### Nuprl Lemma : divides_iff_div_exact

`∀a:ℤ. ∀n:ℤ-o.  (n | a `⇐⇒` ((a ÷ n) * n) = a ∈ ℤ)`

Proof

Definitions occuring in Statement :  divides: `b | a` int_nzero: `ℤ-o` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` divide: `n ÷ m` multiply: `n * m` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` member: `t ∈ T` uall: `∀[x:A]. B[x]` int_nzero: `ℤ-o` prop: `ℙ` rev_implies: `P `` Q` nequal: `a ≠ b ∈ T ` not: `¬A` false: `False` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` subtype_rel: `A ⊆r B` uiff: `uiff(P;Q)` decidable: `Dec(P)` or: `P ∨ Q` divides: `b | a`
Lemmas referenced :  divides_wf int_nzero_properties full-omega-unsat istype-int 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_not_lemma int_subtype_base int_nzero_wf divides_iff_rem_zero add_mono_wrt_eq div_rem_sum decidable__equal_int add-is-int-iff multiply-is-int-iff intformand_wf intformnot_wf intformeq_wf itermMultiply_wf itermVar_wf itermAdd_wf itermConstant_wf int_term_value_mul_lemma int_term_value_add_lemma int_formula_prop_wf false_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  independent_pairFormation cut hypothesis Error :universeIsType,  introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality Error :equalityIsType4,  Error :inhabitedIsType,  multiplyEquality divideEquality because_Cache independent_functionElimination voidElimination independent_isectElimination approximateComputation Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality natural_numberEquality dependent_functionElimination Error :isect_memberEquality_alt,  sqequalRule applyEquality productElimination equalityTransitivity equalitySymmetry unionElimination pointwiseFunctionality promote_hyp baseApply closedConclusion baseClosed

Latex:
\mforall{}a:\mBbbZ{}.  \mforall{}n:\mBbbZ{}\msupminus{}\msupzero{}.    (n  |  a  \mLeftarrow{}{}\mRightarrow{}  ((a  \mdiv{}  n)  *  n)  =  a)

Date html generated: 2019_06_20-PM-02_20_31
Last ObjectModification: 2018_10_03-AM-00_35_42

Theory : num_thy_1

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