### Nuprl Lemma : div_unique2

`∀[a:ℕ]. ∀[n:ℕ+]. ∀[p:ℕ].  uiff((a ÷ n) = p ∈ ℤ;Div(a;n;p))`

Proof

Definitions occuring in Statement :  div_nrel: `Div(a;n;q)` nat_plus: `ℕ+` nat: `ℕ` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` divide: `n ÷ m` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` prop: `ℙ` squash: `↓T` nat: `ℕ` nat_plus: `ℕ+` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` true: `True` div_nrel: `Div(a;n;q)` lelt: `i ≤ j < k` le: `A ≤ B` nequal: `a ≠ b ∈ T ` subtype_rel: `A ⊆r B`
Lemmas referenced :  div_elim div_nrel_wf squash_wf true_wf nat_wf nat_properties nat_plus_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf le_wf less_than'_wf member-less_than equal_wf intformeq_wf intformless_wf int_formula_prop_eq_lemma int_formula_prop_less_lemma equal-wf-base int_subtype_base nat_plus_wf div_unique
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut independent_pairFormation extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality productElimination hypothesis hyp_replacement equalitySymmetry sqequalRule equalityTransitivity applyEquality lambdaEquality imageElimination isectElimination because_Cache dependent_set_memberEquality setElimination rename natural_numberEquality unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality imageMemberEquality baseClosed independent_pairEquality multiplyEquality axiomEquality addEquality Error :universeIsType,  divideEquality lambdaFormation Error :inhabitedIsType

Latex:
\mforall{}[a:\mBbbN{}].  \mforall{}[n:\mBbbN{}\msupplus{}].  \mforall{}[p:\mBbbN{}].    uiff((a  \mdiv{}  n)  =  p;Div(a;n;p))

Date html generated: 2019_06_20-PM-01_14_25
Last ObjectModification: 2018_09_26-PM-02_34_40

Theory : int_2

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