### Nuprl Lemma : rem_2_to_1

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

Proof

Definitions occuring in Statement :  int_lower: `{...i}` nat_plus: `ℕ+` 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}` subtype_rel: `A ⊆r B` nat_plus: `ℕ+` int_nzero: `ℤ-o` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` prop: `ℙ` all: `∀x:A. B[x]` implies: `P `` Q` nequal: `a ≠ b ∈ T ` not: `¬A` false: `False` guard: `{T}` true: `True` squash: `↓T` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` subtract: `n - m` top: `Top`
Lemmas referenced :  nat_plus_wf int_lower_wf subtype_rel_sets less_than_wf nequal_wf less_than_transitivity1 le_weakening less_than_irreflexivity equal_wf equal-wf-base int_subtype_base squash_wf true_wf rem_to_div iff_weakening_equal subtract_wf minus-one-mul mul-associates minus-one-mul-top mul-commutes one-mul minus-add minus-minus div_2_to_1
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut hypothesis extract_by_obid sqequalRule sqequalHypSubstitution isect_memberEquality isectElimination thin hypothesisEquality axiomEquality because_Cache natural_numberEquality intEquality setElimination rename applyEquality lambdaEquality independent_isectElimination setEquality lambdaFormation dependent_functionElimination independent_functionElimination voidElimination baseClosed minusEquality imageElimination equalityTransitivity equalitySymmetry universeEquality imageMemberEquality productElimination multiplyEquality divideEquality voidEquality addEquality

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

Date html generated: 2017_04_14-AM-07_18_22
Last ObjectModification: 2017_02_27-PM-02_52_32

Theory : arithmetic

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