### Nuprl Lemma : rem_3_to_1

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

Proof

Definitions occuring in Statement :  int_lower: `{...i}` uall: `∀[x:A]. B[x]` remainder: `n rem 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` squash: `↓T` prop: `ℙ` int_nzero: `ℤ-o` le: `A ≤ B` and: `P ∧ Q` nequal: `a ≠ b ∈ T ` subtype_rel: `A ⊆r B` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` implies: `P `` Q` false: `False` guard: `{T}` subtract: `n - m` less_than': `less_than'(a;b)` true: `True`
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 sqequalRule isect_memberEquality axiomEquality because_Cache independent_isectElimination dependent_functionElimination voidElimination voidEquality multiplyEquality applyEquality lambdaEquality imageElimination equalityTransitivity equalitySymmetry universeEquality intEquality dependent_set_memberEquality productElimination addEquality unionElimination inlFormation independent_pairFormation lambdaFormation inrFormation independent_functionElimination addLevel orFunctionality imageMemberEquality baseClosed divideEquality

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

Date html generated: 2017_04_14-AM-07_18_26
Last ObjectModification: 2017_02_27-PM-02_53_06

Theory : arithmetic

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