### Nuprl Lemma : div_unique3

`∀a:ℤ. ∀n:ℤ-o.`
`  ∀[p:ℤ]`
`    uiff((a ÷ n) = p ∈ ℤ;∃r:ℤ`
`                          (|r| < |n|`
`                          ∧ (a = ((p * n) + r) ∈ ℤ)`
`                          ∧ ((0 ≤ a) `` (0 ≤ r))`
`                          ∧ (0 < r `` 0 < a)`
`                          ∧ (r < 0 `` a < 0)))`

Proof

Definitions occuring in Statement :  absval: `|i|` int_nzero: `ℤ-o` less_than: `a < b` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` le: `A ≤ B` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` divide: `n ÷ m` multiply: `n * m` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` member: `t ∈ T` subtype_rel: `A ⊆r B` int_nzero: `ℤ-o` so_lambda: `λ2x.t[x]` so_apply: `x[s]` exists: `∃x:A. B[x]` nat: `ℕ` implies: `P `` Q` nequal: `a ≠ b ∈ T ` not: `¬A` false: `False` sq_type: `SQType(T)` guard: `{T}` true: `True` squash: `↓T` prop: `ℙ` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` top: `Top` subtract: `n - m` decidable: `Dec(P)` or: `P ∨ Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` less_than: `a < b` less_than': `less_than'(a;b)` bfalse: `ff` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` le: `A ≤ B` cand: `A c∧ B` ge: `i ≥ j `
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  Error :isect_memberFormation_alt,  independent_pairFormation cut introduction axiomEquality hypothesis thin rename Error :equalityIsType4,  Error :inhabitedIsType,  hypothesisEquality sqequalRule baseApply closedConclusion baseClosed applyEquality extract_by_obid sqequalHypSubstitution isectElimination intEquality Error :lambdaEquality_alt,  natural_numberEquality independent_isectElimination Error :productIsType,  setElimination equalityTransitivity equalitySymmetry Error :functionIsType,  because_Cache Error :universeIsType,  Error :dependent_pairFormation_alt,  remainderEquality independent_functionElimination voidElimination dependent_functionElimination promote_hyp instantiate cumulativity imageElimination universeEquality imageMemberEquality productElimination divideEquality Error :isect_memberEquality_alt,  multiplyEquality addEquality minusEquality unionElimination Error :dependent_set_memberEquality_alt,  equalityElimination lessCases axiomSqEquality Error :isectIsTypeImplies,  Error :equalityIsType1,  applyLambdaEquality

Latex:
\mforall{}a:\mBbbZ{}.  \mforall{}n:\mBbbZ{}\msupminus{}\msupzero{}.
\mforall{}[p:\mBbbZ{}]
uiff((a  \mdiv{}  n)  =  p;\mexists{}r:\mBbbZ{}
(|r|  <  |n|
\mwedge{}  (a  =  ((p  *  n)  +  r))
\mwedge{}  ((0  \mleq{}  a)  {}\mRightarrow{}  (0  \mleq{}  r))
\mwedge{}  (0  <  r  {}\mRightarrow{}  0  <  a)
\mwedge{}  (r  <  0  {}\mRightarrow{}  a  <  0)))

Date html generated: 2019_06_20-AM-11_24_53
Last ObjectModification: 2018_10_18-PM-03_54_42

Theory : arithmetic

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