### Nuprl Lemma : div_3_to_1

`∀[a:{...0}]. ∀[b:{...-1}].  ((a ÷ b) = ((-a) ÷ -b) ∈ ℤ)`

Proof

Definitions occuring in Statement :  int_lower: `{...i}` uall: `∀[x:A]. B[x]` divide: `n ÷ m` minus: `-n` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` int_lower: `{...i}` uimplies: `b supposing a` all: `∀x:A. B[x]` top: `Top` subtype_rel: `A ⊆r B` subtract: `n - m` int_nzero: `ℤ-o` le: `A ≤ B` and: `P ∧ Q` nequal: `a ≠ b ∈ T ` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` decidable: `Dec(P)` or: `P ∨ Q` prop: `ℙ` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` implies: `P `` Q` false: `False` guard: `{T}` less_than': `less_than'(a;b)` true: `True` nat: `ℕ` nat_plus: `ℕ+` exists: `∃x:A. B[x]` cand: `A c∧ B` squash: `↓T` sq_type: `SQType(T)` less_than: `a < b`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin minusEquality natural_numberEquality hypothesisEquality hypothesis setElimination rename because_Cache independent_isectElimination dependent_functionElimination sqequalRule isect_memberEquality axiomEquality multiplyEquality voidElimination voidEquality applyEquality lambdaEquality intEquality dependent_set_memberEquality productElimination addEquality unionElimination inlFormation independent_pairFormation lambdaFormation inrFormation independent_functionElimination addLevel orFunctionality divideEquality dependent_pairFormation productEquality baseApply closedConclusion baseClosed functionEquality imageElimination equalityTransitivity equalitySymmetry imageMemberEquality universeEquality hyp_replacement remainderEquality instantiate cumulativity

Latex:
\mforall{}[a:\{...0\}].  \mforall{}[b:\{...-1\}].    ((a  \mdiv{}  b)  =  ((-a)  \mdiv{}  -b))

Date html generated: 2017_04_14-AM-07_18_14
Last ObjectModification: 2017_02_27-PM-02_54_16

Theory : arithmetic

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