Nuprl Lemma : div_is_zero

`∀[n:{2...}]. ∀[i:ℤ].  i ÷ n ~ 0 supposing |i| < n`

Proof

Definitions occuring in Statement :  absval: `|i|` int_upper: `{i...}` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` divide: `n ÷ m` natural_number: `\$n` int: `ℤ` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` sq_type: `SQType(T)` all: `∀x:A. B[x]` implies: `P `` Q` guard: `{T}` subtype_rel: `A ⊆r B` nat: `ℕ` int_upper: `{i...}` int_nzero: `ℤ-o` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nequal: `a ≠ b ∈ T ` not: `¬A` false: `False` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` decidable: `Dec(P)` or: `P ∨ Q` nat_plus: `ℕ+` squash: `↓T` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` int_lower: `{...i}` gt: `i > j` ge: `i ≥ j ` cand: `A c∧ B` less_than: `a < b` uiff: `uiff(P;Q)`
Lemmas referenced :  subtype_base_sq int_subtype_base istype-less_than absval_wf istype-int istype-int_upper div_rem_sum subtype_rel_sets_simple le_wf nequal_wf full-omega-unsat intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma istype-void int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf istype-le decidable__le rem_bounds_1 int_upper_properties decidable__lt intformnot_wf intformless_wf int_formula_prop_not_lemma int_formula_prop_less_lemma nat_wf set_subtype_base absval-non-neg absval_pos equal_wf squash_wf true_wf istype-universe quotient-is-zero upper_subtype_nat istype-false subtype_rel_self iff_weakening_equal rem_bounds_2 absval_neg itermMinus_wf int_term_value_minus_lemma mul_preserves_le itermMultiply_wf itermAdd_wf int_term_value_mul_lemma int_term_value_add_lemma decidable__equal_int add-is-int-iff multiply-is-int-iff false_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut thin instantiate extract_by_obid sqequalHypSubstitution isectElimination cumulativity intEquality independent_isectElimination hypothesis dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination axiomSqEquality hypothesisEquality applyEquality Error :lambdaEquality_alt,  setElimination rename Error :inhabitedIsType,  sqequalRule Error :isect_memberEquality_alt,  Error :isectIsTypeImplies,  natural_numberEquality Error :lambdaFormation_alt,  approximateComputation Error :dependent_pairFormation_alt,  int_eqEquality voidElimination independent_pairFormation Error :universeIsType,  Error :equalityIstype,  baseClosed sqequalBase because_Cache unionElimination Error :dependent_set_memberEquality_alt,  productElimination imageElimination universeEquality imageMemberEquality minusEquality divideEquality pointwiseFunctionality promote_hyp baseApply closedConclusion

Latex:
\mforall{}[n:\{2...\}].  \mforall{}[i:\mBbbZ{}].    i  \mdiv{}  n  \msim{}  0  supposing  |i|  <  n

Date html generated: 2019_06_20-PM-01_18_50
Last ObjectModification: 2019_02_12-PM-00_26_19

Theory : int_2

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