### Nuprl Lemma : mod_bounds_1

`∀[a:ℤ]. ∀[n:ℤ-o].  ((0 ≤ (a mod n)) ∧ a mod n < |n|)`

Proof

Definitions occuring in Statement :  modulus: `a mod n` absval: `|i|` int_nzero: `ℤ-o` less_than: `a < b` uall: `∀[x:A]. B[x]` le: `A ≤ B` and: `P ∧ Q` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` and: `P ∧ Q` le: `A ≤ B` not: `¬A` implies: `P `` Q` false: `False` subtype_rel: `A ⊆r B` nat: `ℕ` prop: `ℙ` int_nzero: `ℤ-o` uimplies: `b supposing a` modulus: `a mod n` has-value: `(a)↓` nequal: `a ≠ b ∈ T ` all: `∀x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` less_than: `a < b` less_than': `less_than'(a;b)` top: `Top` true: `True` squash: `↓T` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` gt: `i > j`
Lemmas referenced :  less_than'_wf modulus_wf nat_wf member-less_than absval_wf int_nzero_wf zero-le-nat value-type-has-value int-value-type equal_wf lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_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 add_functionality_wrt_lt le_reflexive zero-add add-commutes le_wf absval_pos not-gt-2 iff_weakening_equal rem_bounds_z less_than_transitivity2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation hypothesis sqequalRule sqequalHypSubstitution productElimination thin independent_pairEquality lambdaEquality dependent_functionElimination hypothesisEquality because_Cache extract_by_obid isectElimination applyEquality setElimination rename natural_numberEquality axiomEquality equalityTransitivity equalitySymmetry independent_isectElimination isect_memberEquality voidElimination intEquality callbyvalueReduce remainderEquality lambdaFormation independent_functionElimination unionElimination equalityElimination lessCases sqequalAxiom voidEquality imageMemberEquality baseClosed imageElimination dependent_pairFormation promote_hyp instantiate cumulativity impliesFunctionality dependent_set_memberEquality

Latex:
\mforall{}[a:\mBbbZ{}].  \mforall{}[n:\mBbbZ{}\msupminus{}\msupzero{}].    ((0  \mleq{}  (a  mod  n))  \mwedge{}  a  mod  n  <  |n|)

Date html generated: 2018_05_21-PM-00_02_02
Last ObjectModification: 2018_05_19-AM-07_13_07

Theory : arithmetic

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