### Nuprl Lemma : rem_antisym

`∀[a:ℤ]. ∀[b:ℤ-o].  ((-a rem b) = (-(a rem b)) ∈ ℤ)`

Proof

Definitions occuring in Statement :  int_nzero: `ℤ-o` uall: `∀[x:A]. B[x]` remainder: `n rem m` minus: `-n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` decidable: `Dec(P)` or: `P ∨ Q` squash: `↓T` prop: `ℙ` nat_plus: `ℕ+` le: `A ≤ B` and: `P ∧ Q` nequal: `a ≠ b ∈ T ` not: `¬A` implies: `P `` Q` false: `False` guard: `{T}` uimplies: `b supposing a` true: `True` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` top: `Top` int_lower: `{...i}` uiff: `uiff(P;Q)` subtract: `n - m` less_than': `less_than'(a;b)` int_nzero: `ℤ-o` so_apply: `x[s]` so_lambda: `λ2x.t[x]`
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation hypothesis introduction extract_by_obid intEquality Error :isect_memberFormation_alt,  Error :universeIsType,  sqequalRule sqequalHypSubstitution isect_memberEquality isectElimination thin hypothesisEquality axiomEquality because_Cache dependent_functionElimination natural_numberEquality unionElimination applyEquality lambdaEquality imageElimination equalityTransitivity equalitySymmetry universeEquality minusEquality remainderEquality setElimination rename productElimination independent_isectElimination independent_functionElimination voidElimination imageMemberEquality baseClosed instantiate voidEquality dependent_set_memberEquality multiplyEquality independent_pairFormation addEquality closedConclusion baseApply

Latex:
\mforall{}[a:\mBbbZ{}].  \mforall{}[b:\mBbbZ{}\msupminus{}\msupzero{}].    ((-a  rem  b)  =  (-(a  rem  b)))

Date html generated: 2019_06_20-AM-11_25_22
Last ObjectModification: 2018_09_26-AM-10_58_28

Theory : arithmetic

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