### Nuprl Lemma : div_div

`∀[a:ℤ]. ∀[n,m:ℤ-o].  (a ÷ n ÷ m ~ a ÷ n * m)`

Proof

Definitions occuring in Statement :  int_nzero: `ℤ-o` uall: `∀[x:A]. B[x]` divide: `n ÷ m` multiply: `n * m` int: `ℤ` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` int_nzero: `ℤ-o` sq_type: `SQType(T)` implies: `P `` Q` guard: `{T}` nat: `ℕ` prop: `ℙ` nat_plus: `ℕ+` le: `A ≤ B` and: `P ∧ Q` nequal: `a ≠ b ∈ T ` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` false: `False` uiff: `uiff(P;Q)` less_than': `less_than'(a;b)` true: `True` subtract: `n - m` subtype_rel: `A ⊆r B` top: `Top` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` gt: `i > j` squash: `↓T` int_lower: `{...i}` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  subtype_base_sq int_subtype_base decidable__le int_nzero_wf istype-int div_div_nat le_wf decidable__lt false_wf not-lt-2 not-equal-2 add_functionality_wrt_le zero-add add-zero le-add-cancel condition-implies-le add-commutes minus-add minus-zero less_than_wf add-associates int_entire_a int_nzero_properties satisfiable-full-omega-tt intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformnot_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_not_lemma int_formula_prop_wf equal-wf-base neg_mul_arg_bounds intformless_wf intformle_wf int_formula_prop_less_lemma int_formula_prop_le_lemma gt_wf equal_wf squash_wf true_wf div_4_to_1 itermMultiply_wf int_term_value_mul_lemma iff_weakening_equal decidable__equal_int itermMinus_wf int_term_value_minus_lemma divide_wf not-le-2 minus-one-mul minus-one-mul-top div_2_to_1 div_lbound_1 zero-mul div_anti_sym div_3_to_1 member_wf full-omega-unsat istype-void mul_nat_plus istype-false subtype_rel_self div_bounds_2 div_anti_sym2 pos_mul_arg_bounds div_bounds_3
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 natural_numberEquality hypothesisEquality unionElimination because_Cache setElimination rename equalityTransitivity equalitySymmetry independent_functionElimination axiomSqEquality Error :inhabitedIsType,  sqequalRule Error :isect_memberEquality_alt,  Error :universeIsType,  dependent_set_memberEquality productElimination independent_pairFormation lambdaFormation voidElimination addEquality minusEquality applyEquality lambdaEquality isect_memberEquality voidEquality divideEquality multiplyEquality dependent_pairFormation int_eqEquality computeAll baseClosed inrFormation productEquality imageElimination universeEquality imageMemberEquality baseApply closedConclusion inlFormation Error :lambdaEquality_alt,  Error :lambdaFormation_alt,  approximateComputation Error :dependent_pairFormation_alt,  Error :equalityIsType4,  Error :dependent_set_memberEquality_alt,  Error :inrFormation_alt,  Error :productIsType

Latex:
\mforall{}[a:\mBbbZ{}].  \mforall{}[n,m:\mBbbZ{}\msupminus{}\msupzero{}].    (a  \mdiv{}  n  \mdiv{}  m  \msim{}  a  \mdiv{}  n  *  m)

Date html generated: 2019_06_20-PM-01_14_36
Last ObjectModification: 2018_10_04-PM-01_07_08

Theory : int_2

Home Index