### Nuprl Lemma : modulus_wf

[a:ℤ]. ∀[n:ℤ-o].  (a mod n ∈ ℕ)

Proof

Definitions occuring in Statement :  modulus: mod n int_nzero: -o nat: uall: [x:A]. B[x] member: t ∈ T int:
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T modulus: mod n has-value: (a)↓ uimplies: supposing a int_nzero: -o nequal: a ≠ b ∈  not: ¬A implies:  Q false: False prop: less_than: a < b and: P ∧ Q less_than': less_than'(a;b) true: True squash: T top: Top nat: all: x:A. B[x] uiff: uiff(P;Q) gt: i > j bool: 𝔹 unit: Unit it: btrue: tt bfalse: ff exists: x:A. B[x] or: P ∨ Q sq_type: SQType(T) guard: {T} bnot: ¬bb ifthenelse: if then else fi  assert: b iff: ⇐⇒ Q rev_implies:  Q decidable: Dec(P) subtract: m subtype_rel: A ⊆B le: A ≤ B
Lemmas referenced :  value-type-has-value int-value-type equal_wf less_than_wf not-gt-2 le_wf int_nzero_wf rem_bounds_z lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base iff_transitivity assert_wf bnot_wf not_wf iff_weakening_uiff assert_of_bnot decidable__lt false_wf not-lt-2 less-iff-le condition-implies-le minus-one-mul minus-minus minus-one-mul-top add_functionality_wrt_le add-associates zero-add add-swap add-commutes add-zero le-add-cancel absval_unfold2 add_functionality_wrt_lt le_reflexive less_than_transitivity2 le_weakening2 add-mul-special zero-mul subtract_wf absval_wf nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule callbyvalueReduce extract_by_obid sqequalHypSubstitution isectElimination thin intEquality independent_isectElimination hypothesis remainderEquality because_Cache setElimination rename lambdaFormation independent_functionElimination voidElimination hypothesisEquality natural_numberEquality lessCases independent_pairFormation baseClosed equalityTransitivity equalitySymmetry imageMemberEquality axiomSqEquality isect_memberEquality voidEquality imageElimination productElimination dependent_set_memberEquality dependent_functionElimination axiomEquality unionElimination equalityElimination dependent_pairFormation promote_hyp instantiate cumulativity impliesFunctionality minusEquality applyEquality lambdaEquality addEquality multiplyEquality

Latex:
\mforall{}[a:\mBbbZ{}].  \mforall{}[n:\mBbbZ{}\msupminus{}\msupzero{}].    (a  mod  n  \mmember{}  \mBbbN{})

Date html generated: 2019_06_20-AM-11_25_35
Last ObjectModification: 2018_08_20-PM-09_28_29

Theory : arithmetic

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