[i]*x^i` divides: `b | a` int_seg: `{i..j-}` nat_plus: `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_plus: `ℕ+` eqmod: `a ≡ b mod m` divides: `b | a` exists: `∃x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` implies: `P `` Q` 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` so_lambda: `λ2x.t[x]` so_apply: `x[s]` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` less_than': `less_than'(a;b)` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` sq_type: `SQType(T)` guard: `{T}` squash: `↓T` true: `True` nat: `ℕ` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T `
Lemmas referenced :  int_seg_wf istype-int nat_plus_wf nat_plus_properties decidable__equal_int full-omega-unsat intformnot_wf intformeq_wf itermSubtract_wf itermConstant_wf itermMultiply_wf int_formula_prop_not_lemma istype-void int_formula_prop_eq_lemma int_term_value_subtract_lemma int_term_value_constant_lemma int_term_value_mul_lemma int_formula_prop_wf int_subtype_base power-sum_wf nat_plus_subtype_nat eqmod_wf istype-false decidable__lt intformand_wf intformless_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma le_wf less_than_wf eqmod_functionality_wrt_eqmod power-sum_functionality_wrt_eqmod eqmod_weakening subtype_base_sq isolate_summand exp_wf2 int_seg_subtype_nat equal_wf squash_wf true_wf istype-universe subtype_rel_self iff_weakening_equal exp0_lemma sum_wf nat_wf eq_int_wf eqtt_to_assert assert_of_eq_int eqff_to_assert set_subtype_base lelt_wf bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int zero_ann_a exp-zero not-lt-2 not-equal-2 add_functionality_wrt_le add-associates add-zero zero-add le-add-cancel condition-implies-le add-commutes minus-add minus-zero itermAdd_wf int_term_value_add_lemma add_functionality_wrt_eq sum_constant divides_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut functionIsType universeIsType introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename hypothesisEquality hypothesis dependent_pairFormation_alt dependent_functionElimination because_Cache unionElimination independent_isectElimination approximateComputation independent_functionElimination lambdaEquality_alt isect_memberEquality_alt voidElimination sqequalRule equalityIsType4 inhabitedIsType baseClosed baseApply closedConclusion applyEquality dependent_set_memberEquality_alt independent_pairFormation int_eqEquality productIsType productElimination promote_hyp instantiate cumulativity intEquality equalityTransitivity equalitySymmetry multiplyEquality imageElimination universeEquality imageMemberEquality equalityElimination equalityIsType2 inrFormation_alt addEquality minusEquality equalityIsType1

Latex:
\mforall{}n:\mBbbN{}\msupplus{}.  \mforall{}a:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}.    (2  |  \mSigma{}i<n.a[i]*10\^{}i  \mLeftarrow{}{}\mRightarrow{}  2  |  a[0])

Date html generated: 2019_10_15-AM-11_25_58
Last ObjectModification: 2018_10_09-PM-00_15_01

Theory : general

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